Trigonometric Functions
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Graphs of Trigonometric Functions
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Angles
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Trig Identities
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Advanced Trigonometric Identities
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- Explanation and Examples - Part 1
- Explanation and Examples - Part 2
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9 - Part a
- Exercise 9 - Part b
- Exercise 9 - Part c
- Exercise 9 - Part d
- Exercise 10 - Part a
- Exercise 10 - Part b
- Exercise 10 - Part c
- Exercise 10 - Part d
- Exercise 11

Inverse Trigonometric Functions
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Triangles
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Trigonometric Equations
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- Trigonometric Equations - Part 1
- Trigonometric Equations - Part 2
- Trigonometric Equations - Part 3
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 4c
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30
- Exercise 31
- Exercise 32
- Exercise 33
- Exercise 34
- Exercise 35
- Exercise 36

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[{"Name":"Trigonometric Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Trigonometric Functions - Part 1","Duration":"15m 14s","ChapterTopicVideoID":10418,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10418.jpeg","UploadDate":"2021-06-29T14:05:48.5400000","DurationForVideoObject":"PT15M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"We\u0027re starting a new topic called trigonometric functions."},{"Start":"00:04.995 ","End":"00:12.405","Text":"The first thing I want to introduce is the concept of the unit circle."},{"Start":"00:12.405 ","End":"00:19.080","Text":"When I say the unit circle I mean the circle in the plane whose center is at"},{"Start":"00:19.080 ","End":"00:27.225","Text":"the origin and the radius is equal to 1."},{"Start":"00:27.225 ","End":"00:31.690","Text":"We\u0027ll call this circle U."},{"Start":"00:31.690 ","End":"00:34.505","Text":"It\u0027s got the equation,"},{"Start":"00:34.505 ","End":"00:39.905","Text":"x^2 plus y^2 equals 1."},{"Start":"00:39.905 ","End":"00:44.520","Text":"Here\u0027s a picture of the unit circle."},{"Start":"00:44.520 ","End":"00:46.879","Text":"This 1 point in particular,"},{"Start":"00:46.879 ","End":"00:48.485","Text":"that I\u0027d like to label,"},{"Start":"00:48.485 ","End":"00:50.765","Text":"this is going to be important in a moment,"},{"Start":"00:50.765 ","End":"00:52.804","Text":"is this point here,"},{"Start":"00:52.804 ","End":"00:55.790","Text":"the point where x is 1 and y is 0."},{"Start":"00:55.790 ","End":"00:57.920","Text":"This point here."},{"Start":"00:57.920 ","End":"01:04.100","Text":"Also useful might be the other 3 intersections with the axes."},{"Start":"01:04.100 ","End":"01:06.575","Text":"There we are."},{"Start":"01:06.575 ","End":"01:10.700","Text":"The next thing I want to do is describe an association,"},{"Start":"01:10.700 ","End":"01:16.745","Text":"where given any real number t,"},{"Start":"01:16.745 ","End":"01:22.115","Text":"I want to associate with it a point P,"},{"Start":"01:22.115 ","End":"01:23.935","Text":"you could say P(t),"},{"Start":"01:23.935 ","End":"01:25.670","Text":"since it depends on t,"},{"Start":"01:25.670 ","End":"01:27.360","Text":"it\u0027s a function of t,"},{"Start":"01:27.360 ","End":"01:29.210","Text":"on the unit circle."},{"Start":"01:29.210 ","End":"01:32.320","Text":"Let\u0027s say it\u0027s x, y."},{"Start":"01:32.320 ","End":"01:35.880","Text":"So t is a real number,"},{"Start":"01:35.880 ","End":"01:42.720","Text":"and P(t) is on U. I\u0027m going to describe it as follows."},{"Start":"01:42.720 ","End":"01:45.530","Text":"I\u0027ll have to break it up into cases."},{"Start":"01:45.530 ","End":"01:50.370","Text":"If t is bigger or equal to 0,"},{"Start":"01:50.370 ","End":"01:53.080","Text":"then what we do,"},{"Start":"01:53.390 ","End":"01:57.320","Text":"in this case we simply start from"},{"Start":"01:57.320 ","End":"02:01.220","Text":"this point here and we travel along the circumference in"},{"Start":"02:01.220 ","End":"02:07.700","Text":"a counterclockwise manner until we get"},{"Start":"02:07.700 ","End":"02:15.710","Text":"to a point where this circumference length is t units."},{"Start":"02:15.710 ","End":"02:20.375","Text":"l actually like to write it as absolute value of t units."},{"Start":"02:20.375 ","End":"02:22.430","Text":"If t is positive it doesn\u0027t matter."},{"Start":"02:22.430 ","End":"02:25.415","Text":"You\u0027ll see why I did this in a moment."},{"Start":"02:25.415 ","End":"02:28.820","Text":"This point here will be the point P,"},{"Start":"02:28.820 ","End":"02:31.175","Text":"depending on t,"},{"Start":"02:31.175 ","End":"02:35.855","Text":"and it will have some coordinates, x, y."},{"Start":"02:35.855 ","End":"02:40.280","Text":"Now that\u0027s for t positive,"},{"Start":"02:40.280 ","End":"02:42.635","Text":"and we go in this direction."},{"Start":"02:42.635 ","End":"02:47.885","Text":"Now if t is negative,"},{"Start":"02:47.885 ","End":"02:52.940","Text":"then it\u0027s a similar thing only we go counterclockwise."},{"Start":"02:52.940 ","End":"02:55.820","Text":"Again, we start from this point here and we go"},{"Start":"02:55.820 ","End":"03:01.670","Text":"along until we get to some point where this length"},{"Start":"03:01.670 ","End":"03:06.410","Text":"here is length absolute value"},{"Start":"03:06.410 ","End":"03:10.340","Text":"of t. Now this time it\u0027s clear why I\u0027ve written the absolute value,"},{"Start":"03:10.340 ","End":"03:12.200","Text":"because t is negative,"},{"Start":"03:12.200 ","End":"03:14.130","Text":"and I don\u0027t know,"},{"Start":"03:14.130 ","End":"03:16.800","Text":"t might be say minus 1,"},{"Start":"03:16.800 ","End":"03:20.180","Text":"and I have to go plus 1 unit."},{"Start":"03:20.180 ","End":"03:26.075","Text":"The minus is reflected by the reverse direction, the clockwise direction."},{"Start":"03:26.075 ","End":"03:28.280","Text":"This time in this case,"},{"Start":"03:28.280 ","End":"03:30.775","Text":"this would be P(t),"},{"Start":"03:30.775 ","End":"03:36.145","Text":"which is also equal to some x, y."},{"Start":"03:36.145 ","End":"03:41.165","Text":"By the way, if t equals 0,"},{"Start":"03:41.165 ","End":"03:43.730","Text":"then we travel a distance of 0,"},{"Start":"03:43.730 ","End":"03:49.820","Text":"so maybe I\u0027ll write that separately so that when t equals 0,"},{"Start":"03:49.820 ","End":"03:58.370","Text":"then P(t) is the point 1,0, like P(0)."},{"Start":"03:58.370 ","End":"04:01.550","Text":"Maybe I should write something here."},{"Start":"04:01.550 ","End":"04:03.889","Text":"When t is less than 0,"},{"Start":"04:03.889 ","End":"04:07.915","Text":"then we go clockwise,"},{"Start":"04:07.915 ","End":"04:15.530","Text":"absolute value of t units along the circumference,"},{"Start":"04:15.530 ","End":"04:18.799","Text":"this way, and when t is bigger than 0,"},{"Start":"04:18.799 ","End":"04:24.025","Text":"then anticlockwise or counterclockwise,"},{"Start":"04:24.025 ","End":"04:32.090","Text":"also absolute value of t units along the circumference starting from 1,0."},{"Start":"04:32.930 ","End":"04:36.040","Text":"The picture explains it."},{"Start":"04:36.040 ","End":"04:38.300","Text":"For each value of t,"},{"Start":"04:38.300 ","End":"04:40.295","Text":"we get a point P,"},{"Start":"04:40.295 ","End":"04:43.835","Text":"just to be formal and you can ignore what I\u0027m going to say now,"},{"Start":"04:43.835 ","End":"04:52.054","Text":"mathematically P would be a function from the real numbers to the unit circle."},{"Start":"04:52.054 ","End":"04:56.855","Text":"For each real number I have a point on the unit circle."},{"Start":"04:56.855 ","End":"05:00.800","Text":"Of course if t is big enough I might wrap around several times,"},{"Start":"05:00.800 ","End":"05:03.910","Text":"but I still end up somewhere on the unit circle,"},{"Start":"05:03.910 ","End":"05:06.310","Text":"likewise if t is very negative,"},{"Start":"05:06.310 ","End":"05:12.770","Text":"could maybe go around a few times until you get to the right distance."},{"Start":"05:12.770 ","End":"05:15.290","Text":"Again, you\u0027d settle on the unit circle somewhere."},{"Start":"05:15.290 ","End":"05:18.545","Text":"Whatever the value of t positive or negative,"},{"Start":"05:18.545 ","End":"05:21.400","Text":"we get a point on the unit circle."},{"Start":"05:21.400 ","End":"05:26.975","Text":"Now we\u0027re ready to introduce the definition of the trigonometric functions."},{"Start":"05:26.975 ","End":"05:31.340","Text":"What I\u0027d like to do first is just copy this here,"},{"Start":"05:31.340 ","End":"05:34.130","Text":"so I don\u0027t lose it when I scroll."},{"Start":"05:34.130 ","End":"05:36.424","Text":"Given a real number t,"},{"Start":"05:36.424 ","End":"05:40.280","Text":"we associate with it a point P,"},{"Start":"05:40.280 ","End":"05:45.090","Text":"or P(t), on the unit circle,"},{"Start":"05:45.090 ","End":"05:49.125","Text":"which is some x, y."},{"Start":"05:49.125 ","End":"05:54.640","Text":"Now we\u0027re going to define 6 trigonometric functions."},{"Start":"05:55.250 ","End":"06:05.615","Text":"We\u0027re going to say that sine t is equal to the y of the point."},{"Start":"06:05.615 ","End":"06:10.834","Text":"Cosine t is the second trigonometric function."},{"Start":"06:10.834 ","End":"06:12.770","Text":"They actually have long names,"},{"Start":"06:12.770 ","End":"06:14.600","Text":"and I\u0027ll maybe write those in a moment."},{"Start":"06:14.600 ","End":"06:19.153","Text":"Cosine is equal to the x of the point."},{"Start":"06:19.153 ","End":"06:24.050","Text":"The tangent of t is equal"},{"Start":"06:24.050 ","End":"06:30.394","Text":"to the quotient y over x."},{"Start":"06:30.394 ","End":"06:35.630","Text":"This assumes that x is not equal to 0."},{"Start":"06:35.630 ","End":"06:40.415","Text":"Otherwise we say the tangent of t is undefined."},{"Start":"06:40.415 ","End":"06:42.725","Text":"3 more I want to write."},{"Start":"06:42.725 ","End":"06:48.045","Text":"There is a csc(t),"},{"Start":"06:48.045 ","End":"06:51.845","Text":"and that is equal to 1 over y."},{"Start":"06:51.845 ","End":"06:58.220","Text":"Again, provided that y is not equal to 0 and not dividing by 0, otherwise it\u0027s undefined."},{"Start":"06:58.220 ","End":"07:06.130","Text":"The secant of t is equal to 1 over x,"},{"Start":"07:06.130 ","End":"07:10.350","Text":"and if x is not equal to 0."},{"Start":"07:10.350 ","End":"07:13.295","Text":"The sixth one is the cotangent,"},{"Start":"07:13.295 ","End":"07:19.655","Text":"and the cotangent of t is equal to x over y;"},{"Start":"07:19.655 ","End":"07:24.260","Text":"provided that y is not equal to 0."},{"Start":"07:24.260 ","End":"07:26.060","Text":"We don\u0027t have to,"},{"Start":"07:26.060 ","End":"07:31.520","Text":"but we could put brackets around the argument,"},{"Start":"07:31.520 ","End":"07:34.010","Text":"[inaudible] enough when you do, otherwise you think"},{"Start":"07:34.010 ","End":"07:37.490","Text":"it\u0027s just well 1 long word or something."},{"Start":"07:37.490 ","End":"07:41.105","Text":"I said I\u0027d give you also the long names."},{"Start":"07:41.105 ","End":"07:44.755","Text":"This one is sine."},{"Start":"07:44.755 ","End":"07:47.760","Text":"This one is cosine."},{"Start":"07:47.760 ","End":"07:52.860","Text":"This one is called tangent."},{"Start":"07:52.860 ","End":"07:57.630","Text":"This one is cosecant."},{"Start":"07:57.630 ","End":"08:00.690","Text":"This one is secant."},{"Start":"08:00.690 ","End":"08:05.250","Text":"This one is cotangent."},{"Start":"08:05.250 ","End":"08:08.740","Text":"We do see this, we say csc(t)."},{"Start":"08:08.740 ","End":"08:11.695","Text":"Example."},{"Start":"08:11.695 ","End":"08:14.640","Text":"I\u0027m not giving you what t is,"},{"Start":"08:14.640 ","End":"08:16.080","Text":"but I\u0027m going to give you"},{"Start":"08:16.080 ","End":"08:21.640","Text":"that p(t"},{"Start":"08:21.640 ","End":"08:27.315","Text":") =(-4/5, 3/5)."},{"Start":"08:27.315 ","End":"08:30.165","Text":"Let me get the diagram."},{"Start":"08:30.165 ","End":"08:34.260","Text":"Here\u0027s the picture I took from above."},{"Start":"08:34.260 ","End":"08:38.610","Text":"All I have to do is to replace the general x,"},{"Start":"08:38.610 ","End":"08:44.890","Text":"y that it was written here by -4/5, 3/5."},{"Start":"08:50.030 ","End":"08:55.350","Text":"You just took my word for it that this is on the unit circle."},{"Start":"08:55.350 ","End":"09:02.715","Text":"But you really ought to check if this is my x and this is the y,"},{"Start":"09:02.715 ","End":"09:07.380","Text":"I have to make sure that x squared plus y squared equals 1."},{"Start":"09:07.380 ","End":"09:13.530","Text":"Like x squared would be 16/25 with a plus,"},{"Start":"09:13.530 ","End":"09:15.720","Text":"and the y squared would be this thing squared,"},{"Start":"09:15.720 ","End":"09:18.615","Text":"which is 9/25,"},{"Start":"09:18.615 ","End":"09:22.605","Text":"which is 16 + 9/25,"},{"Start":"09:22.605 ","End":"09:27.920","Text":"which is,25/25, and it is equal to 1 so"},{"Start":"09:27.920 ","End":"09:33.320","Text":"this p(t) really is on u."},{"Start":"09:33.320 ","End":"09:35.800","Text":"Now, the trigonometric functions,"},{"Start":"09:35.800 ","End":"09:44.815","Text":"the sine(t) is y, which is 3/5."},{"Start":"09:44.815 ","End":"09:52.360","Text":"The cosine of this particular t is the x, which is -4/5."},{"Start":"09:52.430 ","End":"10:01.905","Text":"The tangent of t is equal to the y over the x,"},{"Start":"10:01.905 ","End":"10:06.000","Text":"which is 3/5 over"},{"Start":"10:06.000 ","End":"10:10.080","Text":"-4/5 and that"},{"Start":"10:10.080 ","End":"10:16.110","Text":"simplifies to simply -3/4."},{"Start":"10:16.110 ","End":"10:19.920","Text":"Let\u0027s go to the other 3,"},{"Start":"10:19.920 ","End":"10:28.740","Text":"csc(t) is equal to 1 over y."},{"Start":"10:28.740 ","End":"10:32.310","Text":"For 1 over you just have to take"},{"Start":"10:32.310 ","End":"10:36.480","Text":"the fraction upside down so I won\u0027t write the intermediate step."},{"Start":"10:36.480 ","End":"10:39.475","Text":"It\u0027s 5/3."},{"Start":"10:39.475 ","End":"10:49.749","Text":"The secant of t is 1 over x and 1 over you just take it upside down."},{"Start":"10:49.749 ","End":"10:53.685","Text":"It\u0027s -5/4."},{"Start":"10:53.685 ","End":"11:00.375","Text":"The cotangent of t is equal to"},{"Start":"11:00.375 ","End":"11:08.025","Text":"x over y minus 4/5 over 3/5,"},{"Start":"11:08.025 ","End":"11:13.590","Text":"which simplifies to -4/3."},{"Start":"11:13.590 ","End":"11:18.930","Text":"That\u0027s the example of all 6 trigonometric functions for this point,"},{"Start":"11:18.930 ","End":"11:25.230","Text":"where it\u0027s coordinates -4/5 and 3/5,"},{"Start":"11:25.230 ","End":"11:28.140","Text":"or at least it\u0027s the trigonometric functions of the t,"},{"Start":"11:28.140 ","End":"11:33.825","Text":"which is the length of arc in the counterclockwise direction from here to here."},{"Start":"11:33.825 ","End":"11:38.535","Text":"Now I want to say something about symmetries"},{"Start":"11:38.535 ","End":"11:45.225","Text":"involving points on the unit circle."},{"Start":"11:45.225 ","End":"11:48.360","Text":"Suppose I have some value of t,"},{"Start":"11:48.360 ","End":"11:57.630","Text":"which takes me to here and p(t) is equal to some x, y."},{"Start":"11:57.630 ","End":"12:04.485","Text":"If I were to go another complete circle around I\u0027d get to the same point."},{"Start":"12:04.485 ","End":"12:12.340","Text":"This could be expressed as symmetry rule and I could say that p( t"},{"Start":"12:12.340 ","End":"12:21.510","Text":"+ 2Pi) because the circumference of the unit circle is 2Pi."},{"Start":"12:21.510 ","End":"12:26.385","Text":"This would also equal xy,"},{"Start":"12:26.385 ","End":"12:29.835","Text":"or in other words, it\u0027s equal to the original p( t)."},{"Start":"12:29.835 ","End":"12:36.810","Text":"Actually, I could generalize it and I could go to subtract 2Pi I could go 2Pi again."},{"Start":"12:36.810 ","End":"12:42.240","Text":"I could really say here, +2n Pi,"},{"Start":"12:42.240 ","End":"12:44.759","Text":"where n could be,"},{"Start":"12:44.759 ","End":"12:49.890","Text":"any integer, could be positive or negative or 0."},{"Start":"12:49.890 ","End":"12:54.960","Text":"Any the whole number of circles in any direction will bring me back to the same point."},{"Start":"12:54.960 ","End":"12:56.835","Text":"That\u0027s what this is saying."},{"Start":"12:56.835 ","End":"13:01.960","Text":"Maybe I\u0027ll number these as symmetry number 1,"},{"Start":"13:05.420 ","End":"13:07.935","Text":"if instead of taking t,"},{"Start":"13:07.935 ","End":"13:11.805","Text":"let\u0027s say this is my t in this direction."},{"Start":"13:11.805 ","End":"13:19.890","Text":"If I took -t and went the same distance in the other direction,"},{"Start":"13:19.890 ","End":"13:26.835","Text":"then I\u0027d get to a point which is directly below the image of this."},{"Start":"13:26.835 ","End":"13:32.295","Text":"Let me just write it p(-t )is going to be,"},{"Start":"13:32.295 ","End":"13:35.470","Text":"this point will be x,"},{"Start":"13:35.720 ","End":"13:44.490","Text":"-y. I can get"},{"Start":"13:44.490 ","End":"13:50.655","Text":"another symmetry by choosing the point that\u0027s diametrically opposed."},{"Start":"13:50.655 ","End":"13:52.140","Text":"If you think about it,"},{"Start":"13:52.140 ","End":"13:55.245","Text":"the way to get from here to here is just to go"},{"Start":"13:55.245 ","End":"14:00.240","Text":"another Pi units along the circumference so"},{"Start":"14:00.240 ","End":"14:09.400","Text":"we get another rule that if I take the point corresponding to t + Pi,"},{"Start":"14:10.490 ","End":"14:12.840","Text":"both of them will be negative,"},{"Start":"14:12.840 ","End":"14:14.985","Text":"now, -x,"},{"Start":"14:14.985 ","End":"14:23.835","Text":"-y. I think it\u0027s fairly clear if we take the opposite point through the origin."},{"Start":"14:23.835 ","End":"14:28.260","Text":"Finally, what point here would be missing?"},{"Start":"14:28.260 ","End":"14:30.780","Text":"Do you think, I guess it would be this one."},{"Start":"14:30.780 ","End":"14:32.730","Text":"Let\u0027s even label it first."},{"Start":"14:32.730 ","End":"14:35.955","Text":"That\u0027s where I keep the y and negate the x."},{"Start":"14:35.955 ","End":"14:38.520","Text":"And how do I get that?"},{"Start":"14:38.520 ","End":"14:43.875","Text":"This distance here is the same as this distance here,"},{"Start":"14:43.875 ","End":"14:47.369","Text":"so together they make up a half-circle, which is Pi."},{"Start":"14:47.369 ","End":"14:51.620","Text":"This is actually, Pi minus this distance because together they make up Pi,"},{"Start":"14:51.620 ","End":"14:51.621","Text":"so"},{"Start":"14:51.621 ","End":"15:03.920","Text":"P(Pi - t)="},{"Start":"15:03.920 ","End":"15:08.170","Text":"-x,"},{"Start":"15:08.170 ","End":"15:08.610","Text":"y."},{"Start":"15:08.610 ","End":"15:14.230","Text":"That\u0027s all about symmetries of points on the unit circle."}],"ID":10902},{"Watched":false,"Name":"Trigonometric Functions - Part 2","Duration":"8m 29s","ChapterTopicVideoID":10419,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10419.jpeg","UploadDate":"2021-06-29T14:06:24.1430000","DurationForVideoObject":"PT8M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.735","Text":"This clip is a continuation of the previous one."},{"Start":"00:03.735 ","End":"00:06.300","Text":"We\u0027re still on trigonometric functions,"},{"Start":"00:06.300 ","End":"00:11.190","Text":"and this time we\u0027re going to talk about periodicity."},{"Start":"00:11.190 ","End":"00:17.250","Text":"Now, you may or may not know what it means to say that a function f is periodic,"},{"Start":"00:17.250 ","End":"00:22.110","Text":"so this could be just a reminder or you could be learning something new."},{"Start":"00:22.110 ","End":"00:29.190","Text":"A function is called periodic if we have the identity that"},{"Start":"00:29.190 ","End":"00:37.620","Text":"f(t+p) is equal to f(t),"},{"Start":"00:37.620 ","End":"00:39.150","Text":"when I say identity,"},{"Start":"00:39.150 ","End":"00:44.970","Text":"means that it\u0027s for all t. As for p,"},{"Start":"00:44.970 ","End":"00:50.555","Text":"p is going to be some positive real number."},{"Start":"00:50.555 ","End":"00:56.120","Text":"In other words, if there exists such a p that this is always true for every t,"},{"Start":"00:56.120 ","End":"00:58.070","Text":"then f is periodic."},{"Start":"00:58.070 ","End":"01:02.745","Text":"Now, the period of f is"},{"Start":"01:02.745 ","End":"01:10.350","Text":"the smallest such p because there could be several p\u0027s,"},{"Start":"01:10.350 ","End":"01:13.960","Text":"but the smallest one is called the period."},{"Start":"01:16.310 ","End":"01:18.380","Text":"That was in general."},{"Start":"01:18.380 ","End":"01:20.675","Text":"We want to talk about trigonometric functions."},{"Start":"01:20.675 ","End":"01:27.725","Text":"Let\u0027s say f is a trig function."},{"Start":"01:27.725 ","End":"01:29.795","Text":"I write it in brief."},{"Start":"01:29.795 ","End":"01:32.800","Text":"If f is a trigonometric function,"},{"Start":"01:32.800 ","End":"01:41.067","Text":"then f(t+2Pi) is equal"},{"Start":"01:41.067 ","End":"01:45.895","Text":"to f(t) for all t. The reason for that is what we saw earlier,"},{"Start":"01:45.895 ","End":"01:53.000","Text":"is that the point p corresponding to (t+2Pi) is equal to"},{"Start":"01:53.000 ","End":"01:57.550","Text":"the point p on the unit circle corresponding to t."},{"Start":"01:57.550 ","End":"02:02.440","Text":"The trigonometric functions are all derived from this point p. Because of this,"},{"Start":"02:02.440 ","End":"02:08.800","Text":"all 6 of the trigonometric functions are periodic."},{"Start":"02:08.800 ","End":"02:12.380","Text":"Now, does that mean that the period is 2Pi?"},{"Start":"02:12.380 ","End":"02:18.710","Text":"Not necessarily because the period is the smallest such p. Let me just tell you"},{"Start":"02:18.710 ","End":"02:26.535","Text":"that the period is 2Pi for the sine,"},{"Start":"02:26.535 ","End":"02:31.470","Text":"the cosine, the cosecant,"},{"Start":"02:31.470 ","End":"02:34.500","Text":"and the secant,"},{"Start":"02:34.500 ","End":"02:45.090","Text":"but it\u0027s only Pi for tangent and cotangent."},{"Start":"02:45.090 ","End":"02:50.050","Text":"It\u0027s actually smaller than this period."},{"Start":"02:50.150 ","End":"02:54.180","Text":"In formula form I copied this from somewhere,"},{"Start":"02:54.180 ","End":"02:56.460","Text":"that\u0027s why there\u0027s a z instead of a t,"},{"Start":"02:56.460 ","End":"02:59.075","Text":"but the letter doesn\u0027t matter."},{"Start":"02:59.075 ","End":"03:04.280","Text":"Notice that they\u0027re all 2Pi except for tangent and cotangent,"},{"Start":"03:04.280 ","End":"03:07.710","Text":"where the period is only Pi."},{"Start":"03:07.710 ","End":"03:15.260","Text":"Now I just want to make a remark on notation involving powers of trigonometric functions."},{"Start":"03:15.260 ","End":"03:19.900","Text":"If I have sin(t) and suppose I want to square it,"},{"Start":"03:19.900 ","End":"03:24.555","Text":"there\u0027s an easier way of writing this and we just write it,"},{"Start":"03:24.555 ","End":"03:26.930","Text":"I\u0027ll put 3 lines, equivalent to,"},{"Start":"03:26.930 ","End":"03:33.110","Text":"it\u0027s the same thing as sin^2 t. We put the 2 here,"},{"Start":"03:33.110 ","End":"03:38.030","Text":"but it means all of the sin(t)^2."},{"Start":"03:38.030 ","End":"03:44.975","Text":"For example, cos(t)^3 would be written"},{"Start":"03:44.975 ","End":"03:51.930","Text":"as cos^3 t. It\u0027s good for all the trigonometric functions."},{"Start":"03:51.930 ","End":"03:54.110","Text":"I\u0027ll give one more example."},{"Start":"03:54.110 ","End":"03:58.610","Text":"If I have tan(t) and I want to raise it to the power of 4,"},{"Start":"03:58.610 ","End":"04:08.315","Text":"I would write it as tan^4 t. There is one warning however."},{"Start":"04:08.315 ","End":"04:11.870","Text":"These exponents are usually positive."},{"Start":"04:11.870 ","End":"04:15.320","Text":"You cannot do it with minus 1 because"},{"Start":"04:15.320 ","End":"04:22.130","Text":"the notation sin^(minus 1) t is reserved for something special."},{"Start":"04:22.130 ","End":"04:24.469","Text":"Later on we\u0027ll see inverse functions,"},{"Start":"04:24.469 ","End":"04:31.680","Text":"so this is not the same as sin(t)^minus 1."},{"Start":"04:31.790 ","End":"04:36.230","Text":"No. This is a special thing,"},{"Start":"04:36.230 ","End":"04:38.540","Text":"so just be careful."},{"Start":"04:38.540 ","End":"04:43.025","Text":"Next topic is trigonometric identities."},{"Start":"04:43.025 ","End":"04:46.700","Text":"There are many of these. I\u0027m just going to show the more important ones."},{"Start":"04:46.700 ","End":"04:53.470","Text":"One group of identities is known as quotient identities."},{"Start":"04:53.470 ","End":"04:55.370","Text":"There were just 2 of these,"},{"Start":"04:55.370 ","End":"05:02.795","Text":"and they express tangent and cotangent in terms of sine and cosine as follows."},{"Start":"05:02.795 ","End":"05:05.540","Text":"Of course this only holds true wherever it\u0027s defined."},{"Start":"05:05.540 ","End":"05:09.170","Text":"This one is true wherever cosine is not 0,"},{"Start":"05:09.170 ","End":"05:12.110","Text":"and this one is true where sine is not 0."},{"Start":"05:12.110 ","End":"05:14.120","Text":"I borrowed them from elsewhere,"},{"Start":"05:14.120 ","End":"05:16.280","Text":"so that\u0027s why we have Theta instead of t."},{"Start":"05:16.280 ","End":"05:19.800","Text":"That should not worry you whatever letter we use."},{"Start":"05:19.810 ","End":"05:24.800","Text":"The identity means, of course that it\u0027s true for all Theta."},{"Start":"05:24.800 ","End":"05:27.050","Text":"It\u0027s not an equation to find Theta."},{"Start":"05:27.050 ","End":"05:29.345","Text":"It\u0027s a statement that whatever theta is,"},{"Start":"05:29.345 ","End":"05:33.140","Text":"tangent is sine over cosine, for example."},{"Start":"05:33.140 ","End":"05:39.430","Text":"The next group is the reciprocal identities."},{"Start":"05:39.430 ","End":"05:44.030","Text":"There are 3 of these or 6 of these depending how you look at it."},{"Start":"05:44.030 ","End":"05:48.560","Text":"This expresses the 3 less common ones, cosecant, secant,"},{"Start":"05:48.560 ","End":"05:51.950","Text":"and cotangent as reciprocals of sine,"},{"Start":"05:51.950 ","End":"05:54.635","Text":"cosine and tangent respectively."},{"Start":"05:54.635 ","End":"05:58.130","Text":"Of course, you could add 3 more by saying that sine is"},{"Start":"05:58.130 ","End":"06:02.300","Text":"1/ cosecant and cosine is 1/secant and so on,"},{"Start":"06:02.300 ","End":"06:05.465","Text":"but we\u0027ll stick with them in this form."},{"Start":"06:05.465 ","End":"06:12.720","Text":"Another group of identities is the Pythagorean identities."},{"Start":"06:12.910 ","End":"06:15.830","Text":"There are really 3 of them,"},{"Start":"06:15.830 ","End":"06:20.045","Text":"although we have 9 of them because the first one,"},{"Start":"06:20.045 ","End":"06:23.675","Text":"and this is the most important one and you should memorize this one,"},{"Start":"06:23.675 ","End":"06:29.885","Text":"is that the sin^2 of an angle plus the cos^2 of an angle is always equal to1."},{"Start":"06:29.885 ","End":"06:33.590","Text":"Now, in a simple algebraic manipulation,"},{"Start":"06:33.590 ","End":"06:40.675","Text":"we can put sin^2 on one side and everything else on the right and similarly for cos^2."},{"Start":"06:40.675 ","End":"06:45.050","Text":"Actually from this one we derive these 2, this one and this one,"},{"Start":"06:45.050 ","End":"06:47.630","Text":"and once we have these main ones,"},{"Start":"06:47.630 ","End":"06:50.059","Text":"then we can just extract."},{"Start":"06:50.059 ","End":"06:53.255","Text":"Like here we have tan^2 equals this."},{"Start":"06:53.255 ","End":"06:58.730","Text":"You can rewrite it in several forms."},{"Start":"06:58.730 ","End":"07:03.710","Text":"There are 2 forms that are hardly ever used,"},{"Start":"07:03.710 ","End":"07:06.590","Text":"but they\u0027re included for completeness."},{"Start":"07:06.590 ","End":"07:11.870","Text":"That\u0027s 9 Pythagorean identities and 3 main ones from those."},{"Start":"07:11.870 ","End":"07:17.880","Text":"The next group is the identities for negatives."},{"Start":"07:19.360 ","End":"07:26.250","Text":"This tells us what happens if we replace the argument by its negative,"},{"Start":"07:26.250 ","End":"07:30.390","Text":"so instead of Theta we put minus Theta."},{"Start":"07:30.390 ","End":"07:34.550","Text":"In all cases we get essentially the same thing,"},{"Start":"07:34.550 ","End":"07:38.210","Text":"just possibly with a reversed sign."},{"Start":"07:38.210 ","End":"07:47.375","Text":"Actually what that means is that the ones that have a minus are odd functions."},{"Start":"07:47.375 ","End":"07:48.938","Text":"I can just mention that,"},{"Start":"07:48.938 ","End":"07:50.270","Text":"this is an odd function."},{"Start":"07:50.270 ","End":"07:52.519","Text":"Tangent is an odd function,"},{"Start":"07:52.519 ","End":"07:54.440","Text":"this one\u0027s an odd function,"},{"Start":"07:54.440 ","End":"07:56.195","Text":"this one\u0027s an odd function."},{"Start":"07:56.195 ","End":"07:57.650","Text":"Where we have a plus,"},{"Start":"07:57.650 ","End":"07:59.540","Text":"this means the function is"},{"Start":"07:59.540 ","End":"08:03.904","Text":"an even function because when you replace the argument by its negative,"},{"Start":"08:03.904 ","End":"08:05.450","Text":"if it\u0027s the same thing,"},{"Start":"08:05.450 ","End":"08:07.460","Text":"it\u0027s an even function and if you get a minus,"},{"Start":"08:07.460 ","End":"08:08.645","Text":"it\u0027s an odd function."},{"Start":"08:08.645 ","End":"08:11.555","Text":"Anyway, if you just remember that,"},{"Start":"08:11.555 ","End":"08:16.670","Text":"all the functions are odd except for cosine and secant,"},{"Start":"08:16.670 ","End":"08:19.430","Text":"which is the inverse of the cosine."},{"Start":"08:19.430 ","End":"08:20.600","Text":"Those 2 are the even ones,"},{"Start":"08:20.600 ","End":"08:22.340","Text":"the rest are odd ones."},{"Start":"08:22.340 ","End":"08:26.285","Text":"These are the main identities,"},{"Start":"08:26.285 ","End":"08:29.760","Text":"and that concludes this clip."}],"ID":10903},{"Watched":false,"Name":"Exercise 1 Part a","Duration":"2m 42s","ChapterTopicVideoID":10360,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10360.jpeg","UploadDate":"2017-11-02T14:52:43.1330000","DurationForVideoObject":"PT2M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"In this exercise, we have to evaluate the 6 trigonometric functions of 0."},{"Start":"00:05.400 ","End":"00:08.760","Text":"That\u0027s the argument of the functions."},{"Start":"00:08.760 ","End":"00:12.120","Text":"We start off by figuring out where is"},{"Start":"00:12.120 ","End":"00:16.965","Text":"the point P(0) on the circumference of the unit circle."},{"Start":"00:16.965 ","End":"00:18.510","Text":"This is the unit circle."},{"Start":"00:18.510 ","End":"00:22.575","Text":"I didn\u0027t indicate that the radius is 1."},{"Start":"00:22.575 ","End":"00:25.905","Text":"Just take my word for it. This is the unit circle."},{"Start":"00:25.905 ","End":"00:31.320","Text":"What we do with this 0 is we travel from this point,"},{"Start":"00:31.320 ","End":"00:32.520","Text":"it\u0027s always this point,"},{"Start":"00:32.520 ","End":"00:35.723","Text":"the point 1 on the x-axis,"},{"Start":"00:35.723 ","End":"00:41.560","Text":"and we travel anticlockwise on the unit circle by this amount."},{"Start":"00:41.560 ","End":"00:44.389","Text":"Now we\u0027re traveling 0 amount."},{"Start":"00:44.389 ","End":"00:46.715","Text":"We just stay on the spot,"},{"Start":"00:46.715 ","End":"00:48.860","Text":"and that\u0027s P(0),"},{"Start":"00:48.860 ","End":"00:51.420","Text":"which is 1, 0."},{"Start":"00:52.040 ","End":"00:56.125","Text":"That\u0027s our x and our y."},{"Start":"00:56.125 ","End":"01:00.890","Text":"Now there are other standard formulas for all the 6 trigonometric functions."},{"Start":"01:00.890 ","End":"01:08.270","Text":"The sine is just equal to the y in general,"},{"Start":"01:08.270 ","End":"01:11.910","Text":"which in our case is 0."},{"Start":"01:12.040 ","End":"01:20.225","Text":"The cosine of the same argument is the x of the point P,"},{"Start":"01:20.225 ","End":"01:23.050","Text":"which is 1,"},{"Start":"01:23.050 ","End":"01:28.560","Text":"and the tangent is y over x."},{"Start":"01:28.560 ","End":"01:32.255","Text":"All these 6 formulas you should know by heart,"},{"Start":"01:32.255 ","End":"01:36.940","Text":"is y over x, which is 0 over 1,"},{"Start":"01:36.940 ","End":"01:39.555","Text":"which is just 0."},{"Start":"01:39.555 ","End":"01:42.440","Text":"Now we\u0027ve got 3 more to go."},{"Start":"01:42.440 ","End":"01:44.180","Text":"I\u0027ll just do them in the order."},{"Start":"01:44.180 ","End":"01:46.715","Text":"They\u0027re the reciprocals of these."},{"Start":"01:46.715 ","End":"01:55.290","Text":"The cosecant of 0 is 1 over y. Not quite."},{"Start":"01:55.290 ","End":"01:58.355","Text":"It\u0027s 1 over y, provided that y is not 0,"},{"Start":"01:58.355 ","End":"02:02.125","Text":"provided this is defined and this is undefined."},{"Start":"02:02.125 ","End":"02:05.030","Text":"I\u0027ll just write u or maybe I\u0027ll"},{"Start":"02:05.030 ","End":"02:10.170","Text":"write undefined once and then in future I\u0027ll just write u."},{"Start":"02:11.110 ","End":"02:18.735","Text":"Then the secant is 1 over x,"},{"Start":"02:18.735 ","End":"02:20.400","Text":"which is fine,"},{"Start":"02:20.400 ","End":"02:23.325","Text":"which is 1 over 1, which is 1."},{"Start":"02:23.325 ","End":"02:29.285","Text":"The cotangent is x over y."},{"Start":"02:29.285 ","End":"02:35.700","Text":"That\u0027s also undefined because y is 0."},{"Start":"02:35.830 ","End":"02:42.330","Text":"Let\u0027s write u undefined. That\u0027s it."}],"ID":10720},{"Watched":false,"Name":"Exercise 1 Part b","Duration":"3m 27s","ChapterTopicVideoID":10361,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10361.jpeg","UploadDate":"2017-11-02T14:52:55.1070000","DurationForVideoObject":"PT3M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.220","Text":"In this exercise, we have to evaluate all 6 trigonometric functions of pi over 4."},{"Start":"00:08.220 ","End":"00:12.750","Text":"Here I have a diagram of the unit circle that will help us."},{"Start":"00:12.750 ","End":"00:15.690","Text":"What we need is to first of all,"},{"Start":"00:15.690 ","End":"00:19.922","Text":"find where is P of Pi over 4."},{"Start":"00:19.922 ","End":"00:21.780","Text":"Where P, like in the tutorial,"},{"Start":"00:21.780 ","End":"00:24.310","Text":"means the point on the circumference,"},{"Start":"00:24.310 ","End":"00:32.450","Text":"which is Pi over 4 units away in distance from this point here,"},{"Start":"00:32.450 ","End":"00:35.735","Text":"traveling along the circumference anticlockwise."},{"Start":"00:35.735 ","End":"00:39.245","Text":"Now since Pi is half the circumference,"},{"Start":"00:39.245 ","End":"00:43.619","Text":"Pi over 4 would be halfway between this and this."},{"Start":"00:43.619 ","End":"00:45.615","Text":"This says 1, this is 1,"},{"Start":"00:45.615 ","End":"00:52.285","Text":"so this would be P of Pi over 4."},{"Start":"00:52.285 ","End":"00:57.205","Text":"Question is, what are the coordinates of this point?"},{"Start":"00:57.205 ","End":"00:59.540","Text":"Well, because of the symmetry,"},{"Start":"00:59.540 ","End":"01:01.070","Text":"because it\u0027s exactly halfway,"},{"Start":"01:01.070 ","End":"01:09.335","Text":"it\u0027s going to be on the line that goes from the origin at 45 degrees."},{"Start":"01:09.335 ","End":"01:13.460","Text":"This is the line and it\u0027s formula is y equals x."},{"Start":"01:13.460 ","End":"01:18.915","Text":"The unit circle is x squared plus y squared equals 1."},{"Start":"01:18.915 ","End":"01:22.425","Text":"We want the intersection of these 2 curves."},{"Start":"01:22.425 ","End":"01:24.500","Text":"There\u0027s actually 2 of them."},{"Start":"01:24.500 ","End":"01:26.840","Text":"The hits are over here and over here,"},{"Start":"01:26.840 ","End":"01:30.470","Text":"but if you just substitute y equals x in here and solve it,"},{"Start":"01:30.470 ","End":"01:35.440","Text":"and just take the positive solution we\u0027ll get that this point,"},{"Start":"01:35.440 ","End":"01:38.880","Text":"I won\u0027t do all the work for you,"},{"Start":"01:38.880 ","End":"01:43.590","Text":"comes out to be 1 over square root of 2,"},{"Start":"01:43.590 ","End":"01:46.570","Text":"1 over square root of 2."},{"Start":"01:46.570 ","End":"01:50.555","Text":"Briefly I\u0027ll mention if you do substitute this,"},{"Start":"01:50.555 ","End":"01:53.210","Text":"you\u0027ll get 2y squared equals 1,"},{"Start":"01:53.210 ","End":"01:55.130","Text":"y squared equals a half,"},{"Start":"01:55.130 ","End":"01:58.535","Text":"so y is plus or minus the square root of a half."},{"Start":"01:58.535 ","End":"02:02.195","Text":"But you\u0027d have to take the plus because we see that it\u0027s in the first quadrant,"},{"Start":"02:02.195 ","End":"02:06.470","Text":"and if y is 1 over the square root of 2 and x equals y,"},{"Start":"02:06.470 ","End":"02:08.975","Text":"then x is also equal to that."},{"Start":"02:08.975 ","End":"02:11.720","Text":"Now we have the x and the y,"},{"Start":"02:11.720 ","End":"02:14.600","Text":"and now we just apply the standard formulas,"},{"Start":"02:14.600 ","End":"02:17.825","Text":"all 6 of them."},{"Start":"02:17.825 ","End":"02:23.730","Text":"Sine of Pi over 4 is the y of the point,"},{"Start":"02:23.730 ","End":"02:26.910","Text":"which is 1 over square root of 2,"},{"Start":"02:26.910 ","End":"02:34.430","Text":"and then we have the cosine of the argument is x,"},{"Start":"02:34.430 ","End":"02:38.435","Text":"which is also 1 over square root of 2,"},{"Start":"02:38.435 ","End":"02:44.940","Text":"the tangent is given by y over x,"},{"Start":"02:44.940 ","End":"02:47.970","Text":"y equals x or y over x is 1,"},{"Start":"02:47.970 ","End":"02:49.970","Text":"and then to the reciprocals of these,"},{"Start":"02:49.970 ","End":"02:53.760","Text":"the cosecant of Pi over 4."},{"Start":"02:53.760 ","End":"02:59.385","Text":"You either use the formula or it just take 1 over this square root of 2,"},{"Start":"02:59.385 ","End":"03:03.305","Text":"but it\u0027s also equal to 1 over y."},{"Start":"03:03.305 ","End":"03:11.250","Text":"Anyway, the secant is the reciprocal of cosine,"},{"Start":"03:11.250 ","End":"03:15.135","Text":"so it\u0027s also square root of 2 and finally,"},{"Start":"03:15.135 ","End":"03:22.100","Text":"the cotangent of Pi over 4 is 1 over the tangent or x over y,"},{"Start":"03:22.100 ","End":"03:24.965","Text":"whichever you prefer, and it comes out to be 1."},{"Start":"03:24.965 ","End":"03:28.290","Text":"So that\u0027s all 6 of them and we are done."}],"ID":10721},{"Watched":false,"Name":"Exercise 1 Part c","Duration":"2m 45s","ChapterTopicVideoID":10362,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10362.jpeg","UploadDate":"2017-11-02T14:53:03.9000000","DurationForVideoObject":"PT2M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, we\u0027re going to evaluate all the trigonometric functions,"},{"Start":"00:04.470 ","End":"00:08.295","Text":"6 of them of π/2."},{"Start":"00:08.295 ","End":"00:11.640","Text":"Now here\u0027s the unit circle."},{"Start":"00:11.640 ","End":"00:17.410","Text":"I don\u0027t indicate, I should really have written 1 unit circle."},{"Start":"00:17.450 ","End":"00:26.325","Text":"What we want to do is see where is p(π/2) When I write p,"},{"Start":"00:26.325 ","End":"00:33.225","Text":"it means like in the tutorial that we start from this point here and go anticlockwise,"},{"Start":"00:33.225 ","End":"00:36.960","Text":"a distance of π/2."},{"Start":"00:36.960 ","End":"00:39.930","Text":"But the whole circle is 2π."},{"Start":"00:39.930 ","End":"00:45.950","Text":"Half a circle is π and π/2 would be a quarter of the circle."},{"Start":"00:45.950 ","End":"00:49.015","Text":"We\u0027re talking about this point here."},{"Start":"00:49.015 ","End":"00:53.070","Text":"This is the point (0,1)."},{"Start":"00:53.070 ","End":"00:55.980","Text":"That\u0027s the x of it and that\u0027s the y of it."},{"Start":"00:55.980 ","End":"01:02.990","Text":"Now, we can see what is sine."},{"Start":"01:02.990 ","End":"01:09.785","Text":"I\u0027ll write them, sine, cosine, tangent."},{"Start":"01:09.785 ","End":"01:11.815","Text":"I\u0027ll do the first 3."},{"Start":"01:11.815 ","End":"01:13.350","Text":"We have the formulas."},{"Start":"01:13.350 ","End":"01:15.735","Text":"The sine is y,"},{"Start":"01:15.735 ","End":"01:17.120","Text":"the cosine is x,"},{"Start":"01:17.120 ","End":"01:20.885","Text":"and the tangent is y/x."},{"Start":"01:20.885 ","End":"01:29.030","Text":"In our case, y=1,"},{"Start":"01:29.270 ","End":"01:31.880","Text":"x=0."},{"Start":"01:31.880 ","End":"01:33.680","Text":"But y/x can\u0027t be computed."},{"Start":"01:33.680 ","End":"01:36.765","Text":"We can\u0027t divide by 0, so it\u0027s undefined."},{"Start":"01:36.765 ","End":"01:41.735","Text":"I\u0027ll just write u as a custom undefined."},{"Start":"01:41.735 ","End":"01:44.450","Text":"Now the other 3,"},{"Start":"01:44.450 ","End":"01:53.930","Text":"those are cosecant, secant, and cotangent."},{"Start":"01:53.930 ","End":"01:56.270","Text":"We\u0027re still working"},{"Start":"01:56.270 ","End":"02:06.090","Text":"with π/2=."},{"Start":"02:06.090 ","End":"02:10.405","Text":"Now, this one is 1/y."},{"Start":"02:10.405 ","End":"02:15.490","Text":"It\u0027s going to be the reciprocals of these 1/y, 1/x,"},{"Start":"02:15.490 ","End":"02:22.310","Text":"and x/y, and 1/y=1."},{"Start":"02:22.310 ","End":"02:24.020","Text":"Now x=0 here,"},{"Start":"02:24.020 ","End":"02:27.020","Text":"so we have a problem here."},{"Start":"02:27.020 ","End":"02:29.030","Text":"This one is undefined,"},{"Start":"02:29.030 ","End":"02:35.110","Text":"but x/y is not a problem that is 0."},{"Start":"02:35.110 ","End":"02:39.255","Text":"These are the 6 answers,"},{"Start":"02:39.255 ","End":"02:45.340","Text":"2 of them are undefined, and that\u0027s it."}],"ID":10722},{"Watched":false,"Name":"Exercise 1 Part d","Duration":"2m 59s","ChapterTopicVideoID":10363,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10363.jpeg","UploadDate":"2017-11-02T14:53:14.4400000","DurationForVideoObject":"PT2M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.120","Text":"In this exercise, we have to evaluate all 6 trigonometric functions of 3 pi over 4."},{"Start":"00:09.120 ","End":"00:14.355","Text":"I provided a diagram of the unit circle so that this would be 1,"},{"Start":"00:14.355 ","End":"00:15.780","Text":"and this will be 1."},{"Start":"00:15.780 ","End":"00:23.850","Text":"The first thing we have to do is to figure out what is P(3) pi over 4,"},{"Start":"00:23.850 ","End":"00:31.965","Text":"where P is the point we obtained by going from this point counterclockwise,"},{"Start":"00:31.965 ","End":"00:35.130","Text":"a distance of 3 pi over 4."},{"Start":"00:35.130 ","End":"00:37.860","Text":"Now, 1/2 a circle is Pi,"},{"Start":"00:37.860 ","End":"00:41.110","Text":"so that\u0027s 3/4 (1/2) a circle,"},{"Start":"00:41.110 ","End":"00:45.960","Text":"so that would take us halfway between here and here,"},{"Start":"00:45.960 ","End":"00:48.670","Text":"this would be the point"},{"Start":"00:49.310 ","End":"00:56.885","Text":"P. But to help us in the previous exercise,"},{"Start":"00:56.885 ","End":"01:02.590","Text":"let me just write the argument of P here."},{"Start":"01:03.260 ","End":"01:06.535","Text":"As I said in the previous exercise,"},{"Start":"01:06.535 ","End":"01:10.370","Text":"figured out what is P of Pi over 4,"},{"Start":"01:10.370 ","End":"01:13.020","Text":"and so I\u0027m going to use this to help us find this."},{"Start":"01:13.020 ","End":"01:16.800","Text":"This turned out to be 1 over root 2,"},{"Start":"01:16.800 ","End":"01:19.440","Text":"1 over root 2."},{"Start":"01:19.440 ","End":"01:27.250","Text":"Now by symmetry, this is obtained from this by just reversing the x of the point,"},{"Start":"01:27.250 ","End":"01:33.970","Text":"so this will be minus 1 over root 2,"},{"Start":"01:33.970 ","End":"01:36.270","Text":"1 over root 2."},{"Start":"01:36.270 ","End":"01:39.915","Text":"This is our x and this is our y,"},{"Start":"01:39.915 ","End":"01:44.200","Text":"and now we have the standard formulas that"},{"Start":"01:44.200 ","End":"01:50.555","Text":"the sine of this is equal to the y,"},{"Start":"01:50.555 ","End":"01:54.885","Text":"and our y in this case is 1 over root 2."},{"Start":"01:54.885 ","End":"01:59.655","Text":"Let me start writing the others cosine, tangent,"},{"Start":"01:59.655 ","End":"02:01.910","Text":"then we want cosecant,"},{"Start":"02:01.910 ","End":"02:04.955","Text":"then we want the secant,"},{"Start":"02:04.955 ","End":"02:08.320","Text":"and finally the cotangent."},{"Start":"02:08.510 ","End":"02:10.980","Text":"Just wrote these in,"},{"Start":"02:10.980 ","End":"02:13.445","Text":"equals, equals, equals, equals,"},{"Start":"02:13.445 ","End":"02:21.810","Text":"equals the cosine is x and x here is minus 1 over root 2."},{"Start":"02:21.810 ","End":"02:25.290","Text":"The tangent is y/x."},{"Start":"02:25.290 ","End":"02:29.040","Text":"This over this is going to be clearly -1."},{"Start":"02:29.040 ","End":"02:34.260","Text":"The cosecant is 1/y."},{"Start":"02:34.260 ","End":"02:37.020","Text":"1/y is 1/1 over root 2,"},{"Start":"02:37.020 ","End":"02:39.285","Text":"so it\u0027s just going to be root 2."},{"Start":"02:39.285 ","End":"02:42.780","Text":"Secant is 1/x,"},{"Start":"02:42.780 ","End":"02:47.150","Text":"1/ x will be like this, but minus,"},{"Start":"02:47.150 ","End":"02:54.675","Text":"so it\u0027s minus root 2 and the cotangent is x/y,"},{"Start":"02:54.675 ","End":"02:59.980","Text":"again, it\u0027s going to give us minus 1. that\u0027s it."}],"ID":10723},{"Watched":false,"Name":"Exercise 1 Part e","Duration":"3m 10s","ChapterTopicVideoID":10364,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10364.jpeg","UploadDate":"2017-11-02T14:53:25.1370000","DurationForVideoObject":"PT3M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"In this exercise, we\u0027re asked to evaluate all 6 of"},{"Start":"00:03.330 ","End":"00:10.830","Text":"the trigonometric functions of the argument minus Pi over 4 and as usual,"},{"Start":"00:10.830 ","End":"00:13.710","Text":"I provide the picture of the unit circle."},{"Start":"00:13.710 ","End":"00:17.370","Text":"I should label this 1,1."},{"Start":"00:17.370 ","End":"00:26.340","Text":"The way we do it is, we first compute the point p(minus Pi over 4) as in the Tutorial."},{"Start":"00:26.340 ","End":"00:30.400","Text":"The way to do it is to start from this point here."},{"Start":"00:30.950 ","End":"00:37.890","Text":"When it\u0027s minus, we go clockwise and when it\u0027s plus, we go anticlockwise."},{"Start":"00:37.890 ","End":"00:41.365","Text":"In each case, we travel the absolute value distance."},{"Start":"00:41.365 ","End":"00:45.710","Text":"I have to go Pi over 4 in the clockwise direction."},{"Start":"00:45.710 ","End":"00:48.725","Text":"Now Pi would take me to here,"},{"Start":"00:48.725 ","End":"00:50.615","Text":"so Pi over 2 would be here."},{"Start":"00:50.615 ","End":"00:53.600","Text":"Pi over 4 is halfway between here and here."},{"Start":"00:53.600 ","End":"00:54.838","Text":"Let\u0027s say, this is it,"},{"Start":"00:54.838 ","End":"01:00.540","Text":"so that\u0027s p minus Pi over 4."},{"Start":"01:00.540 ","End":"01:02.840","Text":"In terms of coordinates,"},{"Start":"01:02.840 ","End":"01:09.510","Text":"we can use a previous question where we had Pi over 4, which was here."},{"Start":"01:09.560 ","End":"01:12.440","Text":"I\u0027ll just write Pi over 4."},{"Start":"01:12.440 ","End":"01:14.140","Text":"That was the previous problem."},{"Start":"01:14.140 ","End":"01:19.760","Text":"We computed this as 1 over root 2,"},{"Start":"01:19.760 ","End":"01:23.020","Text":"1 over root 2."},{"Start":"01:23.020 ","End":"01:28.910","Text":"By symmetry, this one is going to equal the same,"},{"Start":"01:28.910 ","End":"01:30.170","Text":"1 over root 2."},{"Start":"01:30.170 ","End":"01:32.945","Text":"The x is the same, but the y is negated;"},{"Start":"01:32.945 ","End":"01:35.600","Text":"minus 1 over root 2,"},{"Start":"01:35.600 ","End":"01:39.150","Text":"just by reflection in the x-axis."},{"Start":"01:40.330 ","End":"01:43.220","Text":"I\u0027ll write it again over here."},{"Start":"01:43.220 ","End":"01:45.050","Text":"1 over root 2,"},{"Start":"01:45.050 ","End":"01:49.530","Text":"minus 1 over root 2."},{"Start":"01:49.530 ","End":"01:52.250","Text":"This is the x of the point,"},{"Start":"01:52.250 ","End":"01:54.290","Text":"and that\u0027s the y of the point."},{"Start":"01:54.290 ","End":"01:57.872","Text":"Now we want the six trigonometric functions. What are they?"},{"Start":"01:57.872 ","End":"02:00.745","Text":"Sine, cosine,"},{"Start":"02:00.745 ","End":"02:04.800","Text":"tangent, cosecant,"},{"Start":"02:04.800 ","End":"02:13.795","Text":"secant and cotangent and I want to evaluate each one minus Pi over 4."},{"Start":"02:13.795 ","End":"02:16.780","Text":"I did a quick copy-paste job here."},{"Start":"02:16.780 ","End":"02:20.920","Text":"You should have these memorized that the sine is always y,"},{"Start":"02:20.920 ","End":"02:25.440","Text":"cosine is the x of the point, tan y/x,"},{"Start":"02:25.440 ","End":"02:28.090","Text":"cosecant is 1/y,"},{"Start":"02:28.090 ","End":"02:35.845","Text":"secant is 1/x and cotangent is x/y and we apply it to this point here."},{"Start":"02:35.845 ","End":"02:37.900","Text":"So we\u0027ll get,"},{"Start":"02:37.900 ","End":"02:42.330","Text":"y is minus 1 over root 2."},{"Start":"02:42.330 ","End":"02:45.209","Text":"X is 1 over root 2, that\u0027s the cosine."},{"Start":"02:45.209 ","End":"02:49.520","Text":"The tangent therefore will be this over this is minus 1."},{"Start":"02:49.520 ","End":"02:52.430","Text":"Cosecant is 1/y,"},{"Start":"02:52.430 ","End":"02:57.455","Text":"this is just without the reciprocal."},{"Start":"02:57.455 ","End":"03:01.880","Text":"The secant, 1/x would be root 2."},{"Start":"03:01.880 ","End":"03:07.010","Text":"The cotangent would be x/y is minus 1."},{"Start":"03:07.010 ","End":"03:10.410","Text":"That\u0027s all six of them and so we\u0027re done."}],"ID":10724},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"6m 12s","ChapterTopicVideoID":10365,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10365.jpeg","UploadDate":"2017-11-02T14:53:45.9800000","DurationForVideoObject":"PT6M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"In this exercise, we\u0027re not given the quantity Theta,"},{"Start":"00:03.975 ","End":"00:10.135","Text":"but we\u0027re told that the sine of theta is 2/5 and that Theta is in the first quadrant."},{"Start":"00:10.135 ","End":"00:14.190","Text":"Our task is to evaluate the other 5 trigonometric functions,"},{"Start":"00:14.190 ","End":"00:15.390","Text":"they are 6 altogether."},{"Start":"00:15.390 ","End":"00:17.535","Text":"So all but the sine."},{"Start":"00:17.535 ","End":"00:21.210","Text":"We don\u0027t need a diagram for this exercise,"},{"Start":"00:21.210 ","End":"00:25.380","Text":"but I put one in any way. Sometimes helps."},{"Start":"00:25.380 ","End":"00:29.770","Text":"I\u0027ll draw a ray here."},{"Start":"00:29.770 ","End":"00:34.325","Text":"I just need the bit from the origin to the circumference."},{"Start":"00:34.325 ","End":"00:39.560","Text":"The theta here is the distance along the circumference from"},{"Start":"00:39.560 ","End":"00:46.935","Text":"this point to this point here, P of Theta."},{"Start":"00:46.935 ","End":"00:49.370","Text":"It has a certain x and a certain y,"},{"Start":"00:49.370 ","End":"00:53.755","Text":"but we don\u0027t know what those are yet."},{"Start":"00:53.755 ","End":"00:56.120","Text":"Theta is this distance, like I said,"},{"Start":"00:56.120 ","End":"01:01.100","Text":"in this counterclockwise direction, because it\u0027s positive."},{"Start":"01:01.100 ","End":"01:03.380","Text":"Let\u0027s see what we can tell."},{"Start":"01:03.380 ","End":"01:08.075","Text":"Well, we know already that y is equal to"},{"Start":"01:08.075 ","End":"01:15.440","Text":"2/5 because y is the sine of Theta."},{"Start":"01:15.440 ","End":"01:20.555","Text":"Now we have to use this to find the other five."},{"Start":"01:20.555 ","End":"01:22.665","Text":"Sine is first we have that."},{"Start":"01:22.665 ","End":"01:24.835","Text":"Next one is the cosine."},{"Start":"01:24.835 ","End":"01:28.820","Text":"We want the cosine of Theta,"},{"Start":"01:28.820 ","End":"01:31.655","Text":"and that would be the x of the point."},{"Start":"01:31.655 ","End":"01:33.665","Text":"But how do I find the x?"},{"Start":"01:33.665 ","End":"01:35.240","Text":"There are 2 similar approaches."},{"Start":"01:35.240 ","End":"01:38.120","Text":"One is to use trigonometric identities."},{"Start":"01:38.120 ","End":"01:44.090","Text":"The sine squared of an angle plus cosine squared of an angle is equal to 1."},{"Start":"01:44.090 ","End":"01:51.380","Text":"We can also say that this point is on the unit circle and so x^2 plus y^2 equals 1,"},{"Start":"01:51.380 ","End":"01:57.184","Text":"but I\u0027ll write it as y^2 plus x^2 equals 1 just to correspond to this."},{"Start":"01:57.184 ","End":"01:59.360","Text":"We have y is 2/5,"},{"Start":"01:59.360 ","End":"02:01.010","Text":"and x is cosine Theta."},{"Start":"02:01.010 ","End":"02:04.190","Text":"So we can write that"},{"Start":"02:04.190 ","End":"02:13.170","Text":"2/5^2 plus x^2 is equal to 1."},{"Start":"02:13.170 ","End":"02:14.370","Text":"If that\u0027s the case,"},{"Start":"02:14.370 ","End":"02:20.925","Text":"then x^2 is 1 minus 4/25,"},{"Start":"02:20.925 ","End":"02:25.500","Text":"which would give me 21/25."},{"Start":"02:25.500 ","End":"02:32.110","Text":"Now normally I would say plus or minus,"},{"Start":"02:32.110 ","End":"02:34.385","Text":"but because it\u0027s in the first quadrant,"},{"Start":"02:34.385 ","End":"02:37.320","Text":"the point then x is positive."},{"Start":"02:37.320 ","End":"02:39.060","Text":"So it\u0027s the square root of this."},{"Start":"02:39.060 ","End":"02:41.420","Text":"I can take the square root of the numerator"},{"Start":"02:41.420 ","End":"02:45.080","Text":"separately and the square root of the denominator."},{"Start":"02:45.080 ","End":"02:47.690","Text":"That\u0027s cosine Theta."},{"Start":"02:47.690 ","End":"02:51.885","Text":"Why don\u0027t I highlight each solution as we come to it?"},{"Start":"02:51.885 ","End":"02:54.660","Text":"That\u0027s the cosine."},{"Start":"02:55.700 ","End":"02:58.600","Text":"Next we want the tangent."},{"Start":"02:58.600 ","End":"03:00.640","Text":"Again, there\u0027s 2 ways of doing it."},{"Start":"03:00.640 ","End":"03:04.390","Text":"One way is to say that the tangent there is a quotient identity,"},{"Start":"03:04.390 ","End":"03:08.860","Text":"is the sine over the cosine."},{"Start":"03:08.860 ","End":"03:14.755","Text":"But we can also say that the tangent is simply y/x."},{"Start":"03:14.755 ","End":"03:17.305","Text":"It\u0027ll come out to the same thing."},{"Start":"03:17.305 ","End":"03:23.290","Text":"We get that tangent Theta is y,"},{"Start":"03:23.290 ","End":"03:29.515","Text":"which was given to be 2/5/x,"},{"Start":"03:29.515 ","End":"03:36.730","Text":"which is the square root of 21/5,"},{"Start":"03:36.730 ","End":"03:46.850","Text":"then the 1/5 cancels and this just gives us 2 over the square root of 21."},{"Start":"03:47.280 ","End":"03:49.360","Text":"Let me highlight it."},{"Start":"03:49.360 ","End":"03:50.560","Text":"That\u0027s the second one."},{"Start":"03:50.560 ","End":"03:53.965","Text":"A tangent is this."},{"Start":"03:53.965 ","End":"03:56.020","Text":"What\u0027s going to be next?"},{"Start":"03:56.020 ","End":"03:58.405","Text":"The cosecant."},{"Start":"03:58.405 ","End":"04:01.150","Text":"Once again, there\u0027s 2 ways I can go about it."},{"Start":"04:01.150 ","End":"04:02.740","Text":"I\u0027m just going to go with the simpler one,"},{"Start":"04:02.740 ","End":"04:06.835","Text":"which I think is from the definition of y and x rather than the identities,"},{"Start":"04:06.835 ","End":"04:09.355","Text":"I could say cosecant is 1 over sine."},{"Start":"04:09.355 ","End":"04:13.180","Text":"But I\u0027d rather say that cosecant of"},{"Start":"04:13.180 ","End":"04:20.060","Text":"Theta is 1/y,"},{"Start":"04:20.060 ","End":"04:29.380","Text":"which is 1/sine and 1/ y since y is 2/5 is 5/2."},{"Start":"04:29.720 ","End":"04:34.305","Text":"Highlight that and highlight that."},{"Start":"04:34.305 ","End":"04:37.435","Text":"What\u0027s going to be next?"},{"Start":"04:37.435 ","End":"04:48.170","Text":"Let\u0027s go with secant of Theta which is either 1/cosine or more simply, 1/ x."},{"Start":"04:48.170 ","End":"04:50.195","Text":"Since we have x here,"},{"Start":"04:50.195 ","End":"04:53.435","Text":"1/x is the reciprocal upside down,"},{"Start":"04:53.435 ","End":"04:59.640","Text":"is 5 over the square root of 21."},{"Start":"04:59.900 ","End":"05:04.540","Text":"Highlight it here."},{"Start":"05:04.850 ","End":"05:08.219","Text":"There\u0027s one more to go."},{"Start":"05:08.219 ","End":"05:16.010","Text":"That one will be the cotangent and the cotangent of Theta."},{"Start":"05:16.010 ","End":"05:20.180","Text":"You can either use the identity that it\u0027s cosine/sine,"},{"Start":"05:20.180 ","End":"05:24.380","Text":"or you can just say straight away that it\u0027s x/y root"},{"Start":"05:24.380 ","End":"05:31.865","Text":"21/5 divided by y,"},{"Start":"05:31.865 ","End":"05:36.260","Text":"which is 2/5."},{"Start":"05:36.260 ","End":"05:41.195","Text":"This makes it equal to root 21/2."},{"Start":"05:41.195 ","End":"05:43.730","Text":"Just point out that there is another thing you could"},{"Start":"05:43.730 ","End":"05:46.340","Text":"have done there is also a reciprocal identity,"},{"Start":"05:46.340 ","End":"05:50.000","Text":"that cotangent is 1/tangent. I\u0027ll just write that down."},{"Start":"05:50.000 ","End":"05:53.030","Text":"It\u0027s also equal to 1/tangent Theta."},{"Start":"05:53.030 ","End":"05:55.490","Text":"If we took the tangent and inverted it,"},{"Start":"05:55.490 ","End":"05:58.950","Text":"then we\u0027d get straightaway from here to here."},{"Start":"05:59.200 ","End":"06:01.550","Text":"This is actually the last one."},{"Start":"06:01.550 ","End":"06:02.780","Text":"I just want to highlight it."},{"Start":"06:02.780 ","End":"06:05.135","Text":"The cotangent is this."},{"Start":"06:05.135 ","End":"06:08.120","Text":"You have cotangent, secant, cosecant, tangent,"},{"Start":"06:08.120 ","End":"06:12.690","Text":"and cosine, the original plus 5 others and we\u0027re done."}],"ID":10725},{"Watched":false,"Name":"Exercise 2 Part b","Duration":"5m 3s","ChapterTopicVideoID":10366,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10366.jpeg","UploadDate":"2017-11-02T14:54:03.5330000","DurationForVideoObject":"PT5M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.200","Text":"In this exercise, we are given that cosine of alpha is minus 1/2,"},{"Start":"00:07.200 ","End":"00:16.620","Text":"and we don\u0027t know what alpha is except that it\u0027s in the second quadrant."},{"Start":"00:16.620 ","End":"00:20.970","Text":"We have to evaluate the other 5 trigonometric functions."},{"Start":"00:20.970 ","End":"00:23.235","Text":"The others except for the cosine."},{"Start":"00:23.235 ","End":"00:28.335","Text":"We don\u0027t really need the diagram but I like it, the unit circle."},{"Start":"00:28.335 ","End":"00:31.275","Text":"This is 1, this is 1."},{"Start":"00:31.275 ","End":"00:33.885","Text":"Don\u0027t forget what the quadrants are."},{"Start":"00:33.885 ","End":"00:36.255","Text":"Just a reminder this is quadrant 1,"},{"Start":"00:36.255 ","End":"00:40.995","Text":"2, 3, and Roman 4."},{"Start":"00:40.995 ","End":"00:46.760","Text":"We know that alpha is somewhere in this second quadrant."},{"Start":"00:46.760 ","End":"00:49.790","Text":"Actually, it\u0027s more correct to say that p"},{"Start":"00:49.790 ","End":"00:53.915","Text":"of alpha is in the second quadrant at some point here."},{"Start":"00:53.915 ","End":"00:57.669","Text":"But because the cosine is the x,"},{"Start":"00:57.669 ","End":"00:59.925","Text":"this is minus 1,"},{"Start":"00:59.925 ","End":"01:03.450","Text":"here let\u0027s say is minus 1/2,"},{"Start":"01:03.450 ","End":"01:06.365","Text":"so if I just go straight above,"},{"Start":"01:06.365 ","End":"01:09.260","Text":"that would be maybe here,"},{"Start":"01:09.260 ","End":"01:11.930","Text":"that would be p of alpha."},{"Start":"01:11.930 ","End":"01:15.320","Text":"I\u0027ll just put a dotted line here, minus 1/2."},{"Start":"01:15.320 ","End":"01:22.634","Text":"That\u0027s the x. The point is minus 1/2, and what is the y?"},{"Start":"01:22.634 ","End":"01:30.720","Text":"As usual, we can use the x^2+y^2=1 because it\u0027s the unit circle."},{"Start":"01:30.720 ","End":"01:33.185","Text":"If we do that computation,"},{"Start":"01:33.185 ","End":"01:40.295","Text":"we\u0027ll get that 1/2^2 is a 1/4 plus this is 1, y^2 is 3/4."},{"Start":"01:40.295 ","End":"01:43.330","Text":"y is going to be then,"},{"Start":"01:43.330 ","End":"01:46.335","Text":"plus or minus root 3/2."},{"Start":"01:46.335 ","End":"01:48.540","Text":"Now, which is it, plus or minus?"},{"Start":"01:48.540 ","End":"01:50.990","Text":"Again, second quadrant y is positive,"},{"Start":"01:50.990 ","End":"01:53.720","Text":"so it\u0027s plus, so that\u0027s our point."},{"Start":"01:53.720 ","End":"01:57.550","Text":"That\u0027s the x of the point and the y of the point,"},{"Start":"01:57.550 ","End":"01:59.720","Text":"and once we\u0027ve got that,"},{"Start":"01:59.720 ","End":"02:07.560","Text":"now it\u0027s pretty straightforward to say what all the 5 functions are."},{"Start":"02:07.560 ","End":"02:09.450","Text":"Let\u0027s see, what are we missing?"},{"Start":"02:09.450 ","End":"02:12.160","Text":"Sine of alpha?"},{"Start":"02:12.950 ","End":"02:18.120","Text":"I just wrote out the five others."},{"Start":"02:18.120 ","End":"02:23.175","Text":"Now one way of doing it certainly is just to use the formulas with y and x."},{"Start":"02:23.175 ","End":"02:26.875","Text":"To say sine of alpha is y,"},{"Start":"02:26.875 ","End":"02:32.310","Text":"tan alpha is y over x, etc."},{"Start":"02:32.310 ","End":"02:36.140","Text":"But this time I\u0027d like to do it with the identities."},{"Start":"02:36.140 ","End":"02:37.970","Text":"There\u0027s the Pythagoras identity,"},{"Start":"02:37.970 ","End":"02:42.890","Text":"reciprocal identity, what\u0027s the other?"},{"Start":"02:42.890 ","End":"02:45.800","Text":"The quotient identities."},{"Start":"02:45.800 ","End":"02:49.010","Text":"If we use the Pythagoras identity here,"},{"Start":"02:49.010 ","End":"02:55.500","Text":"we\u0027d say that sine"},{"Start":"02:55.500 ","End":"03:00.915","Text":"squared alpha is 1 minus cosine squared alpha."},{"Start":"03:00.915 ","End":"03:02.900","Text":"If we did this, well,"},{"Start":"03:02.900 ","End":"03:07.370","Text":"exactly the same computation that led us to the root 3/2,"},{"Start":"03:07.370 ","End":"03:12.125","Text":"would lead us to the same thing, root 3/2."},{"Start":"03:12.125 ","End":"03:19.010","Text":"Now for tangent, there is a reciprocal identity that tangent of anything,"},{"Start":"03:19.010 ","End":"03:27.420","Text":"say alpha, is sine over cosine of alpha."},{"Start":"03:28.720 ","End":"03:35.420","Text":"What we get here is just cosine from here, sine from here."},{"Start":"03:35.420 ","End":"03:42.550","Text":"It\u0027s going to be this divided by this,"},{"Start":"03:42.550 ","End":"03:50.550","Text":"and that will make it minus root 3, and highlight it."},{"Start":"03:50.550 ","End":"03:53.690","Text":"Cosecant; for that,"},{"Start":"03:53.690 ","End":"03:56.135","Text":"we\u0027ll use the identity of the cosecant,"},{"Start":"03:56.135 ","End":"03:58.310","Text":"is 1 over the sine."},{"Start":"03:58.310 ","End":"04:01.660","Text":"I didn\u0027t bother putting the alpha or any other letter."},{"Start":"04:01.660 ","End":"04:04.755","Text":"It\u0027s 1 over the sine, so the reciprocal of this,"},{"Start":"04:04.755 ","End":"04:08.445","Text":"it\u0027s 2 over square root of 3,"},{"Start":"04:08.445 ","End":"04:10.740","Text":"and that\u0027s another one,"},{"Start":"04:10.740 ","End":"04:13.675","Text":"that\u0027s 3 out of 5, 2 more to go."},{"Start":"04:13.675 ","End":"04:17.660","Text":"The secant, very similar to the cosecant."},{"Start":"04:17.660 ","End":"04:19.835","Text":"The reciprocal identity,"},{"Start":"04:19.835 ","End":"04:22.410","Text":"it\u0027s 1 over the cosine."},{"Start":"04:23.200 ","End":"04:26.510","Text":"Whereas the cosine that was given,"},{"Start":"04:26.510 ","End":"04:28.555","Text":"1 over that,"},{"Start":"04:28.555 ","End":"04:34.080","Text":"is minus 2, highlight it."},{"Start":"04:34.080 ","End":"04:37.920","Text":"Finally, for the cotangent,"},{"Start":"04:37.920 ","End":"04:40.505","Text":"there\u0027s actually more than one identity."},{"Start":"04:40.505 ","End":"04:43.895","Text":"It\u0027s equal to cosine over sine,"},{"Start":"04:43.895 ","End":"04:49.615","Text":"but it\u0027s also that the reciprocal identity that the cotangent is 1 over the tangent."},{"Start":"04:49.615 ","End":"04:53.775","Text":"Cotangent of alpha is 1 over tangent of alpha."},{"Start":"04:53.775 ","End":"04:59.100","Text":"That\u0027s going to make it minus 1 over root 3."},{"Start":"04:59.100 ","End":"05:01.200","Text":"That\u0027s the last one,"},{"Start":"05:01.200 ","End":"05:04.030","Text":"and we are done."}],"ID":10726},{"Watched":false,"Name":"Exercise 2 Part c","Duration":"5m 37s","ChapterTopicVideoID":10367,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10367.jpeg","UploadDate":"2017-11-02T14:54:22.1930000","DurationForVideoObject":"PT5M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.765","Text":"In this exercise, we\u0027re given that for some t in the fourth quadrant,"},{"Start":"00:06.765 ","End":"00:15.090","Text":"the tangent is minus 3/5 and we have to figure out the other 5, the 6 altogether."},{"Start":"00:15.090 ","End":"00:18.045","Text":"We have the tangent you want the other 5 trigonometric functions of"},{"Start":"00:18.045 ","End":"00:21.720","Text":"t. Don\u0027t need really a diagram,"},{"Start":"00:21.720 ","End":"00:24.060","Text":"just wanted to remind you what quadrant 4 is."},{"Start":"00:24.060 ","End":"00:31.920","Text":"From these 4, this is quadrant 4 and really it\u0027s not t in quadrant 4 but p of t,"},{"Start":"00:31.920 ","End":"00:41.450","Text":"some point corresponds to the argument t. What we do know is that if it\u0027s x,"},{"Start":"00:41.450 ","End":"00:43.100","Text":"y for some particular x and y,"},{"Start":"00:43.100 ","End":"00:49.715","Text":"we do know that x is positive and y is negative."},{"Start":"00:49.715 ","End":"00:52.580","Text":"This will be useful when we have to take"},{"Start":"00:52.580 ","End":"00:56.225","Text":"square roots and things we won\u0027t have to say plus or minus."},{"Start":"00:56.225 ","End":"00:58.130","Text":"I\u0027m getting ahead of myself."},{"Start":"00:58.130 ","End":"01:00.095","Text":"Let\u0027s begin."},{"Start":"01:00.095 ","End":"01:07.400","Text":"We\u0027ll use the identities rather than the definition to get the other."},{"Start":"01:07.400 ","End":"01:11.270","Text":"First, the easiest one is the cotangent."},{"Start":"01:11.270 ","End":"01:15.230","Text":"Cotangent is the reciprocal identity that it\u0027s 1 over"},{"Start":"01:15.230 ","End":"01:20.700","Text":"tangent and that makes it 1 over minus 3/5."},{"Start":"01:20.700 ","End":"01:24.210","Text":"That\u0027s minus 5 over 3."},{"Start":"01:24.210 ","End":"01:31.730","Text":"Next one will go for the cosecant because there is an identity that"},{"Start":"01:31.730 ","End":"01:40.400","Text":"the cosecant squared is equal to the cotangent squared plus 1."},{"Start":"01:40.400 ","End":"01:44.267","Text":"What we get is the cosecant of"},{"Start":"01:44.267 ","End":"01:52.415","Text":"t is equal to,"},{"Start":"01:52.415 ","End":"01:55.985","Text":"the cotangent we have is minus 5/3,"},{"Start":"01:55.985 ","End":"02:04.770","Text":"so it\u0027s going to be 25 over 9 plus 1 and that is equal to,"},{"Start":"02:04.770 ","End":"02:06.510","Text":"this is 9 over 9,"},{"Start":"02:06.510 ","End":"02:11.290","Text":"so it\u0027s 34 over 9."},{"Start":"02:11.780 ","End":"02:15.760","Text":"Well, that\u0027s the cosecant squared."},{"Start":"02:16.430 ","End":"02:24.095","Text":"The cosecant of t is going to equal,"},{"Start":"02:24.095 ","End":"02:27.890","Text":"I would say normally plus or minus the square root,"},{"Start":"02:27.890 ","End":"02:31.780","Text":"but we know the sine of the cosecant."},{"Start":"02:31.780 ","End":"02:35.785","Text":"Remember that the cosecant in terms of x and y,"},{"Start":"02:35.785 ","End":"02:43.335","Text":"the cosecant of the angle is"},{"Start":"02:43.335 ","End":"02:52.890","Text":"1 over the y here and because y is negative,"},{"Start":"02:52.890 ","End":"02:55.930","Text":"1 over y will also be negative."},{"Start":"02:55.930 ","End":"02:58.840","Text":"We want to take minus the square root of this."},{"Start":"02:58.840 ","End":"03:04.510","Text":"I can take the square root of the numerator and the square root of the denominator."},{"Start":"03:04.510 ","End":"03:07.780","Text":"There\u0027s also a lookup table that tells you in each of"},{"Start":"03:07.780 ","End":"03:11.460","Text":"the quadrants which function is positive and negative and if you look it up,"},{"Start":"03:11.460 ","End":"03:17.210","Text":"cosecant in quadrant 4 has to be negative and that\u0027s why we know to take the minus."},{"Start":"03:17.210 ","End":"03:23.825","Text":"Next, I\u0027m going to use the reciprocal identity that sine of t is 1 over"},{"Start":"03:23.825 ","End":"03:30.830","Text":"cosecant of t. That makes it just the inverse of this,"},{"Start":"03:30.830 ","End":"03:36.595","Text":"so it\u0027s minus 3 over square root of 34."},{"Start":"03:36.595 ","End":"03:43.135","Text":"Next, I\u0027ll go for the cosine and use the Pythagorean identity."},{"Start":"03:43.135 ","End":"03:49.215","Text":"The cosine squared is 1 minus the sine squared."},{"Start":"03:49.215 ","End":"03:50.700","Text":"In this case,"},{"Start":"03:50.700 ","End":"03:57.270","Text":"we know the sine so this will equal 1 minus"},{"Start":"03:57.270 ","End":"04:06.540","Text":"the square of this is 9 over 34."},{"Start":"04:06.540 ","End":"04:16.545","Text":"We get that cosine squared of t is 25 over 34."},{"Start":"04:16.545 ","End":"04:23.510","Text":"That means that the cosine of t is plus or minus the square root of this,"},{"Start":"04:23.510 ","End":"04:25.610","Text":"but in the fourth quadrant,"},{"Start":"04:25.610 ","End":"04:31.415","Text":"cosine is positive because cosine is x and x is positive."},{"Start":"04:31.415 ","End":"04:39.275","Text":"It\u0027s plus square root of this 5 over square root of 34."},{"Start":"04:39.275 ","End":"04:41.014","Text":"Now, which one are we missing?"},{"Start":"04:41.014 ","End":"04:43.370","Text":"We\u0027re missing the tangent."},{"Start":"04:43.370 ","End":"04:49.190","Text":"For the tangent, we can use the reciprocal identity."},{"Start":"04:49.190 ","End":"04:53.190","Text":"It\u0027s 1 over the cotangent of t,"},{"Start":"04:53.190 ","End":"04:58.900","Text":"so 1 over minus 5/3 is minus 3/5."},{"Start":"04:58.900 ","End":"05:00.950","Text":"I haven\u0027t highlighted the solutions,"},{"Start":"05:00.950 ","End":"05:02.300","Text":"but they\u0027re all in there somewhere,"},{"Start":"05:02.300 ","End":"05:03.455","Text":"we\u0027ve got all 5."},{"Start":"05:03.455 ","End":"05:10.700","Text":"Let\u0027s see cotangent, cosecant, secant."},{"Start":"05:10.700 ","End":"05:13.190","Text":"Wait a minute, where is secant?"},{"Start":"05:13.190 ","End":"05:17.410","Text":"My apologies, one missing."},{"Start":"05:17.410 ","End":"05:24.200","Text":"We\u0027ll go for the identity that the secant of t from the reciprocal identity is 1 over"},{"Start":"05:24.200 ","End":"05:32.215","Text":"cosine of t. That would be root 34 over 5."},{"Start":"05:32.215 ","End":"05:37.200","Text":"I believe we\u0027ve covered them all now and we are done."}],"ID":10727},{"Watched":false,"Name":"Exercise 3","Duration":"3m 26s","ChapterTopicVideoID":10368,"CourseChapterTopicPlaylistID":257200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10368.jpeg","UploadDate":"2017-11-02T14:54:35.1030000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this exercise, we\u0027re given that p(t) is a point"},{"Start":"00:04.710 ","End":"00:08.640","Text":"on the unit circle with these coordinates."},{"Start":"00:08.640 ","End":"00:16.950","Text":"We have to evaluate all 6 trigonometric functions of t. We don\u0027t have to have a diagram,"},{"Start":"00:16.950 ","End":"00:18.420","Text":"but I thought it would be helpful."},{"Start":"00:18.420 ","End":"00:19.950","Text":"This is the unit circle,"},{"Start":"00:19.950 ","End":"00:26.230","Text":"so that this is 1 and this is 1 and p(t)."},{"Start":"00:26.230 ","End":"00:28.175","Text":"Now, which quadrant it in?"},{"Start":"00:28.175 ","End":"00:30.890","Text":"You see that this is negative and this is negative."},{"Start":"00:30.890 ","End":"00:33.230","Text":"So x is negative and y is negative."},{"Start":"00:33.230 ","End":"00:34.940","Text":"That means the third quadrant."},{"Start":"00:34.940 ","End":"00:37.740","Text":"This is quadrant number 3."},{"Start":"00:37.780 ","End":"00:40.520","Text":"It doesn\u0027t really matter exactly where,"},{"Start":"00:40.520 ","End":"00:45.260","Text":"but I noticed that the y is bigger than the x and absolute value."},{"Start":"00:45.260 ","End":"00:49.890","Text":"Maybe it\u0027s somewhere, say here."},{"Start":"00:49.930 ","End":"00:58.410","Text":"Let me just write that p(t) is minus 8 over 17,"},{"Start":"00:58.410 ","End":"01:03.705","Text":"minus 15 over 17."},{"Start":"01:03.705 ","End":"01:09.770","Text":"T would be going from here all the way along here up to here,"},{"Start":"01:09.770 ","End":"01:12.485","Text":"that length of arc."},{"Start":"01:12.485 ","End":"01:17.975","Text":"Now, the trigonometric functions as a formula if this is,"},{"Start":"01:17.975 ","End":"01:21.785","Text":"call it the x of the point and the y of the point."},{"Start":"01:21.785 ","End":"01:24.845","Text":"This is just a basic exercise."},{"Start":"01:24.845 ","End":"01:30.875","Text":"We just have to remember that sin(t) is y,"},{"Start":"01:30.875 ","End":"01:39.750","Text":"cos(t) is x. Tan(t) is y over x."},{"Start":"01:39.750 ","End":"01:42.740","Text":"We\u0027ll do the computation in a moment."},{"Start":"01:42.740 ","End":"01:44.525","Text":"I\u0027m just reminding you,"},{"Start":"01:44.525 ","End":"01:47.010","Text":"these you should know by heart."},{"Start":"01:47.020 ","End":"01:53.015","Text":"Csc(t) is 1 over y."},{"Start":"01:53.015 ","End":"01:54.890","Text":"That\u0027s why I usually arrange them."},{"Start":"01:54.890 ","End":"01:58.505","Text":"I get the 3 main ones and then the reciprocals of these."},{"Start":"01:58.505 ","End":"02:02.375","Text":"The next one is the sec(t),"},{"Start":"02:02.375 ","End":"02:05.165","Text":"which is 1 over x."},{"Start":"02:05.165 ","End":"02:09.215","Text":"Then the cot(t),"},{"Start":"02:09.215 ","End":"02:11.165","Text":"which is the reciprocal of this,"},{"Start":"02:11.165 ","End":"02:13.445","Text":"is x over y."},{"Start":"02:13.445 ","End":"02:15.050","Text":"Now we have x,"},{"Start":"02:15.050 ","End":"02:17.090","Text":"and we have y,"},{"Start":"02:17.090 ","End":"02:19.100","Text":"so just plug it in."},{"Start":"02:19.100 ","End":"02:24.560","Text":"Y here is minus 15 over 17 and"},{"Start":"02:24.560 ","End":"02:30.795","Text":"x is minus 8 over 17."},{"Start":"02:30.795 ","End":"02:32.720","Text":"That\u0027s the sine and the cosine."},{"Start":"02:32.720 ","End":"02:35.990","Text":"The tangent is y over x."},{"Start":"02:35.990 ","End":"02:38.150","Text":"Negative over negative is positive."},{"Start":"02:38.150 ","End":"02:46.350","Text":"The 17s cancel and so we get 15 over 8."},{"Start":"02:46.350 ","End":"02:49.640","Text":"The cosecant is 1 over y,"},{"Start":"02:49.640 ","End":"02:52.940","Text":"so just flip it upside down."},{"Start":"02:52.940 ","End":"02:57.200","Text":"Got minus 17 over 15."},{"Start":"02:57.200 ","End":"03:03.110","Text":"The secant is the reciprocal of this,"},{"Start":"03:03.110 ","End":"03:08.175","Text":"which would be minus 17 over 8."},{"Start":"03:08.175 ","End":"03:11.510","Text":"The cotangent is 1 over the tangent,"},{"Start":"03:11.510 ","End":"03:14.120","Text":"which also x over y,"},{"Start":"03:14.120 ","End":"03:17.480","Text":"negative and negative give us positive."},{"Start":"03:17.480 ","End":"03:22.920","Text":"That would be 8 over 15."},{"Start":"03:22.920 ","End":"03:27.070","Text":"That\u0027s all there is to it. We are done."}],"ID":10728}],"Thumbnail":null,"ID":257200},{"Name":"Graphs of Trigonometric Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Graphs of Trigonometric Functions - Part 1","Duration":"7m 52s","ChapterTopicVideoID":10420,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10420.jpeg","UploadDate":"2021-06-29T12:36:07.4570000","DurationForVideoObject":"PT7M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this clip, we\u0027ll be talking about"},{"Start":"00:02.640 ","End":"00:07.725","Text":"the graphs of trigonometric functions and how to sketch them."},{"Start":"00:07.725 ","End":"00:13.770","Text":"We\u0027ll start with just 2 of the 6 trigonometric functions;"},{"Start":"00:13.770 ","End":"00:15.465","Text":"the sine and the cosine,"},{"Start":"00:15.465 ","End":"00:18.180","Text":"and their curves or graphs."},{"Start":"00:18.180 ","End":"00:24.100","Text":"I\u0027ll also talk about the basic sine and cosine curves."},{"Start":"00:24.560 ","End":"00:29.340","Text":"Here are the 2 graphs."},{"Start":"00:29.340 ","End":"00:39.375","Text":"This one is the graph of y equals sine x. I didn\u0027t label the axes because"},{"Start":"00:39.375 ","End":"00:43.520","Text":"some people would say u equals"},{"Start":"00:43.520 ","End":"00:50.247","Text":"sine t and other letters might be used in any event."},{"Start":"00:50.247 ","End":"00:52.070","Text":"This would be in our case,"},{"Start":"00:52.070 ","End":"00:55.040","Text":"the y-axis and the x-axis."},{"Start":"00:55.040 ","End":"01:01.475","Text":"The second graph is the graph of y equals cosine x."},{"Start":"01:01.475 ","End":"01:03.740","Text":"Both of these were done by computer."},{"Start":"01:03.740 ","End":"01:07.160","Text":"Of course, they extend infinitely in both directions,"},{"Start":"01:07.160 ","End":"01:11.015","Text":"but we only have enough room for a finite portion."},{"Start":"01:11.015 ","End":"01:14.240","Text":"Now both these functions are defined on"},{"Start":"01:14.240 ","End":"01:18.740","Text":"all real numbers so I can say that the domain is all real numbers,"},{"Start":"01:18.740 ","End":"01:26.285","Text":"which we indicate by this R. The range,"},{"Start":"01:26.285 ","End":"01:33.045","Text":"the set of values that sine and cosine can take is from minus 1 to 1."},{"Start":"01:33.045 ","End":"01:36.335","Text":"I would write it as the interval from minus 1 to 1,"},{"Start":"01:36.335 ","End":"01:40.340","Text":"square brackets mean including the 1 and the minus 1."},{"Start":"01:40.340 ","End":"01:47.495","Text":"Another important parameter is the period."},{"Start":"01:47.495 ","End":"01:50.510","Text":"We already discussed this earlier."},{"Start":"01:50.510 ","End":"01:54.035","Text":"The period is 2Pi."},{"Start":"01:54.035 ","End":"01:57.155","Text":"Every time we move along 2Pi,"},{"Start":"01:57.155 ","End":"02:00.190","Text":"the curve repeats itself."},{"Start":"02:00.190 ","End":"02:07.520","Text":"There is a maximum and minimum value which you can get from the range."},{"Start":"02:07.520 ","End":"02:12.995","Text":"The max is 1 and the minimum is minus 1."},{"Start":"02:12.995 ","End":"02:19.980","Text":"From these two, we extract another quantity called the amplitude."},{"Start":"02:20.260 ","End":"02:26.975","Text":"The amplitude is the maximum minus the minimum over 2,"},{"Start":"02:26.975 ","End":"02:29.940","Text":"so that is equal to 1."},{"Start":"02:30.770 ","End":"02:33.630","Text":"It\u0027s 1 minus,"},{"Start":"02:33.630 ","End":"02:37.045","Text":"minus 1 over 2."},{"Start":"02:37.045 ","End":"02:40.470","Text":"Maximum minus minimum 1/2."},{"Start":"02:40.600 ","End":"02:45.229","Text":"Exactly the same for the cosine."},{"Start":"02:45.229 ","End":"02:47.720","Text":"I\u0027ll just write as above,"},{"Start":"02:47.720 ","End":"02:52.430","Text":"I could do a copy-paste but let me just leave it like that."},{"Start":"02:52.430 ","End":"03:01.250","Text":"Notice that the cosine can be thought of as the sine just shifted left by Pi over 2."},{"Start":"03:01.250 ","End":"03:08.895","Text":"This here would be Pi over 2 and this would be Pi and so on."},{"Start":"03:08.895 ","End":"03:12.090","Text":"If you just shifted left, you\u0027d get the cosine."},{"Start":"03:12.130 ","End":"03:15.770","Text":"Now come to the term basic."},{"Start":"03:15.770 ","End":"03:23.255","Text":"The basic sine and cosine curves is just part of the sine or cosine one period\u0027s worth,"},{"Start":"03:23.255 ","End":"03:26.420","Text":"but specifically from zero to 2Pi."},{"Start":"03:26.420 ","End":"03:28.440","Text":"I\u0027ve shaded the part here."},{"Start":"03:28.440 ","End":"03:35.765","Text":"This part is the basic sine curve and from here to here is the basic cosine curve."},{"Start":"03:35.765 ","End":"03:38.885","Text":"I\u0027d just like to add some points here."},{"Start":"03:38.885 ","End":"03:43.370","Text":"Usually, we break up the basic curve into"},{"Start":"03:43.370 ","End":"03:49.920","Text":"4 pieces for the purposes of sketching and we\u0027ll see this later on."},{"Start":"03:50.740 ","End":"03:53.745","Text":"A copy of it to here also."},{"Start":"03:53.745 ","End":"04:00.515","Text":"The difference is with the basic form is that for basic,"},{"Start":"04:00.515 ","End":"04:06.454","Text":"then the domain is not all of the real numbers,"},{"Start":"04:06.454 ","End":"04:10.040","Text":"it\u0027s just from 0 to 2Pi."},{"Start":"04:10.040 ","End":"04:11.735","Text":"Again, for both of these,"},{"Start":"04:11.735 ","End":"04:14.735","Text":"the sine and the cosine."},{"Start":"04:14.735 ","End":"04:19.820","Text":"Next, we\u0027ll talk about a couple of other concepts which generalize this a bit."},{"Start":"04:19.820 ","End":"04:28.480","Text":"We\u0027ll talk about standard sine and cosine curves and upside-down sine and cosine curves."},{"Start":"04:28.480 ","End":"04:33.215","Text":"These will be the graphs of instead of y equals sine x,"},{"Start":"04:33.215 ","End":"04:38.450","Text":"y equals some number A times sine x."},{"Start":"04:38.450 ","End":"04:47.120","Text":"We\u0027ll also have y equals A cosine x."},{"Start":"04:47.120 ","End":"04:53.418","Text":"Now the amplitude turns out to be the absolute value of A,"},{"Start":"04:53.418 ","End":"05:01.645","Text":"so let me write that."},{"Start":"05:01.645 ","End":"05:05.025","Text":"If A is bigger than 0,"},{"Start":"05:05.025 ","End":"05:07.610","Text":"I should have said it should not be 0."},{"Start":"05:07.610 ","End":"05:09.080","Text":"Otherwise, it\u0027s not interesting."},{"Start":"05:09.080 ","End":"05:10.810","Text":"It\u0027s just the 0 function."},{"Start":"05:10.810 ","End":"05:12.910","Text":"If A is bigger than 0,"},{"Start":"05:12.910 ","End":"05:22.420","Text":"then it\u0027s called a standard sine curve or a cosine curve."},{"Start":"05:22.420 ","End":"05:25.915","Text":"If A is less than 0,"},{"Start":"05:25.915 ","End":"05:32.740","Text":"then it\u0027s considered to be an upside-down sine curve or cosine curve."},{"Start":"05:32.740 ","End":"05:36.970","Text":"I\u0027ll give a couple of examples and that will make it clearer."},{"Start":"05:36.970 ","End":"05:44.140","Text":"For example, let\u0027s take y equals and that A be 2 and will take"},{"Start":"05:44.140 ","End":"05:47.980","Text":"the cosine 2 cosine x so that"},{"Start":"05:47.980 ","End":"05:55.720","Text":"the amplitude is equal to absolute value of 2, which is 2."},{"Start":"05:55.720 ","End":"06:00.160","Text":"Here\u0027s the sketch to a different scale,"},{"Start":"06:00.160 ","End":"06:03.380","Text":"but it\u0027s just like the previous one but stretched"},{"Start":"06:03.380 ","End":"06:07.515","Text":"upwards by a factor of 2 so that the peak is 2,"},{"Start":"06:07.515 ","End":"06:10.485","Text":"maximum is 2, the minimum is minus 2,"},{"Start":"06:10.485 ","End":"06:13.350","Text":"and the amplitude is 2."},{"Start":"06:13.350 ","End":"06:17.040","Text":"Still the same period of 2Pi."},{"Start":"06:17.040 ","End":"06:21.900","Text":"Because this a which is 2 is bigger than 0,"},{"Start":"06:21.900 ","End":"06:28.505","Text":"then it say standard cosine curve."},{"Start":"06:28.505 ","End":"06:31.640","Text":"Now I\u0027ll give an example of an upside-down."},{"Start":"06:31.640 ","End":"06:41.970","Text":"The upside-down one will take y equals minus 3 sine x."},{"Start":"06:41.970 ","End":"06:51.020","Text":"Here the amplitude is going to equal absolute value of minus 3, which is 3."},{"Start":"06:51.020 ","End":"06:55.520","Text":"I guess also the maximum and minimum change as above,"},{"Start":"06:55.520 ","End":"06:56.690","Text":"maximum is 3;"},{"Start":"06:56.690 ","End":"06:58.865","Text":"minimum is minus 3 anyway,"},{"Start":"06:58.865 ","End":"07:01.885","Text":"and the period is still 2Pi."},{"Start":"07:01.885 ","End":"07:05.550","Text":"But because the minus 3,"},{"Start":"07:05.550 ","End":"07:08.245","Text":"which is this A is less than 0,"},{"Start":"07:08.245 ","End":"07:13.870","Text":"then it\u0027s going to be an upside-down sine curve."},{"Start":"07:13.870 ","End":"07:16.510","Text":"I\u0027ll bring you the picture."},{"Start":"07:16.510 ","End":"07:19.280","Text":"Here\u0027s what it looks like."},{"Start":"07:19.280 ","End":"07:21.320","Text":"The amplitude is 3."},{"Start":"07:21.320 ","End":"07:26.255","Text":"It goes from 3 to minus 3 or vice versa."},{"Start":"07:26.255 ","End":"07:29.315","Text":"But notice that it\u0027s upside down."},{"Start":"07:29.315 ","End":"07:32.855","Text":"It\u0027s like reflected in the x-axis."},{"Start":"07:32.855 ","End":"07:39.070","Text":"Because usually from the 0 we start going up and then down here and then to here."},{"Start":"07:39.070 ","End":"07:41.030","Text":"This one is upside down."},{"Start":"07:41.030 ","End":"07:43.445","Text":"Just take a look at the other one."},{"Start":"07:43.445 ","End":"07:50.360","Text":"We\u0027ll take a break here and then we\u0027ll continue with the other trigonometric functions."},{"Start":"07:50.360 ","End":"07:53.520","Text":"There are 6 of them and we\u0027ve just covered 2."}],"ID":10891},{"Watched":false,"Name":"Graphs of Trigonometric Functions - Part 2","Duration":"8m 30s","ChapterTopicVideoID":10474,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10474.jpeg","UploadDate":"2021-06-29T12:36:40.2770000","DurationForVideoObject":"PT8M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"Next we want to generalize this."},{"Start":"00:02.430 ","End":"00:04.710","Text":"If we just take sine x and cosine x,"},{"Start":"00:04.710 ","End":"00:07.214","Text":"not very interesting is just 2 functions."},{"Start":"00:07.214 ","End":"00:09.210","Text":"You want to modify these,"},{"Start":"00:09.210 ","End":"00:13.650","Text":"essentially what we\u0027re going to do is stretch them vertically and horizontally and"},{"Start":"00:13.650 ","End":"00:18.180","Text":"shift them up and down and still retain the basic sine shape."},{"Start":"00:18.180 ","End":"00:21.720","Text":"What we\u0027re going to do is generalization and we\u0027ll"},{"Start":"00:21.720 ","End":"00:25.650","Text":"get standard and upside down sine and cosine curves."},{"Start":"00:25.650 ","End":"00:28.755","Text":"I\u0027m not going to go for the full generality right away."},{"Start":"00:28.755 ","End":"00:36.375","Text":"We\u0027re going to talk about y equals some number A times sine x."},{"Start":"00:36.375 ","End":"00:44.120","Text":"Similarly, the function A cosine x and assume A is not 0,"},{"Start":"00:44.120 ","End":"00:46.205","Text":"otherwise it\u0027s not interesting."},{"Start":"00:46.205 ","End":"00:52.400","Text":"It turns out that when we have A positive,"},{"Start":"00:52.400 ","End":"00:55.985","Text":"then we get a standard curve,"},{"Start":"00:55.985 ","End":"01:02.275","Text":"this will be either a sine or cosine curved."},{"Start":"01:02.275 ","End":"01:05.385","Text":"But when A is negative,"},{"Start":"01:05.385 ","End":"01:08.010","Text":"then it\u0027s not standard,"},{"Start":"01:08.010 ","End":"01:11.990","Text":"it\u0027s called the upside down."},{"Start":"01:11.990 ","End":"01:16.070","Text":"Again, sine or cosine as appropriate."},{"Start":"01:16.070 ","End":"01:18.860","Text":"Let me just give you a sneak preview of where we\u0027re heading."},{"Start":"01:18.860 ","End":"01:20.420","Text":"I just didn\u0027t want to do it all at once."},{"Start":"01:20.420 ","End":"01:23.960","Text":"In general, we\u0027re going to head for 4 constants."},{"Start":"01:23.960 ","End":"01:26.480","Text":"We\u0027re going to have A sine of Bx minus C plus"},{"Start":"01:26.480 ","End":"01:35.585","Text":"D. Similarly for cosine, same applies here."},{"Start":"01:35.585 ","End":"01:37.760","Text":"When A is bigger than 0,"},{"Start":"01:37.760 ","End":"01:43.205","Text":"it will be a standard sine or cosine curve and A less than 0,"},{"Start":"01:43.205 ","End":"01:45.094","Text":"it\u0027s going to be upside down."},{"Start":"01:45.094 ","End":"01:46.910","Text":"That\u0027s just a sneak preview."},{"Start":"01:46.910 ","End":"01:53.505","Text":"I almost forgot to say instead of the amplitude being 1,"},{"Start":"01:53.505 ","End":"01:55.305","Text":"the amplitude is A."},{"Start":"01:55.305 ","End":"01:57.845","Text":"Not quite because if A is negative,"},{"Start":"01:57.845 ","End":"02:00.020","Text":"we need to make it positive."},{"Start":"02:00.020 ","End":"02:02.600","Text":"The effective and negative A makes"},{"Start":"02:02.600 ","End":"02:06.585","Text":"it what we call upside down and you\u0027ll see when you see the picture."},{"Start":"02:06.585 ","End":"02:13.005","Text":"The example I\u0027m going to start off with is y=2cosine x,"},{"Start":"02:13.005 ","End":"02:15.340","Text":"which is this, where A is 2,"},{"Start":"02:15.340 ","End":"02:20.475","Text":"it\u0027s positive, so it\u0027s a standard cosine curve."},{"Start":"02:20.475 ","End":"02:25.840","Text":"The amplitude will be equal to 2."},{"Start":"02:25.840 ","End":"02:30.840","Text":"What it will look like is this,"},{"Start":"02:30.840 ","End":"02:35.270","Text":"very similar to the original y=cosine x."},{"Start":"02:35.270 ","End":"02:39.650","Text":"But the 2 here means that there\u0027s a stretch by a factor of 2."},{"Start":"02:39.650 ","End":"02:41.575","Text":"Instead of going from minus 1 to 1,"},{"Start":"02:41.575 ","End":"02:43.715","Text":"it goes from minus 2 to 2."},{"Start":"02:43.715 ","End":"02:51.380","Text":"We\u0027ll go to the full generalization and I\u0027ll do it on a new page."},{"Start":"02:51.380 ","End":"02:59.260","Text":"To remind you, we\u0027re now going to generalize to y=A sin,"},{"Start":"02:59.260 ","End":"03:07.100","Text":"not just x, but Bx it could be plus C or it could be minus C. Some professors,"},{"Start":"03:07.100 ","End":"03:09.605","Text":"some books do it 1 way, some the other."},{"Start":"03:09.605 ","End":"03:14.870","Text":"In 1 case the C will come out minus to compensate for the minus."},{"Start":"03:14.870 ","End":"03:23.600","Text":"Anyway we\u0027ll take it as a minus and then plus another constant D. Likewise,"},{"Start":"03:23.600 ","End":"03:27.380","Text":"we\u0027ll also talk about the cosine."},{"Start":"03:27.380 ","End":"03:34.100","Text":"Just to remind you that when A is bigger than 0,"},{"Start":"03:34.100 ","End":"03:42.900","Text":"then it\u0027s called a basic sine or cosine curve."},{"Start":"03:42.900 ","End":"03:46.160","Text":"When A is less than 0,"},{"Start":"03:46.160 ","End":"03:54.470","Text":"then we call it upside down sine curve or cosine curve,"},{"Start":"03:54.470 ","End":"03:56.730","Text":"as the case may be."},{"Start":"03:57.050 ","End":"04:01.385","Text":"I\u0027ll explain what the meaning of these constants is."},{"Start":"04:01.385 ","End":"04:05.165","Text":"Talk about each of the constants separately, I\u0027ll say something."},{"Start":"04:05.165 ","End":"04:07.565","Text":"A, we\u0027ve already mentioned,"},{"Start":"04:07.565 ","End":"04:15.340","Text":"causes a vertical stretch that\u0027s informally."},{"Start":"04:15.370 ","End":"04:17.895","Text":"But I think you know what I mean,"},{"Start":"04:17.895 ","End":"04:25.522","Text":"like a change of scale vertically and the amplitude changes,"},{"Start":"04:25.522 ","End":"04:29.840","Text":"instead of 1, it becomes A."},{"Start":"04:29.840 ","End":"04:31.295","Text":"Well, not quite,"},{"Start":"04:31.295 ","End":"04:33.065","Text":"absolute value of A."},{"Start":"04:33.065 ","End":"04:35.570","Text":"But if it\u0027s negative A,"},{"Start":"04:35.570 ","End":"04:38.675","Text":"it\u0027s an upside down."},{"Start":"04:38.675 ","End":"04:40.130","Text":"When you see the first picture,"},{"Start":"04:40.130 ","End":"04:42.125","Text":"you\u0027ll know what I mean by upside down."},{"Start":"04:42.125 ","End":"04:50.575","Text":"B causes a horizontal stretch,"},{"Start":"04:50.575 ","End":"04:56.360","Text":"a compression, sometimes we say."},{"Start":"04:56.360 ","End":"05:03.185","Text":"This horizontal compression or compression in the x-direction changes the period."},{"Start":"05:03.185 ","End":"05:08.710","Text":"The period, which previously was 2Pi,"},{"Start":"05:08.710 ","End":"05:11.750","Text":"is now divided by,"},{"Start":"05:11.750 ","End":"05:14.640","Text":"because of the horizontal compression,"},{"Start":"05:14.640 ","End":"05:18.965","Text":"factor B, the period is now 2Pi over B."},{"Start":"05:18.965 ","End":"05:21.625","Text":"Because a regular curve sine x,"},{"Start":"05:21.625 ","End":"05:23.550","Text":"B is implicitly 1,"},{"Start":"05:23.550 ","End":"05:25.800","Text":"so it doesn\u0027t change the period."},{"Start":"05:25.800 ","End":"05:33.095","Text":"I just forgot to add that in this case we can always assume that B is positive."},{"Start":"05:33.095 ","End":"05:37.980","Text":"If it\u0027s 0, it\u0027s not a sine function at all."},{"Start":"05:38.060 ","End":"05:41.279","Text":"What it is, it\u0027s a constant."},{"Start":"05:41.279 ","End":"05:43.070","Text":"If B is negative,"},{"Start":"05:43.070 ","End":"05:46.880","Text":"we can always use the formula for sine of minus an angle and"},{"Start":"05:46.880 ","End":"05:50.840","Text":"the cosine of minus an angle to get B to be positive,"},{"Start":"05:50.840 ","End":"05:53.465","Text":"so the period will be positive."},{"Start":"05:53.465 ","End":"05:55.910","Text":"Now on to the next constant,"},{"Start":"05:55.910 ","End":"05:58.600","Text":"C. C"},{"Start":"05:58.600 ","End":"06:06.590","Text":"affects the horizontal placement it shifted,"},{"Start":"06:06.590 ","End":"06:09.120","Text":"but not by the amount C,"},{"Start":"06:09.120 ","End":"06:13.520","Text":"so C produces a phase shift."},{"Start":"06:13.520 ","End":"06:20.345","Text":"The amount of the phase shift is actually C over B."},{"Start":"06:20.345 ","End":"06:24.215","Text":"Well, its absolute value of C over B."},{"Start":"06:24.215 ","End":"06:30.200","Text":"It\u0027s in this direction or this direction."},{"Start":"06:30.200 ","End":"06:34.280","Text":"The right phase shift is for when C is positive and"},{"Start":"06:34.280 ","End":"06:41.285","Text":"the left phase shift is when C is negative and this is the amount it shifted by."},{"Start":"06:41.285 ","End":"06:44.525","Text":"The last constant, D,"},{"Start":"06:44.525 ","End":"06:49.070","Text":"affects the up and down movement."},{"Start":"06:49.070 ","End":"06:54.530","Text":"I guess I\u0027ll call it a vertical shift."},{"Start":"06:54.530 ","End":"07:00.305","Text":"The vertical shift is always by the absolute value of D,"},{"Start":"07:00.305 ","End":"07:09.500","Text":"but it\u0027s upwards if D is positive and it\u0027s downwards,"},{"Start":"07:09.500 ","End":"07:11.480","Text":"if D is negative,"},{"Start":"07:11.480 ","End":"07:20.420","Text":"but the amount of shift is absolute value of D. Let\u0027s take the example"},{"Start":"07:20.420 ","End":"07:23.491","Text":"y=minus"},{"Start":"07:23.491 ","End":"07:30.950","Text":"3 sine of 2x."},{"Start":"07:30.950 ","End":"07:34.550","Text":"Let\u0027s compute some of the parameters for this."},{"Start":"07:34.550 ","End":"07:41.215","Text":"The amplitude is absolute value of A, which is 3."},{"Start":"07:41.215 ","End":"07:44.505","Text":"This is A and this is B."},{"Start":"07:44.505 ","End":"07:49.406","Text":"The period is 2Pi over B,"},{"Start":"07:49.406 ","End":"07:51.045","Text":"it\u0027s still up here,"},{"Start":"07:51.045 ","End":"07:55.960","Text":"is 2Pi over 2, which is Pi."},{"Start":"07:55.960 ","End":"07:58.610","Text":"There is no shift,"},{"Start":"07:58.610 ","End":"08:02.705","Text":"either phase shift or vertical shift."},{"Start":"08:02.705 ","End":"08:06.160","Text":"The sketch of it, here it is."},{"Start":"08:06.160 ","End":"08:10.880","Text":"I should have said it\u0027s because of the negativity of A,"},{"Start":"08:10.880 ","End":"08:15.260","Text":"that this one is an upside down sine curve."},{"Start":"08:15.260 ","End":"08:23.285","Text":"Notice that the amplitude being 3 means that it goes up to 3 and down to minus 3."},{"Start":"08:23.285 ","End":"08:25.370","Text":"Notice that the period is Pi,"},{"Start":"08:25.370 ","End":"08:30.180","Text":"1 complete cycle ends at Pi."}],"ID":10892},{"Watched":false,"Name":"Graphs of Trigonometric Functions - Part 3","Duration":"12m ","ChapterTopicVideoID":10472,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10472.jpeg","UploadDate":"2021-06-29T12:37:30.8270000","DurationForVideoObject":"PT12M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.910","Text":"Now we\u0027re still heading for the goal of the more general y equals"},{"Start":"00:05.910 ","End":"00:13.950","Text":"a sine of bx minus c plus d. Similarly,"},{"Start":"00:13.950 ","End":"00:16.650","Text":"for cosine that\u0027s just taken one at a time,"},{"Start":"00:16.650 ","End":"00:23.490","Text":"I\u0027d prefer to switch from y and x to u and t. You\u0027ll see why in a moment."},{"Start":"00:23.490 ","End":"00:28.995","Text":"But I wanted to go back once again to the simple case of u equals"},{"Start":"00:28.995 ","End":"00:36.085","Text":"sine t. Go into this a bit deeper and then go to the general case."},{"Start":"00:36.085 ","End":"00:42.804","Text":"We\u0027ve already seen earlier that if we want to draw just the basic part,"},{"Start":"00:42.804 ","End":"00:48.410","Text":"1 period from 0 to 2 Pi then it looks like this but this was computer sketched."},{"Start":"00:48.410 ","End":"00:51.860","Text":"I\u0027m going to try and show you how you might do this if"},{"Start":"00:51.860 ","End":"00:55.460","Text":"you had to freehand sketch it may be on a test."},{"Start":"00:55.460 ","End":"00:58.100","Text":"How would we go about it?"},{"Start":"00:58.100 ","End":"01:00.710","Text":"Let me remove the graph."},{"Start":"01:00.710 ","End":"01:05.540","Text":"Here, I borrowed the picture from one of the previous clips,"},{"Start":"01:05.540 ","End":"01:11.090","Text":"which explained how we find sine and cosine and all the other functions."},{"Start":"01:11.090 ","End":"01:15.455","Text":"Remember first we find a point p on the circle."},{"Start":"01:15.455 ","End":"01:22.355","Text":"From 0-2 Pi will mean going counterclockwise until we get back to the same point."},{"Start":"01:22.355 ","End":"01:26.150","Text":"For each point, we have the sine and"},{"Start":"01:26.150 ","End":"01:29.900","Text":"cosine and all the other is defined in terms of the x and y here."},{"Start":"01:29.900 ","End":"01:37.955","Text":"In particular, the sine of t is equal to y."},{"Start":"01:37.955 ","End":"01:41.465","Text":"The cosine of t is x."},{"Start":"01:41.465 ","End":"01:43.055","Text":"We\u0027re concerned with the sine."},{"Start":"01:43.055 ","End":"01:46.140","Text":"After we\u0027ve done the sine we go back to the cosine."},{"Start":"01:46.570 ","End":"01:49.820","Text":"If we take a look at what happens to y as"},{"Start":"01:49.820 ","End":"01:52.490","Text":"we go around the circle will be able to get the graph."},{"Start":"01:52.490 ","End":"01:56.000","Text":"Now notice that when we go from here all the way around,"},{"Start":"01:56.000 ","End":"02:01.310","Text":"there are 3 intermediary stations or if you like,"},{"Start":"02:01.310 ","End":"02:05.945","Text":"we\u0027ve divided the journey up into 4 legs of the journey."},{"Start":"02:05.945 ","End":"02:08.390","Text":"We\u0027ll do that first of all here."},{"Start":"02:08.390 ","End":"02:10.115","Text":"If I divide it up, well,"},{"Start":"02:10.115 ","End":"02:13.010","Text":"I have from 0-2 Pi."},{"Start":"02:13.010 ","End":"02:15.995","Text":"Then halfway it\u0027s going to be Pi,"},{"Start":"02:15.995 ","End":"02:21.910","Text":"and then halfway is going to be Pi over 2 and 3 Pi over 2."},{"Start":"02:21.910 ","End":"02:25.204","Text":"I\u0027m just guessing I\u0027m not measuring exactly."},{"Start":"02:25.204 ","End":"02:30.725","Text":"Then we know that we go from minus 1-1."},{"Start":"02:30.725 ","End":"02:33.515","Text":"If we look at the y it goes from minus 1-1."},{"Start":"02:33.515 ","End":"02:37.955","Text":"The most we can go is 1 and the least is minus 1."},{"Start":"02:37.955 ","End":"02:42.380","Text":"It\u0027s also important to put it in the midpoint of these two."},{"Start":"02:42.380 ","End":"02:45.815","Text":"Let\u0027s say here, this would be like the maximum,"},{"Start":"02:45.815 ","End":"02:48.335","Text":"the minimum, and the middle value."},{"Start":"02:48.335 ","End":"02:50.630","Text":"Now once we have these,"},{"Start":"02:50.630 ","End":"02:52.835","Text":"what you do is this."},{"Start":"02:52.835 ","End":"02:57.230","Text":"When we look here at 0, we have a 0."},{"Start":"02:57.230 ","End":"03:00.170","Text":"Then when we go to Pi over 2,"},{"Start":"03:00.170 ","End":"03:04.625","Text":"we are here so the y is equal to 1 somewhere here."},{"Start":"03:04.625 ","End":"03:06.770","Text":"Then when we\u0027re up to Pi,"},{"Start":"03:06.770 ","End":"03:11.365","Text":"we\u0027re here and the y is again 0."},{"Start":"03:11.365 ","End":"03:14.185","Text":"Looking at the y here."},{"Start":"03:14.185 ","End":"03:17.240","Text":"Then at 3 Pi over 2,"},{"Start":"03:17.240 ","End":"03:21.314","Text":"y is minus 1 which is here."},{"Start":"03:21.314 ","End":"03:24.005","Text":"Back here at 2 Pi,"},{"Start":"03:24.005 ","End":"03:28.565","Text":"at the starting point, it\u0027s the same as the starting point here where it\u0027s 0."},{"Start":"03:28.565 ","End":"03:31.700","Text":"In general, when we do this,"},{"Start":"03:31.700 ","End":"03:41.510","Text":"full-blown case will also be drawing 5 points for intervals and getting some points."},{"Start":"03:41.510 ","End":"03:46.010","Text":"Then we draw a smooth line through these,"},{"Start":"03:46.010 ","End":"03:47.450","Text":"not a straight line,"},{"Start":"03:47.450 ","End":"03:50.600","Text":"don\u0027t make it a zigzag. We know it."},{"Start":"03:50.600 ","End":"03:53.915","Text":"One thing you know is that it has a hump here."},{"Start":"03:53.915 ","End":"03:55.400","Text":"We go like this,"},{"Start":"03:55.400 ","End":"04:01.460","Text":"maybe it down here and then through here, and then here."},{"Start":"04:01.460 ","End":"04:07.400","Text":"You\u0027re not expected to get it very accurate without a computer but something like this."},{"Start":"04:07.400 ","End":"04:09.590","Text":"Then we\u0027ve got a graph."},{"Start":"04:09.590 ","End":"04:14.165","Text":"This is the standard sine curve as opposed to the upside-down."},{"Start":"04:14.165 ","End":"04:19.550","Text":"If we had a minus here then we would do the similar process,"},{"Start":"04:19.550 ","End":"04:22.760","Text":"except that we do the mirror image."},{"Start":"04:22.760 ","End":"04:23.960","Text":"We\u0027d get this point,"},{"Start":"04:23.960 ","End":"04:27.115","Text":"and then we\u0027d get this point, and this point."},{"Start":"04:27.115 ","End":"04:31.550","Text":"Get minus the value of whatever it is,"},{"Start":"04:31.550 ","End":"04:34.250","Text":"which means reflection in the x-axis."},{"Start":"04:34.250 ","End":"04:38.510","Text":"Then we would sketch something like this,"},{"Start":"04:38.510 ","End":"04:44.720","Text":"and like this, and this is your typical upside-down sine curve."},{"Start":"04:44.720 ","End":"04:46.925","Text":"If I quickly summarize,"},{"Start":"04:46.925 ","End":"04:52.760","Text":"what we need is the domain which we had from 0-2 Pi in this case."},{"Start":"04:52.760 ","End":"04:56.915","Text":"We broke it up into 4 Pieces and put points."},{"Start":"04:56.915 ","End":"04:58.370","Text":"The halfway, the quarter,"},{"Start":"04:58.370 ","End":"04:59.750","Text":"and the 3/4 mark."},{"Start":"04:59.750 ","End":"05:03.975","Text":"We also knew the maximum and the minimum."},{"Start":"05:03.975 ","End":"05:07.670","Text":"The average of these would be the middle."},{"Start":"05:07.670 ","End":"05:12.480","Text":"At each of these 5 points,"},{"Start":"05:12.480 ","End":"05:16.700","Text":"we took either the middle or the maximum or"},{"Start":"05:16.700 ","End":"05:23.225","Text":"the minimum according to as to whether it was standard or the upside-down."},{"Start":"05:23.225 ","End":"05:26.240","Text":"Similarly, we\u0027ll get to the cosine."},{"Start":"05:26.240 ","End":"05:34.110","Text":"Just for emphasis, let me draw a little box around this."},{"Start":"05:34.360 ","End":"05:40.850","Text":"The bounds of this box on the horizontal scale,"},{"Start":"05:40.850 ","End":"05:43.280","Text":"it\u0027s from 0-2 Pi."},{"Start":"05:43.280 ","End":"05:50.825","Text":"This we divide into 4 Pieces,"},{"Start":"05:50.825 ","End":"05:52.795","Text":"whatever you call them."},{"Start":"05:52.795 ","End":"05:58.615","Text":"We also know that we had a maximum, which was 1."},{"Start":"05:58.615 ","End":"06:01.770","Text":"A minimum was minus 1."},{"Start":"06:01.770 ","End":"06:05.330","Text":"Also, our interest was the middle I\u0027ll call it mid,"},{"Start":"06:05.330 ","End":"06:07.790","Text":"which was equal to 0."},{"Start":"06:07.790 ","End":"06:14.210","Text":"Now I want to generalize this to the case where we have a, b,"},{"Start":"06:14.210 ","End":"06:16.655","Text":"c, and d. In this case,"},{"Start":"06:16.655 ","End":"06:21.169","Text":"we start from the phase shift."},{"Start":"06:21.169 ","End":"06:25.129","Text":"The phase shift is c over b."},{"Start":"06:25.129 ","End":"06:28.325","Text":"Then we have to add 1 period."},{"Start":"06:28.325 ","End":"06:31.535","Text":"The period is 2 Pi over b."},{"Start":"06:31.535 ","End":"06:34.030","Text":"If I add 2 Pi over b,"},{"Start":"06:34.030 ","End":"06:40.750","Text":"it will be C plus 2 Pi over b because it\u0027s all over b."},{"Start":"06:40.900 ","End":"06:46.320","Text":"Once again, we would divide it into 4 bits."},{"Start":"06:46.550 ","End":"06:49.310","Text":"The middle is d,"},{"Start":"06:49.310 ","End":"06:50.720","Text":"that\u0027s the vertical shift,"},{"Start":"06:50.720 ","End":"06:52.460","Text":"that\u0027s where the 0 goes up to."},{"Start":"06:52.460 ","End":"07:00.530","Text":"The maximum is d plus the amplitude which is absolute value of a."},{"Start":"07:00.530 ","End":"07:06.845","Text":"The minimum is d minus the absolute value of a."},{"Start":"07:06.845 ","End":"07:10.790","Text":"Once we have these numbers in any given case,"},{"Start":"07:10.790 ","End":"07:15.770","Text":"then we can do the same thing here."},{"Start":"07:15.770 ","End":"07:20.045","Text":"When you are going to draw this,"},{"Start":"07:20.045 ","End":"07:26.690","Text":"I did this not by hand but you might mark a rectangle on the graph paper,"},{"Start":"07:26.690 ","End":"07:30.574","Text":"where here it would be c over b."},{"Start":"07:30.574 ","End":"07:34.940","Text":"Here, c plus 2 Pi over b."},{"Start":"07:34.940 ","End":"07:36.490","Text":"Now you would be given a, b, c,"},{"Start":"07:36.490 ","End":"07:38.360","Text":"and d. You could compute these."},{"Start":"07:38.360 ","End":"07:41.660","Text":"Then you would compute the halfway mark of"},{"Start":"07:41.660 ","End":"07:44.840","Text":"these and maybe also split the difference again,"},{"Start":"07:44.840 ","End":"07:46.774","Text":"you\u0027d get these points."},{"Start":"07:46.774 ","End":"07:48.590","Text":"Then you would compute,"},{"Start":"07:48.590 ","End":"07:53.150","Text":"maybe I\u0027ll write them over here where there\u0027s more room."},{"Start":"07:53.150 ","End":"07:57.560","Text":"Here you would put d. This side doesn\u0027t matter."},{"Start":"07:57.560 ","End":"08:02.840","Text":"This would be d minus absolute value of a,"},{"Start":"08:02.840 ","End":"08:07.160","Text":"and this height would be d plus absolute value of a."},{"Start":"08:07.160 ","End":"08:11.700","Text":"Once we have these points then we can"},{"Start":"08:11.700 ","End":"08:16.940","Text":"sketch the sine which starts off being in the midpoint."},{"Start":"08:16.940 ","End":"08:18.935","Text":"Then at the next station,"},{"Start":"08:18.935 ","End":"08:20.675","Text":"we are at the maximum."},{"Start":"08:20.675 ","End":"08:24.005","Text":"The next station we\u0027re back at the midpoint again,"},{"Start":"08:24.005 ","End":"08:26.255","Text":"the next station we\u0027re at the minimum,"},{"Start":"08:26.255 ","End":"08:28.445","Text":"and then we\u0027re back at the midpoint."},{"Start":"08:28.445 ","End":"08:33.215","Text":"Then we just draw a curve through it,"},{"Start":"08:33.215 ","End":"08:37.340","Text":"trying to make sure that it flattens out to a hump."},{"Start":"08:37.340 ","End":"08:40.955","Text":"Here and here. Then we\u0027ll say, yes,"},{"Start":"08:40.955 ","End":"08:45.990","Text":"this is the sine curve but we have these values marked on."},{"Start":"08:46.090 ","End":"08:51.785","Text":"But this would only be valid for when a is bigger than 0."},{"Start":"08:51.785 ","End":"08:58.250","Text":"If we had a less than 0 then it\u0027s an upside-down sign that we have this point."},{"Start":"08:58.250 ","End":"09:02.390","Text":"Then we go down to the minimum then to the middle then to the max,"},{"Start":"09:02.390 ","End":"09:03.605","Text":"and then to the middle."},{"Start":"09:03.605 ","End":"09:10.300","Text":"We would get something like this."},{"Start":"09:10.910 ","End":"09:20.810","Text":"The final step would be to replicate this a few times because this is only 1 period."},{"Start":"09:22.000 ","End":"09:30.215","Text":"Let\u0027s see if I can scroll back to the beginning."},{"Start":"09:30.215 ","End":"09:32.795","Text":"Yeah, just like in the beginning,"},{"Start":"09:32.795 ","End":"09:39.580","Text":"we had 1 period\u0027s worth and we just replicate it."},{"Start":"09:39.580 ","End":"09:42.735","Text":"So back to where we were."},{"Start":"09:42.735 ","End":"09:45.740","Text":"This takes care of the sine."},{"Start":"09:45.740 ","End":"09:48.055","Text":"Also want to do the cosine."},{"Start":"09:48.055 ","End":"09:52.970","Text":"I just copy-paste it but now I have to erase the graphs."},{"Start":"09:52.970 ","End":"09:56.405","Text":"Let\u0027s just quickly go over the case for cosine,"},{"Start":"09:56.405 ","End":"10:05.330","Text":"where it\u0027s a cosine of bx minus c plus d. Once again,"},{"Start":"10:05.330 ","End":"10:10.100","Text":"you have to distinguish cases where a is bigger than 0, a is less than 0."},{"Start":"10:10.100 ","End":"10:13.910","Text":"Let\u0027s start with a bigger than 0 with the cosine,"},{"Start":"10:13.910 ","End":"10:16.805","Text":"we start at the maximum."},{"Start":"10:16.805 ","End":"10:18.740","Text":"Then at the next stop,"},{"Start":"10:18.740 ","End":"10:22.355","Text":"we\u0027re at the middle then we\u0027re at the minimum,"},{"Start":"10:22.355 ","End":"10:25.215","Text":"then we\u0027re at the middle,"},{"Start":"10:25.215 ","End":"10:27.905","Text":"then at the maximum again,"},{"Start":"10:27.905 ","End":"10:31.310","Text":"as I said, don\u0027t just draw a straight line."},{"Start":"10:31.310 ","End":"10:37.085","Text":"Here it starts off flat and then gets steeper and then flattens out again."},{"Start":"10:37.085 ","End":"10:39.275","Text":"Something like this."},{"Start":"10:39.275 ","End":"10:41.015","Text":"That\u0027s for the cosine."},{"Start":"10:41.015 ","End":"10:49.820","Text":"If a was negative then the opposite reflected in the middle line."},{"Start":"10:49.820 ","End":"10:55.970","Text":"We start at the minimum get to the middle then get to the top then to the middle,"},{"Start":"10:55.970 ","End":"10:57.515","Text":"then to the minimum."},{"Start":"10:57.515 ","End":"11:03.920","Text":"The upside-down cosine looks something like this."},{"Start":"11:03.920 ","End":"11:07.805","Text":"Down through here and flattening out."},{"Start":"11:07.805 ","End":"11:09.830","Text":"For freehand is not bad."},{"Start":"11:09.830 ","End":"11:14.735","Text":"That\u0027s why we want to do it with a computer."},{"Start":"11:14.735 ","End":"11:21.215","Text":"All this will make a lot more sense in the solved examples with actual numbers here."},{"Start":"11:21.215 ","End":"11:27.199","Text":"This is just to give you the general idea of the full possible shapes to be encountered,"},{"Start":"11:27.199 ","End":"11:30.680","Text":"and how we just really do 5 points."},{"Start":"11:30.680 ","End":"11:34.485","Text":"We know the general shape and we freehand it,"},{"Start":"11:34.485 ","End":"11:39.540","Text":"if we have to or use a computer, if we\u0027re able to."},{"Start":"11:41.630 ","End":"11:46.125","Text":"As I said, this is just for 1 period."},{"Start":"11:46.125 ","End":"11:50.240","Text":"As I mentioned earlier, we then duplicate it a few times,"},{"Start":"11:50.240 ","End":"11:52.310","Text":"maybe a couple more times."},{"Start":"11:52.310 ","End":"11:54.200","Text":"Maybe once after, once before."},{"Start":"11:54.200 ","End":"11:55.460","Text":"This is just 1 period."},{"Start":"11:55.460 ","End":"11:57.040","Text":"We just extend it a bit."},{"Start":"11:57.040 ","End":"12:00.370","Text":"I\u0027m done."}],"ID":10893},{"Watched":false,"Name":"Graphs of Trigonometric Functions - Part 4","Duration":"14m 33s","ChapterTopicVideoID":10473,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10473.jpeg","UploadDate":"2021-06-29T12:38:32.6900000","DurationForVideoObject":"PT14M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"Next, after sine and cosine,"},{"Start":"00:02.130 ","End":"00:04.470","Text":"we come to the tangent."},{"Start":"00:04.470 ","End":"00:11.080","Text":"The most basic form is y= tangent x."},{"Start":"00:11.180 ","End":"00:18.135","Text":"Here\u0027s a sketch of what it looks like."},{"Start":"00:18.135 ","End":"00:21.795","Text":"Computer drawn, of course."},{"Start":"00:21.795 ","End":"00:28.200","Text":"It actually continues to infinity and to minus infinity."},{"Start":"00:28.200 ","End":"00:31.275","Text":"These dotted lines are asymptotes."},{"Start":"00:31.275 ","End":"00:33.600","Text":"It\u0027s also periodic."},{"Start":"00:33.600 ","End":"00:36.380","Text":"Notice that this part here repeats itself."},{"Start":"00:36.380 ","End":"00:40.170","Text":"Let me just highlight that part."},{"Start":"00:40.610 ","End":"00:46.790","Text":"This part corresponds to x being"},{"Start":"00:46.790 ","End":"00:53.955","Text":"between minus Pi over 2 and plus Pi over 2."},{"Start":"00:53.955 ","End":"00:59.555","Text":"It\u0027s not defined at these points where there\u0027s an asymptote."},{"Start":"00:59.555 ","End":"01:08.540","Text":"Those points are actually where the cosine is 0 because tangent is sine over cosine."},{"Start":"01:08.540 ","End":"01:11.095","Text":"Wherever cosine is 0,"},{"Start":"01:11.095 ","End":"01:17.800","Text":"like Pi over 2 and then multiples of Pi, either way, added."},{"Start":"01:19.040 ","End":"01:21.760","Text":"For one period, this is what it is."},{"Start":"01:21.760 ","End":"01:25.840","Text":"The period is the length of this is Pi,"},{"Start":"01:25.840 ","End":"01:27.670","Text":"as opposed to sine and cosine,"},{"Start":"01:27.670 ","End":"01:30.385","Text":"which had a period of 2Pi."},{"Start":"01:30.385 ","End":"01:34.990","Text":"When sketching it, there\u0027s always 3 points we can count on."},{"Start":"01:34.990 ","End":"01:37.075","Text":"There\u0027s the origin."},{"Start":"01:37.075 ","End":"01:44.035","Text":"There\u0027s also the point Pi over 2,1 because tangent of Pi over"},{"Start":"01:44.035 ","End":"01:53.820","Text":"2=1 and also tangent of minus Pi over 2 is minus 1."},{"Start":"01:53.820 ","End":"01:55.680","Text":"Perhaps I\u0027ll mark those."},{"Start":"01:55.680 ","End":"02:00.765","Text":"This is Pi over 2,1,"},{"Start":"02:00.765 ","End":"02:05.235","Text":"and this is minus Pi over 2, minus 1."},{"Start":"02:05.235 ","End":"02:07.140","Text":"This is the origin."},{"Start":"02:07.140 ","End":"02:09.140","Text":"We have these 3 points,"},{"Start":"02:09.140 ","End":"02:16.345","Text":"and then there\u0027s the asymptotes at x= Pi over 2,"},{"Start":"02:16.345 ","End":"02:19.820","Text":"where it goes to plus infinity,"},{"Start":"02:19.820 ","End":"02:24.430","Text":"and x= minus Pi over 2,"},{"Start":"02:24.430 ","End":"02:27.730","Text":"where it goes to minus infinity."},{"Start":"02:27.730 ","End":"02:30.610","Text":"Also notice that it has asymmetry."},{"Start":"02:30.610 ","End":"02:35.890","Text":"It\u0027s an odd function that if we rotated 180 degrees,"},{"Start":"02:35.890 ","End":"02:38.205","Text":"then it\u0027s onto itself."},{"Start":"02:38.205 ","End":"02:44.265","Text":"Tangent of minus x is minus tangent x."},{"Start":"02:44.265 ","End":"02:45.600","Text":"Here\u0027s another graph."},{"Start":"02:45.600 ","End":"02:51.525","Text":"This is the upside-down tangent y= minus tangent of x."},{"Start":"02:51.525 ","End":"02:56.110","Text":"This one would be upside down."},{"Start":"02:57.020 ","End":"03:01.160","Text":"Similar here is the same domain,"},{"Start":"03:01.160 ","End":"03:04.985","Text":"the same period, just the asymptotes."},{"Start":"03:04.985 ","End":"03:09.710","Text":"The opposite, here it goes minus Pi over 2,"},{"Start":"03:09.710 ","End":"03:11.300","Text":"it goes to infinity,"},{"Start":"03:11.300 ","End":"03:14.870","Text":"and here it goes to minus infinity."},{"Start":"03:14.870 ","End":"03:21.626","Text":"I guess the points are also correspondingly negated,"},{"Start":"03:21.626 ","End":"03:24.810","Text":"but we do have 3 points that we can always sketch."},{"Start":"03:24.810 ","End":"03:27.460","Text":"Then we also have the asymptotes."},{"Start":"03:27.460 ","End":"03:31.185","Text":"Okay, I won\u0027t say any more about the minus tangent."},{"Start":"03:31.185 ","End":"03:33.770","Text":"I\u0027d like to generalize a little bit like we did with"},{"Start":"03:33.770 ","End":"03:38.990","Text":"the sine and cosine to the case where y=A"},{"Start":"03:38.990 ","End":"03:43.788","Text":"tangent of Bx minus C"},{"Start":"03:43.788 ","End":"03:49.460","Text":"plus D. I\u0027m not going to do it in as much detail as the sine and cosine."},{"Start":"03:49.460 ","End":"03:55.910","Text":"But there\u0027s no amplitude in the case of the tangent because it goes off to infinity."},{"Start":"03:55.910 ","End":"04:00.080","Text":"I\u0027m going to assume that B is positive."},{"Start":"04:00.080 ","End":"04:01.160","Text":"If it isn\u0027t positive,"},{"Start":"04:01.160 ","End":"04:07.430","Text":"we can use this rule of the oddness to rearrange it so that B does come out positive."},{"Start":"04:07.430 ","End":"04:11.585","Text":"Also, I\u0027d like to look at this."},{"Start":"04:11.585 ","End":"04:19.340","Text":"Sometimes it\u0027s better to look at it as B times x minus C over B."},{"Start":"04:19.340 ","End":"04:22.380","Text":"You\u0027ll see this in the moment."},{"Start":"04:22.430 ","End":"04:27.410","Text":"What B is, B is the property that when"},{"Start":"04:27.410 ","End":"04:31.658","Text":"you multiply the independent variable by a number,"},{"Start":"04:31.658 ","End":"04:35.660","Text":"that changes the graph so that B is a"},{"Start":"04:35.660 ","End":"04:39.750","Text":"horizontal and whether to say stretch or compression."},{"Start":"04:39.750 ","End":"04:41.555","Text":"I\u0027m going to write compression,"},{"Start":"04:41.555 ","End":"04:45.785","Text":"but it could also be thought of as a stretch."},{"Start":"04:45.785 ","End":"04:49.595","Text":"If B is less than 1, then it expands."},{"Start":"04:49.595 ","End":"04:51.270","Text":"Let\u0027s think of stretch as being more,"},{"Start":"04:51.270 ","End":"04:52.700","Text":"getting wider, and compression."},{"Start":"04:52.700 ","End":"04:59.520","Text":"That depends if B is bigger than 1 or B is less than 1."},{"Start":"05:00.020 ","End":"05:07.550","Text":"The period therefore changes."},{"Start":"05:07.550 ","End":"05:09.665","Text":"Because of this compression,"},{"Start":"05:09.665 ","End":"05:12.170","Text":"the period is no longer Pi."},{"Start":"05:12.170 ","End":"05:16.055","Text":"The period is Pi over B."},{"Start":"05:16.055 ","End":"05:21.570","Text":"Let us write down what some of the other variables are."},{"Start":"05:21.870 ","End":"05:31.440","Text":"A is the vertical stretch."},{"Start":"05:31.440 ","End":"05:35.420","Text":"If it\u0027s bigger than 1, then it expands."},{"Start":"05:35.420 ","End":"05:37.400","Text":"If it\u0027s less than 1, it contracts."},{"Start":"05:37.400 ","End":"05:41.900","Text":"I really should have written the absolute value of A. I\u0027m not going to write that,"},{"Start":"05:41.900 ","End":"05:45.815","Text":"but if A is positive,"},{"Start":"05:45.815 ","End":"05:48.160","Text":"then we get one of these,"},{"Start":"05:48.160 ","End":"05:51.860","Text":"and if A is negative,"},{"Start":"05:51.860 ","End":"05:54.365","Text":"you get an upside down one."},{"Start":"05:54.365 ","End":"05:55.850","Text":"Depending on the sign,"},{"Start":"05:55.850 ","End":"06:00.470","Text":"we know we have a basic tangent curve or an upside-down tangent curve."},{"Start":"06:00.470 ","End":"06:05.975","Text":"Other than that, really the absolute value of A is the vertical stretch."},{"Start":"06:05.975 ","End":"06:11.370","Text":"B, we\u0027ve already covered."},{"Start":"06:11.590 ","End":"06:14.300","Text":"C, not exactly."},{"Start":"06:14.300 ","End":"06:21.275","Text":"C over B is the horizontal shift,"},{"Start":"06:21.275 ","End":"06:25.955","Text":"or sometimes called the phase shift."},{"Start":"06:25.955 ","End":"06:30.285","Text":"When it\u0027s a horizontal, phase shift in this direction."},{"Start":"06:30.285 ","End":"06:32.810","Text":"Of course, if it\u0027s negative,"},{"Start":"06:32.810 ","End":"06:36.800","Text":"then it\u0027s in the other direction with the magnitude, the absolute value."},{"Start":"06:36.800 ","End":"06:38.900","Text":"I\u0027m not writing all that, but a lot of things that are"},{"Start":"06:38.900 ","End":"06:41.794","Text":"one way or the other way if they\u0027re negative."},{"Start":"06:41.794 ","End":"06:45.845","Text":"What else do we have?"},{"Start":"06:45.845 ","End":"06:51.920","Text":"D is the vertical shift,"},{"Start":"06:51.920 ","End":"06:54.810","Text":"which means a shift upwards,"},{"Start":"06:54.810 ","End":"06:58.100","Text":"again, not providing that D is positive,"},{"Start":"06:58.100 ","End":"06:59.600","Text":"and if D is negative,"},{"Start":"06:59.600 ","End":"07:03.868","Text":"then it\u0027s a downward shift."},{"Start":"07:03.868 ","End":"07:07.240","Text":"Let\u0027s do a couple of examples of"},{"Start":"07:07.240 ","End":"07:10.570","Text":"a more generalized tangent and you\u0027ll see I\u0027ll give"},{"Start":"07:10.570 ","End":"07:14.695","Text":"an example of a right way up on an example of an upside-down."},{"Start":"07:14.695 ","End":"07:16.690","Text":"The first example I\u0027ll"},{"Start":"07:16.690 ","End":"07:26.380","Text":"take is y=tan(2x-Pi/3),"},{"Start":"07:26.380 ","End":"07:30.985","Text":"so 2 is the B"},{"Start":"07:30.985 ","End":"07:37.540","Text":"and C is the Pi/3."},{"Start":"07:37.540 ","End":"07:40.195","Text":"There\u0027s no A and there\u0027s no D,"},{"Start":"07:40.195 ","End":"07:49.675","Text":"so what we can say is that the period is instead of Pi,"},{"Start":"07:49.675 ","End":"07:52.870","Text":"it\u0027s Pi/B, which is 2 into Pi/2,"},{"Start":"07:52.870 ","End":"07:57.085","Text":"it\u0027s a compression,"},{"Start":"07:57.085 ","End":"08:02.635","Text":"and there is also a horizontal shift,"},{"Start":"08:02.635 ","End":"08:07.255","Text":"or phase shift and that is what we said."},{"Start":"08:07.255 ","End":"08:13.750","Text":"Whereas at C/B Pi/3/2 is Pi/6."},{"Start":"08:13.750 ","End":"08:18.625","Text":"It\u0027s this way because it\u0027s positive."},{"Start":"08:18.625 ","End":"08:20.875","Text":"What we do is we take this,"},{"Start":"08:20.875 ","End":"08:24.970","Text":"we squish it inwards by a factor of 2,"},{"Start":"08:24.970 ","End":"08:31.010","Text":"and then shift it Pi/6 to the right."},{"Start":"08:31.440 ","End":"08:33.940","Text":"Here\u0027s what it looks like."},{"Start":"08:33.940 ","End":"08:37.840","Text":"I\u0027m not going into any great detail how to get this."},{"Start":"08:37.840 ","End":"08:42.535","Text":"Probably you would just compute these 3 points,"},{"Start":"08:42.535 ","End":"08:48.860","Text":"but you would first of all compress and shift."},{"Start":"08:50.670 ","End":"08:53.260","Text":"First of all, if you shifted them,"},{"Start":"08:53.260 ","End":"08:56.905","Text":"this would be Pi/4 and this would be minus Pi/4,"},{"Start":"08:56.905 ","End":"09:00.160","Text":"and then you would add Pi/6."},{"Start":"09:00.160 ","End":"09:03.520","Text":"Pi/6 would be this quantity here."},{"Start":"09:03.520 ","End":"09:05.440","Text":"That\u0027s where the origin is shifted to,"},{"Start":"09:05.440 ","End":"09:08.889","Text":"so you might compute this point and compute"},{"Start":"09:08.889 ","End":"09:13.180","Text":"where it\u0027s equal to 1 and where it\u0027s equal to -1."},{"Start":"09:13.180 ","End":"09:17.830","Text":"Anyway, I\u0027m not doing all the details,"},{"Start":"09:17.830 ","End":"09:20.210","Text":"just want to give you the idea."},{"Start":"09:20.550 ","End":"09:23.245","Text":"The second example,"},{"Start":"09:23.245 ","End":"09:24.850","Text":"I\u0027ll take"},{"Start":"09:24.850 ","End":"09:37.015","Text":"y=-2tanx+1."},{"Start":"09:37.015 ","End":"09:47.095","Text":"This time it\u0027s going to be upside down because the A is -2 is negative."},{"Start":"09:47.095 ","End":"09:56.755","Text":"and we have a vertical stretch by a factor of 2,"},{"Start":"09:56.755 ","End":"09:59.275","Text":"the minus just makes it upside down,"},{"Start":"09:59.275 ","End":"10:01.840","Text":"and we also have the 1,"},{"Start":"10:01.840 ","End":"10:08.560","Text":"which gives us a vertical shift of 1."},{"Start":"10:08.560 ","End":"10:13.360","Text":"We go upwards 1, it\u0027s positive."},{"Start":"10:13.360 ","End":"10:17.240","Text":"The period doesn\u0027t change."},{"Start":"10:18.690 ","End":"10:22.825","Text":"The period because there\u0027s no horizontal stretch,"},{"Start":"10:22.825 ","End":"10:26.690","Text":"the period is still equal to Pi."},{"Start":"10:27.240 ","End":"10:34.795","Text":"What we could do from this would be to stretch it by a factor of 2."},{"Start":"10:34.795 ","End":"10:37.420","Text":"The minus is taking care of by the upside-down,"},{"Start":"10:37.420 ","End":"10:39.924","Text":"stretch it upwards and downwards,"},{"Start":"10:39.924 ","End":"10:43.880","Text":"and then lift the whole thing 1 up."},{"Start":"10:44.550 ","End":"10:49.765","Text":"If I look at what happens to these 3 points stretching by 2,"},{"Start":"10:49.765 ","End":"10:51.700","Text":"the origin stays the origin,"},{"Start":"10:51.700 ","End":"10:55.165","Text":"this one goes up to 2 and this one goes down to -2."},{"Start":"10:55.165 ","End":"10:57.100","Text":"Then when I raise it by 1,"},{"Start":"10:57.100 ","End":"10:58.750","Text":"instead of -2 and 2,"},{"Start":"10:58.750 ","End":"11:00.993","Text":"I\u0027ll get -1 and 3,"},{"Start":"11:00.993 ","End":"11:09.010","Text":"so that it\u0027ll get this and will get the minus 1 here."},{"Start":"11:09.010 ","End":"11:12.475","Text":"The 3 over here,"},{"Start":"11:12.475 ","End":"11:19.910","Text":"just over the Pi/2 and the minus Pi/-2."},{"Start":"11:22.050 ","End":"11:25.840","Text":"That\u0027s it for tangent."},{"Start":"11:25.840 ","End":"11:29.005","Text":"I\u0027m going to conclude with the graphs of the rest of them."},{"Start":"11:29.005 ","End":"11:32.060","Text":"Well, we\u0027ve done tangent,"},{"Start":"11:32.490 ","End":"11:37.735","Text":"but I\u0027ll just briefly show you the other three."},{"Start":"11:37.735 ","End":"11:44.350","Text":"I put the tangent together with the secant on the same graph because they share"},{"Start":"11:44.350 ","End":"11:51.240","Text":"the same asymptote and they don\u0027t interfere with each other."},{"Start":"11:51.240 ","End":"11:54.910","Text":"The tan we\u0027ve talked about,"},{"Start":"11:55.440 ","End":"12:01.180","Text":"and its period, as you recall, is Pi."},{"Start":"12:01.180 ","End":"12:05.830","Text":"The secant, if you look at it,"},{"Start":"12:05.830 ","End":"12:10.720","Text":"doesn\u0027t repeat every Pi because the secant has 2 humps."},{"Start":"12:10.720 ","End":"12:12.906","Text":"It\u0027s 2 parts."},{"Start":"12:12.906 ","End":"12:17.620","Text":"It\u0027s like this cup that\u0027s holding water and the cup that\u0027s spilling,"},{"Start":"12:17.620 ","End":"12:20.360","Text":"however you want to call it."},{"Start":"12:20.520 ","End":"12:31.470","Text":"Only every 2Pi do we return to ourselves."},{"Start":"12:31.470 ","End":"12:33.735","Text":"That\u0027s one difference."},{"Start":"12:33.735 ","End":"12:40.360","Text":"They have the asymptotes in the same place because this is sine over cosine,"},{"Start":"12:40.360 ","End":"12:49.165","Text":"this is 1/cosine, so the asymptotes are everywhere the cosine is 0 like with tangent."},{"Start":"12:49.165 ","End":"12:52.180","Text":"What else should I say?"},{"Start":"12:52.180 ","End":"12:56.125","Text":"The minimum here is at the point"},{"Start":"12:56.125 ","End":"13:05.260","Text":"where y=1 and the high point here is also where y=-1."},{"Start":"13:05.260 ","End":"13:08.905","Text":"That\u0027s because the secant is the reciprocal of the cosine."},{"Start":"13:08.905 ","End":"13:17.410","Text":"The cosine here is 1 and downward."},{"Start":"13:17.410 ","End":"13:22.105","Text":"Here the secant is 1 and upwards and similarly for this thing."},{"Start":"13:22.105 ","End":"13:25.165","Text":"Anyway, just wanted to briefly get you acquainted."},{"Start":"13:25.165 ","End":"13:29.155","Text":"Now, that\u0027s the tangent, that\u0027s the secant."},{"Start":"13:29.155 ","End":"13:31.255","Text":"Now the other two;"},{"Start":"13:31.255 ","End":"13:34.573","Text":"cotangent and co-secant,"},{"Start":"13:34.573 ","End":"13:41.230","Text":"I also put them on the same graph because they also have the same asymptotes,"},{"Start":"13:41.230 ","End":"13:43.090","Text":"only the asymptotes here,"},{"Start":"13:43.090 ","End":"13:49.070","Text":"wherever the sin is 0 because this is cosine/sin, this is 1/sin."},{"Start":"13:50.790 ","End":"14:01.720","Text":"The period here is also Pi because you can see this basic shape repeats every Pi."},{"Start":"14:01.720 ","End":"14:06.415","Text":"But with the co-secant it\u0027s got the 2 humps or whatever you call them,"},{"Start":"14:06.415 ","End":"14:14.290","Text":"so the period is 2Pi and this minimum point is where y=1,"},{"Start":"14:14.290 ","End":"14:19.495","Text":"and this maximum point is where y=-1."},{"Start":"14:19.495 ","End":"14:24.010","Text":"Just for reference generally to be familiar with these graphs,"},{"Start":"14:24.010 ","End":"14:27.610","Text":"so that\u0027s the cotangent and that\u0027s the co-secant."},{"Start":"14:27.610 ","End":"14:33.980","Text":"We\u0027ve already the sine and cosine talked a lot about so that\u0027s it."}],"ID":10894},{"Watched":false,"Name":"Exercise 1","Duration":"3m 7s","ChapterTopicVideoID":10369,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10369.jpeg","UploadDate":"2017-11-02T15:45:28.9670000","DurationForVideoObject":"PT3M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.470","Text":"In this exercise, we have to sketch the graph of y=4 sin(1/2x)."},{"Start":"00:08.540 ","End":"00:13.950","Text":"This is going to be a standard sine curve."},{"Start":"00:13.950 ","End":"00:15.900","Text":"We have the quantities A, B,"},{"Start":"00:15.900 ","End":"00:22.005","Text":"C and D. 4 is what we call A and the 1/2 is B."},{"Start":"00:22.005 ","End":"00:27.870","Text":"The A affects the amplitude, the B affects the horizontal scale,"},{"Start":"00:27.870 ","End":"00:29.385","Text":"the stretching or squishing,"},{"Start":"00:29.385 ","End":"00:34.050","Text":"but there\u0027s no shifting up or down or no phase shift here."},{"Start":"00:34.050 ","End":"00:36.660","Text":"Let\u0027s compute the quantities we need."},{"Start":"00:36.660 ","End":"00:38.565","Text":"We need the amplitude,"},{"Start":"00:38.565 ","End":"00:41.775","Text":"and that would be just A which is 4."},{"Start":"00:41.775 ","End":"00:46.770","Text":"We need the period and that\u0027s equal to 2 Pi over"},{"Start":"00:46.770 ","End":"00:52.650","Text":"B so it\u0027s 2 Pi over 1/2 which makes it 4 Pi."},{"Start":"00:52.650 ","End":"00:56.150","Text":"Because of the period and there\u0027s no shifting it\u0027s"},{"Start":"00:56.150 ","End":"01:00.520","Text":"going to be defined on the interval from 0-4 Pi."},{"Start":"01:00.520 ","End":"01:03.830","Text":"I\u0027m going to break this into 4 bits but"},{"Start":"01:03.830 ","End":"01:07.310","Text":"the range of values is going to be according to the amplitude,"},{"Start":"01:07.310 ","End":"01:09.485","Text":"4 times 1 and minus 1."},{"Start":"01:09.485 ","End":"01:13.610","Text":"In other words, it\u0027s going to go from minus 4 is going to be"},{"Start":"01:13.610 ","End":"01:18.780","Text":"the minimum and 4 is going to be the maximum."},{"Start":"01:20.210 ","End":"01:25.175","Text":"Here I have a bit of a graph paper."},{"Start":"01:25.175 ","End":"01:28.430","Text":"This is the x-axis,"},{"Start":"01:28.430 ","End":"01:32.020","Text":"and this here is the y-axis,"},{"Start":"01:32.020 ","End":"01:34.115","Text":"and this is 0,"},{"Start":"01:34.115 ","End":"01:36.770","Text":"and this will be 4 Pi."},{"Start":"01:36.770 ","End":"01:41.370","Text":"If you multiply 4 times Pi, it\u0027s about 12.5."},{"Start":"01:42.170 ","End":"01:45.225","Text":"Here we have Pi,"},{"Start":"01:45.225 ","End":"01:48.600","Text":"2 Pi, 3 Pi,"},{"Start":"01:48.600 ","End":"01:50.720","Text":"so that\u0027s 4 pieces."},{"Start":"01:50.720 ","End":"01:54.325","Text":"Then we know we want to go from minus 4-4."},{"Start":"01:54.325 ","End":"01:56.219","Text":"Each little square is 1 unit,"},{"Start":"01:56.219 ","End":"01:58.710","Text":"4 and minus 4."},{"Start":"01:58.710 ","End":"02:02.430","Text":"Because it\u0027s a standard sine curve,"},{"Start":"02:02.430 ","End":"02:05.825","Text":"we know that the general shape is this shape."},{"Start":"02:05.825 ","End":"02:09.905","Text":"We can put a point here."},{"Start":"02:09.905 ","End":"02:13.442","Text":"We can put a point above the Pi,"},{"Start":"02:13.442 ","End":"02:15.705","Text":"that would be here."},{"Start":"02:15.705 ","End":"02:19.275","Text":"Then down to 0 again at 2 Pi,"},{"Start":"02:19.275 ","End":"02:22.280","Text":"then down to minus 4,"},{"Start":"02:22.280 ","End":"02:23.810","Text":"that would be here,"},{"Start":"02:23.810 ","End":"02:26.855","Text":"and then back to 0 again."},{"Start":"02:26.855 ","End":"02:30.349","Text":"We\u0027ll just do a freehand curve here,"},{"Start":"02:30.349 ","End":"02:32.630","Text":"down to here, through here,"},{"Start":"02:32.630 ","End":"02:34.770","Text":"and ending up here."},{"Start":"02:35.030 ","End":"02:38.205","Text":"Now, that\u0027s just one period."},{"Start":"02:38.205 ","End":"02:41.895","Text":"All you have to do now is replicate this."},{"Start":"02:41.895 ","End":"02:47.825","Text":"There\u0027s no exact rules of how many times I\u0027ll put one after and one before."},{"Start":"02:47.825 ","End":"02:51.410","Text":"Then we get something like this where this part here is"},{"Start":"02:51.410 ","End":"02:54.320","Text":"one period and then I put another period here,"},{"Start":"02:54.320 ","End":"02:55.470","Text":"another period before,"},{"Start":"02:55.470 ","End":"02:57.920","Text":"and I guess I should really label them."},{"Start":"02:57.920 ","End":"02:59.705","Text":"This is the y-axis,"},{"Start":"02:59.705 ","End":"03:01.955","Text":"and this is the x-axis,"},{"Start":"03:01.955 ","End":"03:03.403","Text":"and this is 0,"},{"Start":"03:03.403 ","End":"03:07.380","Text":"and so on. I think this will do."}],"ID":10729},{"Watched":false,"Name":"Exercise 2","Duration":"3m 51s","ChapterTopicVideoID":10370,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10370.jpeg","UploadDate":"2017-11-02T15:45:41.8600000","DurationForVideoObject":"PT3M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.440","Text":"In this exercise, we\u0027re asked to sketch the graph of u=2 cosine pi t plus 1."},{"Start":"00:07.440 ","End":"00:11.925","Text":"This is a standard cosine curve."},{"Start":"00:11.925 ","End":"00:16.170","Text":"I use u and t sometimes instead of y and x,"},{"Start":"00:16.170 ","End":"00:20.220","Text":"we don\u0027t want always want to get used to the same letters."},{"Start":"00:20.220 ","End":"00:27.810","Text":"From this we want to derive certain quantities and I usually start with the amplitude."},{"Start":"00:27.810 ","End":"00:29.730","Text":"The amplitude is this 2,"},{"Start":"00:29.730 ","End":"00:34.770","Text":"we called it A and I think we called this B and this we called"},{"Start":"00:34.770 ","End":"00:39.890","Text":"D. There\u0027s also a C which is not present here,"},{"Start":"00:39.890 ","End":"00:42.170","Text":"which means there\u0027s no phase shift."},{"Start":"00:42.170 ","End":"00:44.675","Text":"Let\u0027s just write down the important stuff."},{"Start":"00:44.675 ","End":"00:49.615","Text":"The amplitude, which is the A is 2."},{"Start":"00:49.615 ","End":"00:59.025","Text":"The period is 2 pi over B is 2 pi over pi,"},{"Start":"00:59.025 ","End":"01:01.080","Text":"which is 2,"},{"Start":"01:01.080 ","End":"01:08.630","Text":"so the interval that we\u0027re defined on for 1 period is from 0-2."},{"Start":"01:08.630 ","End":"01:11.030","Text":"If there was a phase shift,"},{"Start":"01:11.030 ","End":"01:16.445","Text":"we would shift this along and there is a D=1."},{"Start":"01:16.445 ","End":"01:20.930","Text":"That\u0027s like an upward shift of 1."},{"Start":"01:20.930 ","End":"01:26.225","Text":"Normally the amplitude being 2 it would go from minus 2-2,"},{"Start":"01:26.225 ","End":"01:30.635","Text":"but it\u0027s not from minus 2-2 because we have to add 1,"},{"Start":"01:30.635 ","End":"01:36.990","Text":"and if we add 1 and we\u0027re going from minus 1-3."},{"Start":"01:37.270 ","End":"01:45.010","Text":"Here I have a bit of a piece of graph paper and I have minus 1-3."},{"Start":"01:45.010 ","End":"01:47.175","Text":"In this graph as you can see,"},{"Start":"01:47.175 ","End":"01:48.930","Text":"each square is 1/2 a unit."},{"Start":"01:48.930 ","End":"01:51.015","Text":"This is minus 1,"},{"Start":"01:51.015 ","End":"01:52.395","Text":"this is 1,"},{"Start":"01:52.395 ","End":"01:54.525","Text":"this is 2, and this is 3."},{"Start":"01:54.525 ","End":"02:00.230","Text":"We\u0027re from minus 1-3 and the interval we\u0027re considering is from 0-2,"},{"Start":"02:00.230 ","End":"02:03.805","Text":"that\u0027s already marked, and we divide that into 4."},{"Start":"02:03.805 ","End":"02:06.089","Text":"That works out nicely."},{"Start":"02:06.089 ","End":"02:08.850","Text":"This would be 1/2,"},{"Start":"02:08.850 ","End":"02:12.825","Text":"1, 1 1/2, and 2."},{"Start":"02:12.825 ","End":"02:15.300","Text":"Now, this is a standard cosine curve."},{"Start":"02:15.300 ","End":"02:24.260","Text":"We know the general shape is like this and the middle line is not always 0."},{"Start":"02:24.260 ","End":"02:29.870","Text":"In this case, the middle line is determined by the D which is 1."},{"Start":"02:29.870 ","End":"02:32.910","Text":"We go up to 3,"},{"Start":"02:32.910 ","End":"02:34.670","Text":"down to minus 1 and the middle,"},{"Start":"02:34.670 ","End":"02:37.175","Text":"the average of these 2 in fact is 1."},{"Start":"02:37.175 ","End":"02:42.210","Text":"The first point would go here."},{"Start":"02:42.340 ","End":"02:48.455","Text":"Then I go down to the middle line here,"},{"Start":"02:48.455 ","End":"02:52.355","Text":"then down to the bottom here,"},{"Start":"02:52.355 ","End":"02:55.655","Text":"then to the middle line again,"},{"Start":"02:55.655 ","End":"02:59.090","Text":"and then to the maximum here."},{"Start":"02:59.090 ","End":"03:03.035","Text":"Then I\u0027ll do it free hand,"},{"Start":"03:03.035 ","End":"03:04.460","Text":"something like this,"},{"Start":"03:04.460 ","End":"03:10.490","Text":"you go down and then flatten out and then up again and then flatten out."},{"Start":"03:10.490 ","End":"03:13.565","Text":"That\u0027s one period of the graph."},{"Start":"03:13.565 ","End":"03:16.280","Text":"I forgot to label the variables."},{"Start":"03:16.280 ","End":"03:18.170","Text":"It\u0027s not x and y this time,"},{"Start":"03:18.170 ","End":"03:22.425","Text":"it\u0027s t and it\u0027s u."},{"Start":"03:22.425 ","End":"03:24.205","Text":"That was one period."},{"Start":"03:24.205 ","End":"03:26.870","Text":"I\u0027ll just duplicate it a few times."},{"Start":"03:26.870 ","End":"03:30.110","Text":"I\u0027ll just do one after and one before."},{"Start":"03:30.110 ","End":"03:32.720","Text":"Here we are, it looks nicer."},{"Start":"03:32.720 ","End":"03:34.760","Text":"It was done by computer."},{"Start":"03:34.760 ","End":"03:40.045","Text":"Once again, I should really label t-axis and"},{"Start":"03:40.045 ","End":"03:46.625","Text":"the u-axis and probably should also label some and like these the minus 1 and the 3."},{"Start":"03:46.625 ","End":"03:51.780","Text":"Okay, that\u0027s it. We\u0027re done."}],"ID":10730},{"Watched":false,"Name":"Exercise 3","Duration":"3m 29s","ChapterTopicVideoID":10371,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10371.jpeg","UploadDate":"2017-11-02T15:45:53.2370000","DurationForVideoObject":"PT3M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.200","Text":"In this exercise, we want to sketch the graph of y=2sin^(4x-π)."},{"Start":"00:08.200 ","End":"00:12.360","Text":"Let\u0027s write some of the things we need, some of the parameters."},{"Start":"00:12.360 ","End":"00:14.905","Text":"We\u0027ll need the amplitude."},{"Start":"00:14.905 ","End":"00:21.445","Text":"The amplitude is simply the absolute value of this number here, which is 2."},{"Start":"00:21.445 ","End":"00:24.260","Text":"Then we want the period."},{"Start":"00:24.260 ","End":"00:33.165","Text":"The period we get by taking 2 Pi and dividing it by this 4."},{"Start":"00:33.165 ","End":"00:36.720","Text":"We also have a phase shift,"},{"Start":"00:36.720 ","End":"00:39.935","Text":"because of this minus Pi."},{"Start":"00:39.935 ","End":"00:47.345","Text":"We have a phase shift and it\u0027s the reverse of the sign here so it\u0027s plus Pi,"},{"Start":"00:47.345 ","End":"00:51.420","Text":"but divided by this, which is Pi/4."},{"Start":"00:51.920 ","End":"00:54.890","Text":"There\u0027s no up or down shift."},{"Start":"00:54.890 ","End":"01:00.260","Text":"There\u0027s no what we call d. That means that if we want to draw 1 period,"},{"Start":"01:00.260 ","End":"01:05.780","Text":"we could take that period starting from the phase shift Pi/4"},{"Start":"01:05.780 ","End":"01:13.400","Text":"and extending it 1 period\u0027s worth 2Pi/4 of course is Pi/2."},{"Start":"01:13.400 ","End":"01:16.070","Text":"If I add Pi/2 to Pi/4,"},{"Start":"01:16.070 ","End":"01:21.930","Text":"that will give me 3Pi/4."},{"Start":"01:21.930 ","End":"01:24.800","Text":"The minimum and the maximum are plus or"},{"Start":"01:24.800 ","End":"01:29.690","Text":"minus the amplitude because there\u0027s no up-down shifts,"},{"Start":"01:29.690 ","End":"01:34.020","Text":"so we\u0027re going from minus 2 to 2."},{"Start":"01:34.510 ","End":"01:39.185","Text":"Here\u0027s a bit of graph paper that can fit this on."},{"Start":"01:39.185 ","End":"01:43.690","Text":"We have 2 and minus 2. Let\u0027s see."},{"Start":"01:43.690 ","End":"01:45.150","Text":"The period we want,"},{"Start":"01:45.150 ","End":"01:47.325","Text":"we\u0027ll start at Pi/4,"},{"Start":"01:47.325 ","End":"01:49.905","Text":"which is roughly 3/4."},{"Start":"01:49.905 ","End":"01:54.600","Text":"Let\u0027s see, this would be 1, and let\u0027s see,"},{"Start":"01:54.600 ","End":"02:01.020","Text":"Pi/4 is somewhere around here and the other side, 3Pi/4."},{"Start":"02:01.020 ","End":"02:05.580","Text":"It comes out somewhere over here."},{"Start":"02:07.550 ","End":"02:11.630","Text":"Anyway, I divided the interval up into 4,"},{"Start":"02:11.630 ","End":"02:13.040","Text":"starting from Pi/4,"},{"Start":"02:13.040 ","End":"02:14.420","Text":"ending in 3Pi/4,"},{"Start":"02:14.420 ","End":"02:16.730","Text":"and you can do the computations."},{"Start":"02:16.730 ","End":"02:20.095","Text":"What these points exactly are."},{"Start":"02:20.095 ","End":"02:25.460","Text":"Now we can make a rough sketch because we know"},{"Start":"02:25.460 ","End":"02:31.880","Text":"the general shape of a standard sine curve, something like this."},{"Start":"02:31.880 ","End":"02:34.555","Text":"We start here,"},{"Start":"02:34.555 ","End":"02:40.325","Text":"then we reach the maximum somewhere here,"},{"Start":"02:40.325 ","End":"02:44.430","Text":"then down here again,"},{"Start":"02:47.180 ","End":"02:52.380","Text":"down to the minus 2 somewhere here,"},{"Start":"02:52.380 ","End":"02:56.175","Text":"then back to 0 here."},{"Start":"02:56.175 ","End":"02:58.595","Text":"Then I\u0027ll freehand it."},{"Start":"02:58.595 ","End":"03:03.080","Text":"See up here, down here, it looks awful."},{"Start":"03:03.080 ","End":"03:05.615","Text":"I know, that\u0027s why we need a computer."},{"Start":"03:05.615 ","End":"03:08.580","Text":"That\u0027s 1 period."},{"Start":"03:10.250 ","End":"03:13.915","Text":"Here you have a computer sketch."},{"Start":"03:13.915 ","End":"03:17.390","Text":"Should have used the same color, never mind."},{"Start":"03:17.390 ","End":"03:21.305","Text":"I threw in a couple of extra periods,"},{"Start":"03:21.305 ","End":"03:29.040","Text":"and so you get the idea of the shape and that\u0027s basically it. We\u0027re done."}],"ID":10731},{"Watched":false,"Name":"Exercise 4","Duration":"5m 43s","ChapterTopicVideoID":10372,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10372.jpeg","UploadDate":"2017-11-02T15:46:12.4970000","DurationForVideoObject":"PT5M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this exercise, we have to sketch the graph."},{"Start":"00:04.005 ","End":"00:06.390","Text":"Well, I won\u0027t read it. It\u0027s as follows."},{"Start":"00:06.390 ","End":"00:10.680","Text":"This time we have u and t not y and x,"},{"Start":"00:10.680 ","End":"00:14.280","Text":"and this is a good example because it has all the constants here."},{"Start":"00:14.280 ","End":"00:16.680","Text":"We have the constant A,"},{"Start":"00:16.680 ","End":"00:18.704","Text":"we have a constant B,"},{"Start":"00:18.704 ","End":"00:21.090","Text":"we have a constant C,"},{"Start":"00:21.090 ","End":"00:26.804","Text":"and we have a D. So we can compute all the things that we need."},{"Start":"00:26.804 ","End":"00:28.560","Text":"Let\u0027s start with the amplitude."},{"Start":"00:28.560 ","End":"00:30.630","Text":"That\u0027s what we usually start with."},{"Start":"00:30.630 ","End":"00:37.120","Text":"The amplitude is the absolute value of A, so it\u0027s 2/3."},{"Start":"00:37.120 ","End":"00:39.620","Text":"Next, we want the period."},{"Start":"00:39.620 ","End":"00:42.930","Text":"The period is 2 Pi/B,"},{"Start":"00:42.930 ","End":"00:49.105","Text":"or 2 Pi over the absolute value of B."},{"Start":"00:49.105 ","End":"00:54.590","Text":"So it\u0027s 2 Pi/3,"},{"Start":"00:54.590 ","End":"00:57.350","Text":"and that\u0027s our period."},{"Start":"00:57.350 ","End":"00:59.780","Text":"Now, because of the constant C,"},{"Start":"00:59.780 ","End":"01:01.595","Text":"we have a phase shift."},{"Start":"01:01.595 ","End":"01:04.025","Text":"Let\u0027s compute what that is."},{"Start":"01:04.025 ","End":"01:12.250","Text":"That is equal to Pi/4/3, which is Pi/12."},{"Start":"01:12.250 ","End":"01:15.860","Text":"Assuming there\u0027s a minus there,"},{"Start":"01:15.860 ","End":"01:18.500","Text":"it\u0027s this over this, and if it\u0027s a plus, then it\u0027s minus."},{"Start":"01:18.500 ","End":"01:21.000","Text":"Anyway, you reverse the sign."},{"Start":"01:21.040 ","End":"01:24.050","Text":"Finally, we have a D,"},{"Start":"01:24.050 ","End":"01:27.110","Text":"which means that we have a vertical shift."},{"Start":"01:27.110 ","End":"01:31.340","Text":"I\u0027ll just put a vertical arrow here and say plus 1,"},{"Start":"01:31.340 ","End":"01:34.220","Text":"meaning the whole graph is shifted upwards."},{"Start":"01:34.220 ","End":"01:42.150","Text":"Now, what we do first is we figure out the domain of 1 period and the range,"},{"Start":"01:42.150 ","End":"01:45.720","Text":"like the t from where to where,"},{"Start":"01:45.720 ","End":"01:47.960","Text":"and then u from where to where,"},{"Start":"01:47.960 ","End":"01:50.795","Text":"and we draw 1 period."},{"Start":"01:50.795 ","End":"01:54.800","Text":"Also note that this is not a standard cosine."},{"Start":"01:54.800 ","End":"01:56.734","Text":"Perhaps I should write that down."},{"Start":"01:56.734 ","End":"02:01.970","Text":"This is an upside-down cosine,"},{"Start":"02:01.970 ","End":"02:04.280","Text":"which means that,"},{"Start":"02:04.280 ","End":"02:08.495","Text":"the usual cosine looks something like this,"},{"Start":"02:08.495 ","End":"02:13.730","Text":"and the upside down cosine is something like this."},{"Start":"02:13.730 ","End":"02:15.445","Text":"I\u0027m just [inaudible] roughly,"},{"Start":"02:15.445 ","End":"02:18.875","Text":"the shape. I erase those."},{"Start":"02:18.875 ","End":"02:24.230","Text":"What we need is to see where we\u0027re going from and to as far as t goes."},{"Start":"02:24.230 ","End":"02:26.735","Text":"We start with the phase shift,"},{"Start":"02:26.735 ","End":"02:28.750","Text":"which is Pi/12,"},{"Start":"02:28.750 ","End":"02:32.035","Text":"and we go along 1 period\u0027s worth."},{"Start":"02:32.035 ","End":"02:36.575","Text":"Question is, what is 1/12 plus 2/3?"},{"Start":"02:36.575 ","End":"02:42.585","Text":"2/3 is 8/12 plus 1/12 is 9/12, which is 3/4."},{"Start":"02:42.585 ","End":"02:46.440","Text":"So it\u0027s up to 3 Pi/4."},{"Start":"02:46.440 ","End":"02:49.505","Text":"That\u0027s going to be the interval for 1 period."},{"Start":"02:49.505 ","End":"02:51.680","Text":"Now, we need a range of values,"},{"Start":"02:51.680 ","End":"02:54.380","Text":"a minimum and maximum for u."},{"Start":"02:54.380 ","End":"03:01.460","Text":"That is going to be the 1 from here plus or minus the amplitude."},{"Start":"03:01.460 ","End":"03:03.200","Text":"This is the center line,"},{"Start":"03:03.200 ","End":"03:05.360","Text":"and this is how much above and below."},{"Start":"03:05.360 ","End":"03:13.610","Text":"We\u0027re going to go from 1/3 to 1 and 2/3."},{"Start":"03:13.610 ","End":"03:15.890","Text":"Here I have a piece of graph paper."},{"Start":"03:15.890 ","End":"03:23.190","Text":"I guess I should label u and t. What I want to mark,"},{"Start":"03:23.190 ","End":"03:25.035","Text":"this is 1,"},{"Start":"03:25.035 ","End":"03:29.850","Text":"I\u0027m going to put in on the vertical scale 1/3,"},{"Start":"03:29.850 ","End":"03:31.800","Text":"and 1 and 2/3."},{"Start":"03:31.800 ","End":"03:34.755","Text":"1/3 would be somewhere around here,"},{"Start":"03:34.755 ","End":"03:41.180","Text":"and 1 and 2/3 would be somewhere around here."},{"Start":"03:41.180 ","End":"03:42.810","Text":"That\u0027s our maximum and minimum,"},{"Start":"03:42.810 ","End":"03:45.585","Text":"and 1 is the center line."},{"Start":"03:45.585 ","End":"03:51.435","Text":"Then I want to take Pi/12 and 3 Pi/4."},{"Start":"03:51.435 ","End":"03:54.150","Text":"Pi/12 is about 1/4,"},{"Start":"03:54.150 ","End":"03:58.175","Text":"so somewhere here, Pi/12,"},{"Start":"03:58.175 ","End":"04:02.990","Text":"and 3 Pi/4 and something,"},{"Start":"04:02.990 ","End":"04:08.265","Text":"let\u0027s say here is our 3 Pi/4."},{"Start":"04:08.265 ","End":"04:14.190","Text":"Then we split this interval into 4,"},{"Start":"04:14.190 ","End":"04:16.800","Text":"let\u0027s say the halfway is here."},{"Start":"04:16.800 ","End":"04:25.215","Text":"Then we might want to go here and roughly here."},{"Start":"04:25.215 ","End":"04:28.285","Text":"So it\u0027s 4 pieces."},{"Start":"04:28.285 ","End":"04:35.635","Text":"Now, we want to sketch that upside down cosine that\u0027s very hard to draw freehand."},{"Start":"04:35.635 ","End":"04:45.610","Text":"But we start at Pi/12 and the 1/3, this part here."},{"Start":"04:45.610 ","End":"04:52.191","Text":"Then we\u0027re going to go up to the center line above this mark here,"},{"Start":"04:52.191 ","End":"04:57.265","Text":"then to the peak here above this."},{"Start":"04:57.265 ","End":"05:00.770","Text":"So we have a point here."},{"Start":"05:01.190 ","End":"05:06.594","Text":"Then down to the center line again,"},{"Start":"05:06.594 ","End":"05:10.830","Text":"then down to the minimum here."},{"Start":"05:10.830 ","End":"05:15.295","Text":"Then a bit of a free hand here,"},{"Start":"05:15.295 ","End":"05:18.525","Text":"here, down here, and so on."},{"Start":"05:18.525 ","End":"05:23.555","Text":"That\u0027s very roughly what 1 period looks like."},{"Start":"05:23.555 ","End":"05:26.540","Text":"Then we want to just replicate this a few times."},{"Start":"05:26.540 ","End":"05:29.840","Text":"I\u0027m going to bring you a computer-drawn sketch."},{"Start":"05:29.840 ","End":"05:34.955","Text":"Here we are, and how much better it looks when it\u0027s computer drawn."},{"Start":"05:34.955 ","End":"05:39.650","Text":"I took this period and 1 beyond it and 1 before it,"},{"Start":"05:39.650 ","End":"05:43.140","Text":"and that\u0027s it. We\u0027re done."}],"ID":10732},{"Watched":false,"Name":"Exercise 5","Duration":"2m 55s","ChapterTopicVideoID":10373,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10373.jpeg","UploadDate":"2017-11-02T15:46:22.8270000","DurationForVideoObject":"PT2M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.615","Text":"In this exercise, we want to sketch the graph of y equals absolute value of sin(x)."},{"Start":"00:06.615 ","End":"00:09.975","Text":"This is definitely not the standard sine curve."},{"Start":"00:09.975 ","End":"00:16.080","Text":"I\u0027d like to approach it with a very naive way of solving it."},{"Start":"00:16.080 ","End":"00:20.160","Text":"What I\u0027m going to do is, we\u0027re going to sketch sin(x) and then"},{"Start":"00:20.160 ","End":"00:24.940","Text":"take the absolute value by making everything negative into a positive."},{"Start":"00:24.940 ","End":"00:26.570","Text":"Here\u0027s what I have in mind."},{"Start":"00:26.570 ","End":"00:30.020","Text":"Let\u0027s suppose we take some axes,"},{"Start":"00:30.020 ","End":"00:31.790","Text":"y against x,"},{"Start":"00:31.790 ","End":"00:37.640","Text":"and then we take 1 period of a sine function."},{"Start":"00:37.640 ","End":"00:44.535","Text":"Then instead of this part that goes below the axis,"},{"Start":"00:44.535 ","End":"00:48.110","Text":"we\u0027ll make it positive, take the reflection."},{"Start":"00:48.110 ","End":"00:52.294","Text":"Altogether we\u0027ll get this and this."},{"Start":"00:52.294 ","End":"00:55.130","Text":"Now here I have a bit of graph paper."},{"Start":"00:55.130 ","End":"00:58.490","Text":"We don\u0027t need to compute all the amplitudes and periods."},{"Start":"00:58.490 ","End":"01:00.500","Text":"We know the sine function,"},{"Start":"01:00.500 ","End":"01:02.090","Text":"the basic one very well."},{"Start":"01:02.090 ","End":"01:06.285","Text":"This is 0 Pi, 2 Pi."},{"Start":"01:06.285 ","End":"01:08.880","Text":"Normally, if we took the,"},{"Start":"01:08.880 ","End":"01:11.040","Text":"prop the sine, the regular sine,"},{"Start":"01:11.040 ","End":"01:12.960","Text":"it would go from 1,"},{"Start":"01:12.960 ","End":"01:17.270","Text":"the top and minus 1 at the bottom."},{"Start":"01:17.270 ","End":"01:21.170","Text":"But because of the absolute value,"},{"Start":"01:21.170 ","End":"01:23.435","Text":"it\u0027s just from 0 to 1."},{"Start":"01:23.435 ","End":"01:25.910","Text":"This part is inverted."},{"Start":"01:25.910 ","End":"01:31.130","Text":"Let\u0027s say that we have the 0 and then Pi would"},{"Start":"01:31.130 ","End":"01:37.520","Text":"be just a bit above 3 and the 2 Pi would be maybe here."},{"Start":"01:37.520 ","End":"01:38.833","Text":"So we have 0,"},{"Start":"01:38.833 ","End":"01:41.160","Text":"Pi, 2 Pi."},{"Start":"01:41.160 ","End":"01:46.495","Text":"We also need Pi over 2, which is,"},{"Start":"01:46.495 ","End":"01:53.820","Text":"I\u0027m not sure exactly somewhere halfway and 3"},{"Start":"01:53.820 ","End":"02:00.390","Text":"Pi over 2, somewhere here."},{"Start":"02:00.390 ","End":"02:02.995","Text":"Yeah, I should mark the 1 here."},{"Start":"02:02.995 ","End":"02:05.780","Text":"Sine starts out regular,"},{"Start":"02:05.780 ","End":"02:07.250","Text":"we have a point here,"},{"Start":"02:07.250 ","End":"02:12.995","Text":"then we have a point here which is 1 and then at Pi down to here again."},{"Start":"02:12.995 ","End":"02:15.145","Text":"Then here we do the inverse."},{"Start":"02:15.145 ","End":"02:17.265","Text":"I\u0027ll use the other color."},{"Start":"02:17.265 ","End":"02:19.100","Text":"Same point here,"},{"Start":"02:19.100 ","End":"02:26.225","Text":"then here we go up to 1 and then down back to 0 again."},{"Start":"02:26.225 ","End":"02:35.369","Text":"Then freehand something like this and like this."},{"Start":"02:35.390 ","End":"02:38.300","Text":"That was 1 period of sine,"},{"Start":"02:38.300 ","End":"02:41.120","Text":"it\u0027s actually 2 periods of absolute value of sine,"},{"Start":"02:41.120 ","End":"02:44.780","Text":"because we see the repetition is already at Pi."},{"Start":"02:44.780 ","End":"02:48.160","Text":"Anyway, I could take this and replicate it a bit,"},{"Start":"02:48.160 ","End":"02:50.840","Text":"and we would end up with something like this,"},{"Start":"02:50.840 ","End":"02:55.200","Text":"computer drawn, and that\u0027s it."}],"ID":10733},{"Watched":false,"Name":"Exercise 6","Duration":"4m 11s","ChapterTopicVideoID":10374,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10374.jpeg","UploadDate":"2017-11-02T15:46:37.5470000","DurationForVideoObject":"PT4M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.680","Text":"In this exercise,"},{"Start":"00:01.680 ","End":"00:07.590","Text":"we want to sketch the graph of y=tan( x+Pi/3)."},{"Start":"00:07.590 ","End":"00:10.320","Text":"Now, in general,"},{"Start":"00:10.320 ","End":"00:11.775","Text":"when you have a function,"},{"Start":"00:11.775 ","End":"00:15.960","Text":"and in this case we could start with y= tangent x."},{"Start":"00:15.960 ","End":"00:22.830","Text":"Then if you replace x by x+Pi/3,"},{"Start":"00:22.830 ","End":"00:25.635","Text":"or in general x+-a,"},{"Start":"00:25.635 ","End":"00:31.635","Text":"this has the effect of taking the graph of this"},{"Start":"00:31.635 ","End":"00:38.685","Text":"and shifting it Pi over 3 units to the left."},{"Start":"00:38.685 ","End":"00:39.960","Text":"When it\u0027s a plus,"},{"Start":"00:39.960 ","End":"00:42.255","Text":"it\u0027s a left shift of Pi over 3."},{"Start":"00:42.255 ","End":"00:47.070","Text":"All we have to do is take the basic tangent of x and shift it."},{"Start":"00:47.070 ","End":"00:49.425","Text":"Now the basic tangent function,"},{"Start":"00:49.425 ","End":"00:51.740","Text":"let\u0027s do a quick freehand sketch."},{"Start":"00:51.740 ","End":"00:56.930","Text":"We have the main branch or"},{"Start":"00:56.930 ","End":"01:02.890","Text":"period is from minus Pi over 2,"},{"Start":"01:02.890 ","End":"01:06.695","Text":"where we have an asymptote to plus Pi over 2,"},{"Start":"01:06.695 ","End":"01:13.710","Text":"where we have an asymptote remember this is Pi over 2 minus Pi over 2."},{"Start":"01:13.710 ","End":"01:21.455","Text":"The main points we could sketch would be the origin at Pi/4=1."},{"Start":"01:21.455 ","End":"01:26.180","Text":"At minus Pi over 4 it\u0027s equal to minus 1."},{"Start":"01:26.180 ","End":"01:30.440","Text":"Then we go towards the asymptote, something like this."},{"Start":"01:30.440 ","End":"01:32.210","Text":"Then we repeat it."},{"Start":"01:32.210 ","End":"01:35.940","Text":"This is one branch and it repeats itself."},{"Start":"01:37.370 ","End":"01:41.420","Text":"Here\u0027s a bit of graph paper and what we want to do is take"},{"Start":"01:41.420 ","End":"01:46.100","Text":"this and shift it to the left, Pi over 3."},{"Start":"01:46.100 ","End":"01:51.860","Text":"Now, Pi is a bit more than 3 so Pi over 3 is a over 1,"},{"Start":"01:51.860 ","End":"01:54.695","Text":"but it\u0027s to the left, so it\u0027s minus Pi over 3."},{"Start":"01:54.695 ","End":"02:03.600","Text":"We want to take this graph and just make it so that this vertical axis happens here."},{"Start":"02:06.020 ","End":"02:09.215","Text":"Let\u0027s see what happens to this interval."},{"Start":"02:09.215 ","End":"02:14.730","Text":"We just subtract Pi over"},{"Start":"02:14.730 ","End":"02:20.550","Text":"3 from these Pi over 2 minus Pi over 3 is Pi over 6,"},{"Start":"02:20.550 ","End":"02:24.470","Text":"which is roughly a half,"},{"Start":"02:24.470 ","End":"02:29.310","Text":"just a bit over 0.5."},{"Start":"02:30.760 ","End":"02:34.880","Text":"Maybe I didn\u0027t leave enough room exactly."},{"Start":"02:34.880 ","End":"02:37.940","Text":"I think a half would be somewhere around here, just over a half."},{"Start":"02:37.940 ","End":"02:43.130","Text":"Then in the other direction, somewhere."},{"Start":"02:43.130 ","End":"02:49.280","Text":"Let\u0027s see, minus a 5 sixth Pi"},{"Start":"02:49.280 ","End":"02:54.885","Text":"would be about 2.5 or a bit more so somewhere here."},{"Start":"02:54.885 ","End":"02:59.010","Text":"Here\u0027s where we would get the asymptotes."},{"Start":"03:01.210 ","End":"03:05.220","Text":"Let\u0027s see, we have a point here."},{"Start":"03:05.270 ","End":"03:09.090","Text":"Then we have a point if we take,"},{"Start":"03:09.090 ","End":"03:14.475","Text":"now go 1 unit across or 1 unit up somewhere."},{"Start":"03:14.475 ","End":"03:18.130","Text":"Let\u0027s say here."},{"Start":"03:19.820 ","End":"03:26.050","Text":"Also 1 unit across and 1 unit down, something here."},{"Start":"03:26.150 ","End":"03:30.830","Text":"Then we have an asymptote somewhere."},{"Start":"03:30.830 ","End":"03:34.970","Text":"We just freehand sketch something like this."},{"Start":"03:34.970 ","End":"03:38.870","Text":"I should have maybe draw a dotted line here for the asymptote."},{"Start":"03:38.870 ","End":"03:41.030","Text":"That\u0027s basically it."},{"Start":"03:41.030 ","End":"03:49.870","Text":"Then just replicate it a few times and it\u0027s best done not by hand by computer."},{"Start":"03:49.870 ","End":"03:55.190","Text":"Here we are. This branch here is here."},{"Start":"03:55.190 ","End":"03:59.600","Text":"There\u0027s an asymptote somewhere vertically here"},{"Start":"03:59.600 ","End":"04:04.355","Text":"and here and I draw on extra branch on each side."},{"Start":"04:04.355 ","End":"04:11.220","Text":"This is what we get and that will do."}],"ID":10734},{"Watched":false,"Name":"Exercise 7","Duration":"3m 13s","ChapterTopicVideoID":10375,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10375.jpeg","UploadDate":"2017-11-02T15:46:48.9000000","DurationForVideoObject":"PT3M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, we have to say what is the amplitude and"},{"Start":"00:05.160 ","End":"00:09.510","Text":"what is the period of the function u=cosine Pi t,"},{"Start":"00:09.510 ","End":"00:12.970","Text":"and after that we want to sketch the graph."},{"Start":"00:15.200 ","End":"00:22.170","Text":"Let\u0027s just put some ones in we have 1 cosine of Pi t,"},{"Start":"00:22.170 ","End":"00:27.720","Text":"and the amplitude is the absolute value of 1."},{"Start":"00:27.720 ","End":"00:33.880","Text":"I\u0027ll just write that amplitude = 1."},{"Start":"00:36.110 ","End":"00:46.510","Text":"The period= 2Pi/P,"},{"Start":"00:46.880 ","End":"00:50.770","Text":"so that leaves us with 2."},{"Start":"00:51.950 ","End":"00:56.360","Text":"There is no phase shift and there is no up-down shift."},{"Start":"00:56.360 ","End":"00:58.340","Text":"What we can say,"},{"Start":"00:58.340 ","End":"00:59.690","Text":"because the period is 2,"},{"Start":"00:59.690 ","End":"01:05.065","Text":"we could take the interval to be from 0-2."},{"Start":"01:05.065 ","End":"01:08.540","Text":"It\u0027s a standard cosine curve,"},{"Start":"01:08.540 ","End":"01:12.335","Text":"by the way, there\u0027s no negative here or anything."},{"Start":"01:12.335 ","End":"01:17.550","Text":"The maximum and minimum are just plus or"},{"Start":"01:17.550 ","End":"01:22.640","Text":"minus the amplitude because there\u0027s no up-down shifting,"},{"Start":"01:22.640 ","End":"01:25.730","Text":"so we go from -1 to 1."},{"Start":"01:25.730 ","End":"01:30.395","Text":"Let\u0027s draw one period\u0027s worth of the graph."},{"Start":"01:30.395 ","End":"01:34.325","Text":"Here\u0027s a bit of graph paper, so to speak."},{"Start":"01:34.325 ","End":"01:36.710","Text":"Let\u0027s just label the axis."},{"Start":"01:36.710 ","End":"01:39.745","Text":"This time it\u0027s not y and x it\u0027s u,"},{"Start":"01:39.745 ","End":"01:45.050","Text":"and the horizontal is t. Now we have the 1 and"},{"Start":"01:45.050 ","End":"01:51.870","Text":"-1 already on the axis labeled,"},{"Start":"01:51.870 ","End":"01:54.930","Text":"and 0 and 2 are also there."},{"Start":"01:54.930 ","End":"01:58.330","Text":"We just have to divide this up into 4 bits."},{"Start":"01:58.330 ","End":"02:00.360","Text":"That\u0027s straightforward enough."},{"Start":"02:00.360 ","End":"02:03.660","Text":"0 and 2 will take halfway is here,"},{"Start":"02:03.660 ","End":"02:06.970","Text":"and halfway of each of these."},{"Start":"02:07.130 ","End":"02:14.855","Text":"We know that the general cosine curve is of this shape,"},{"Start":"02:14.855 ","End":"02:20.060","Text":"so we start off at the high point."},{"Start":"02:20.060 ","End":"02:26.660","Text":"Then we go down to the middle line,"},{"Start":"02:26.660 ","End":"02:29.760","Text":"the 0 here,"},{"Start":"02:29.810 ","End":"02:36.180","Text":"then to the low point back to the 0,"},{"Start":"02:36.180 ","End":"02:38.655","Text":"up to the high point."},{"Start":"02:38.655 ","End":"02:43.950","Text":"Then just free hand curve here."},{"Start":"02:43.950 ","End":"02:47.059","Text":"It\u0027s not going to come out very good freehand,"},{"Start":"02:47.059 ","End":"02:50.315","Text":"but there we are, that\u0027s one period."},{"Start":"02:50.315 ","End":"02:54.370","Text":"Then just replicate it to several times."},{"Start":"02:54.370 ","End":"02:58.160","Text":"Of course it\u0027s best done with a computer and not freehand,"},{"Start":"02:58.160 ","End":"03:01.264","Text":"and so here\u0027s this bit here."},{"Start":"03:01.264 ","End":"03:06.560","Text":"I put in an extra period before and after you decide how many of these you want,"},{"Start":"03:06.560 ","End":"03:09.660","Text":"there\u0027s no set rule."},{"Start":"03:09.700 ","End":"03:13.020","Text":"I think we\u0027ll leave it at that."}],"ID":10735},{"Watched":false,"Name":"Exercise 8","Duration":"4m 6s","ChapterTopicVideoID":10376,"CourseChapterTopicPlaylistID":257201,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10376.jpeg","UploadDate":"2017-11-02T15:47:03.7170000","DurationForVideoObject":"PT4M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.910","Text":"In this exercise, we want to state the amplitude and the period of this function."},{"Start":"00:05.910 ","End":"00:09.420","Text":"Y is 2 sine x minus 3."},{"Start":"00:09.420 ","End":"00:12.360","Text":"After that, we want to sketch the graph."},{"Start":"00:12.360 ","End":"00:15.390","Text":"Let\u0027s start with the amplitude."},{"Start":"00:15.390 ","End":"00:17.745","Text":"That\u0027s the most straightforward."},{"Start":"00:17.745 ","End":"00:22.305","Text":"That\u0027s just the absolute value of this 2 here."},{"Start":"00:22.305 ","End":"00:24.525","Text":"It\u0027s just 2."},{"Start":"00:24.525 ","End":"00:26.880","Text":"Let me write it in sine 1x,"},{"Start":"00:26.880 ","End":"00:29.190","Text":"instead of sine x,"},{"Start":"00:29.190 ","End":"00:35.295","Text":"because the period is equal to 2 pi over what\u0027s written here."},{"Start":"00:35.295 ","End":"00:37.560","Text":"It\u0027s 2 pi over 1,"},{"Start":"00:37.560 ","End":"00:40.225","Text":"which is just 2 pi."},{"Start":"00:40.225 ","End":"00:46.010","Text":"Now, this is not enough knowledge for us to do the sketch."},{"Start":"00:46.010 ","End":"00:47.720","Text":"There\u0027s 1 other thing we have to relate to that,"},{"Start":"00:47.720 ","End":"00:49.775","Text":"is this minus 3."},{"Start":"00:49.775 ","End":"00:52.355","Text":"Although there is no phase shift,"},{"Start":"00:52.355 ","End":"00:58.200","Text":"there is a downward shift of minus 3."},{"Start":"00:58.430 ","End":"01:05.525","Text":"Now, the period is 2 pi and there\u0027s no phase shift,"},{"Start":"01:05.525 ","End":"01:08.600","Text":"which means that if we want to draw 1 period,"},{"Start":"01:08.600 ","End":"01:13.435","Text":"we could take the interval from 0 to 2 pi."},{"Start":"01:13.435 ","End":"01:21.740","Text":"But the maximum and the minimum are not the amplitude plus or minus 2."},{"Start":"01:21.740 ","End":"01:24.350","Text":"But we have to take the minus 3 into account,"},{"Start":"01:24.350 ","End":"01:29.760","Text":"so you want minus 3 plus or minus 2."},{"Start":"01:29.760 ","End":"01:34.605","Text":"That gives us, let\u0027s see,"},{"Start":"01:34.605 ","End":"01:43.050","Text":"minus 3 plus 2 is minus 1 and minus 3 minus 2 is minus 5."},{"Start":"01:43.050 ","End":"01:45.660","Text":"Here\u0027s a bit of graph paper."},{"Start":"01:45.660 ","End":"01:48.920","Text":"Now let\u0027s see, horizontally we want from 0,"},{"Start":"01:48.920 ","End":"01:52.840","Text":"we have 0 here to 2 pi."},{"Start":"01:52.840 ","End":"01:54.915","Text":"Pi is 3.142,"},{"Start":"01:54.915 ","End":"01:57.180","Text":"2 pi is 6.28."},{"Start":"01:57.180 ","End":"02:06.230","Text":"I know somewhere around here will be the 2 pi."},{"Start":"02:06.230 ","End":"02:10.070","Text":"Horizontally, we want to go from minus 1,"},{"Start":"02:10.070 ","End":"02:11.675","Text":"which would be here."},{"Start":"02:11.675 ","End":"02:18.525","Text":"Minus 3, minus 5 or minus 1 to minus 5,"},{"Start":"02:18.525 ","End":"02:23.295","Text":"with minus 3 being the center line."},{"Start":"02:23.295 ","End":"02:27.130","Text":"I\u0027ll usually indicate it by y naught."},{"Start":"02:27.170 ","End":"02:29.700","Text":"Now we want to take this 0,"},{"Start":"02:29.700 ","End":"02:32.705","Text":"2 pi and divide it up into 4 intervals."},{"Start":"02:32.705 ","End":"02:35.270","Text":"Let\u0027s see, the half-way will be 1 pi,"},{"Start":"02:35.270 ","End":"02:37.655","Text":"which is just over 3,"},{"Start":"02:37.655 ","End":"02:42.760","Text":"is the 0, 2 pi."},{"Start":"02:42.760 ","End":"02:50.640","Text":"Then just halfway, we can do the computation that we could just by i to halfway between."},{"Start":"02:51.400 ","End":"02:57.080","Text":"This is an ordinary sine curves,"},{"Start":"02:57.080 ","End":"03:01.330","Text":"so it\u0027s going to look something like this."},{"Start":"03:01.370 ","End":"03:06.665","Text":"Let\u0027s say we start at the middle point end here."},{"Start":"03:06.665 ","End":"03:12.455","Text":"Then we want to go up to the maximum here."},{"Start":"03:12.455 ","End":"03:18.665","Text":"Then back down to the middle center line here."},{"Start":"03:18.665 ","End":"03:24.100","Text":"Then here we want to go to the minimum which is the minus 5."},{"Start":"03:24.100 ","End":"03:29.530","Text":"Then finally back up to the center line which is minus 3."},{"Start":"03:29.530 ","End":"03:35.930","Text":"Then I\u0027ll freehand a sine curve."},{"Start":"03:35.930 ","End":"03:39.375","Text":"That\u0027s what 1 period looks like."},{"Start":"03:39.375 ","End":"03:42.140","Text":"I\u0027ll just replicate it a bit."},{"Start":"03:42.140 ","End":"03:45.130","Text":"Computer sketch, something like this."},{"Start":"03:45.130 ","End":"03:46.510","Text":"What we have here was"},{"Start":"03:46.510 ","End":"03:55.510","Text":"just 1 period up to the middle line,"},{"Start":"03:55.510 ","End":"03:57.435","Text":"so that would be the 3."},{"Start":"03:57.435 ","End":"04:06.020","Text":"This is this and then repeating forever in both directions. That\u0027s about it."}],"ID":10736}],"Thumbnail":null,"ID":257201},{"Name":"Angles","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Angles - Part 1","Duration":"5m 3s","ChapterTopicVideoID":10467,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10467.jpeg","UploadDate":"2021-06-28T15:49:37.5600000","DurationForVideoObject":"PT5M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"Our next topic is angles,"},{"Start":"00:01.950 ","End":"00:03.630","Text":"which you\u0027ve probably seen before."},{"Start":"00:03.630 ","End":"00:07.530","Text":"I\u0027m certain you have in geometry and in the Euclidean plane."},{"Start":"00:07.530 ","End":"00:12.080","Text":"Here we\u0027ll talk about angles in the Cartesian plane,"},{"Start":"00:12.080 ","End":"00:19.535","Text":"the xy-plane, where we have a y-axis and an x-axis."},{"Start":"00:19.535 ","End":"00:25.275","Text":"An angle consists of 2 rays."},{"Start":"00:25.275 ","End":"00:29.025","Text":"There will be an initial side and a terminal side,"},{"Start":"00:29.025 ","End":"00:35.617","Text":"and we rotate the initial side by a certain amount until we get to the terminal side,"},{"Start":"00:35.617 ","End":"00:39.660","Text":"and this is called trigonometric angle."},{"Start":"00:39.660 ","End":"00:44.430","Text":"If we turn this in a counterclockwise way,"},{"Start":"00:44.430 ","End":"00:47.085","Text":"it\u0027s called a positive angle."},{"Start":"00:47.085 ","End":"00:51.045","Text":"But we could have gotten from here to here clockwise,"},{"Start":"00:51.045 ","End":"00:55.110","Text":"and then it would\u0027ve been a negative angle."},{"Start":"00:55.110 ","End":"01:03.010","Text":"I could go all the way round and up to here."},{"Start":"01:04.010 ","End":"01:08.940","Text":"There\u0027s more than 1 way of getting from the initial side to the terminal side,"},{"Start":"01:08.940 ","End":"01:12.400","Text":"and we\u0027ll see more of this later."},{"Start":"01:12.950 ","End":"01:15.120","Text":"Let\u0027s put it at the side."},{"Start":"01:15.120 ","End":"01:17.543","Text":"This looks a bit messy,"},{"Start":"01:17.543 ","End":"01:20.760","Text":"but the point is to remember that there\u0027s"},{"Start":"01:20.760 ","End":"01:25.485","Text":"more than 1 way of rotating the initial side to get to the terminal side."},{"Start":"01:25.485 ","End":"01:28.080","Text":"If it\u0027s counterclockwise,"},{"Start":"01:28.080 ","End":"01:31.665","Text":"it\u0027s positive and clockwise it\u0027s considered negative."},{"Start":"01:31.665 ","End":"01:34.080","Text":"Now we actually give a measure to the angle,"},{"Start":"01:34.080 ","End":"01:35.250","Text":"but we\u0027ll see this later."},{"Start":"01:35.250 ","End":"01:38.055","Text":"It will be like degrees or radians."},{"Start":"01:38.055 ","End":"01:40.260","Text":"Perhaps I\u0027ll give 1 more example."},{"Start":"01:40.260 ","End":"01:43.605","Text":"Here\u0027s the initial,"},{"Start":"01:43.605 ","End":"01:45.330","Text":"and I put it in green,"},{"Start":"01:45.330 ","End":"01:46.470","Text":"it\u0027s where we start,"},{"Start":"01:46.470 ","End":"01:49.440","Text":"and here\u0027s the terminal side,"},{"Start":"01:49.440 ","End":"01:51.705","Text":"it\u0027s in red where we stop."},{"Start":"01:51.705 ","End":"01:57.600","Text":"Here\u0027s the angle and we put a little arrow on it to show which way we\u0027re going."},{"Start":"01:57.600 ","End":"02:03.225","Text":"This would be a positive trigonometric angle."},{"Start":"02:03.225 ","End":"02:08.220","Text":"Mostly in this clip I\u0027ll be using the Greek letter Theta for an angle,"},{"Start":"02:08.220 ","End":"02:10.140","Text":"but it could be anything."},{"Start":"02:10.140 ","End":"02:11.595","Text":"I could be using Alpha,"},{"Start":"02:11.595 ","End":"02:13.470","Text":"Beta, even x,"},{"Start":"02:13.470 ","End":"02:17.520","Text":"anything, I just happen to like the Greek letter Theta."},{"Start":"02:17.520 ","End":"02:22.110","Text":"The next concept is an angle in standard position,"},{"Start":"02:22.110 ","End":"02:25.395","Text":"and I\u0027ll bring in a picture."},{"Start":"02:25.395 ","End":"02:30.330","Text":"The standard position means that 1 side is on the x-axis,"},{"Start":"02:30.330 ","End":"02:32.160","Text":"but not just on the x-axis,"},{"Start":"02:32.160 ","End":"02:36.330","Text":"but the vertex has to be at the origin."},{"Start":"02:36.330 ","End":"02:38.580","Text":"In fact, usually,"},{"Start":"02:38.580 ","End":"02:49.140","Text":"we take it that the initial side is the one on the x-axis and the terminal is elsewhere."},{"Start":"02:49.140 ","End":"02:51.060","Text":"In this particular case,"},{"Start":"02:51.060 ","End":"02:55.605","Text":"it\u0027s a positive angle because I\u0027m indicating that this is my angle."},{"Start":"02:55.605 ","End":"02:57.283","Text":"I forgot to say earlier,"},{"Start":"02:57.283 ","End":"03:00.315","Text":"if the terminal side coincides with the initial side,"},{"Start":"03:00.315 ","End":"03:02.655","Text":"then it\u0027s called an angle of 0,"},{"Start":"03:02.655 ","End":"03:04.710","Text":"and it just looks like an array,"},{"Start":"03:04.710 ","End":"03:06.060","Text":"you don\u0027t see the angle."},{"Start":"03:06.060 ","End":"03:12.068","Text":"The next concept I\u0027m going to introduce is that of a quadrantal angle,"},{"Start":"03:12.068 ","End":"03:14.003","Text":"not sure how to pronounce that,"},{"Start":"03:14.003 ","End":"03:16.950","Text":"and then we\u0027ll see what this second part is."},{"Start":"03:16.950 ","End":"03:20.520","Text":"Now I\u0027m still in angles in standard position,"},{"Start":"03:20.520 ","End":"03:23.490","Text":"which means that you actually don\u0027t have to sketch"},{"Start":"03:23.490 ","End":"03:30.153","Text":"the initial side because it\u0027s always the positive x-axis,"},{"Start":"03:30.153 ","End":"03:31.935","Text":"so we just have to show the terminal."},{"Start":"03:31.935 ","End":"03:35.085","Text":"Here I\u0027ve shown 4 different angles."},{"Start":"03:35.085 ","End":"03:42.030","Text":"A quadrantal angle is one which happens to lie on 1 of the axes."},{"Start":"03:42.030 ","End":"03:43.560","Text":"There\u0027s 4 ways that can be,"},{"Start":"03:43.560 ","End":"03:46.380","Text":"it could be as the positive x, the positive y,"},{"Start":"03:46.380 ","End":"03:48.467","Text":"the negative x, or negative y,"},{"Start":"03:48.467 ","End":"03:52.800","Text":"and these are the 4 quadrantal angles."},{"Start":"03:52.800 ","End":"03:57.075","Text":"If it doesn\u0027t fall on top of 1 of the axes,"},{"Start":"03:57.075 ","End":"04:00.900","Text":"then it\u0027s got to fall into 1 of 4 quadrants,"},{"Start":"04:00.900 ","End":"04:03.060","Text":"where this is the first quadrant,"},{"Start":"04:03.060 ","End":"04:05.950","Text":"the second, the third, and the fourth."},{"Start":"04:06.530 ","End":"04:10.020","Text":"Here\u0027s the picture, but I should have explained,"},{"Start":"04:10.020 ","End":"04:11.685","Text":"when I say n here,"},{"Start":"04:11.685 ","End":"04:15.705","Text":"that\u0027s just my shortcut way of saying either I,"},{"Start":"04:15.705 ","End":"04:20.520","Text":"II, III or IV,"},{"Start":"04:20.520 ","End":"04:26.055","Text":"and we just traditionally use Roman numerals for the 4 quadrants."},{"Start":"04:26.055 ","End":"04:31.050","Text":"This picture is an example of a quadrant II angle."},{"Start":"04:31.050 ","End":"04:38.925","Text":"Quadrant II angle means that the terminal side is inside quadrant II."},{"Start":"04:38.925 ","End":"04:41.985","Text":"There\u0027s 4 possibilities."},{"Start":"04:41.985 ","End":"04:46.830","Text":"Here\u0027s another picture showing an examples of all 4,"},{"Start":"04:46.830 ","End":"04:48.300","Text":"quadrant I angle,"},{"Start":"04:48.300 ","End":"04:49.350","Text":"a quadrant II angle,"},{"Start":"04:49.350 ","End":"04:50.490","Text":"a quadrant III angle,"},{"Start":"04:50.490 ","End":"04:53.080","Text":"and quadrant IV angle."},{"Start":"04:53.560 ","End":"04:59.370","Text":"I repeat all these are angles in standard position."},{"Start":"05:00.580 ","End":"05:03.870","Text":"Let\u0027s continue."}],"ID":10886},{"Watched":false,"Name":"Angles - Part 2","Duration":"7m 53s","ChapterTopicVideoID":10468,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10468.jpeg","UploadDate":"2021-06-28T15:50:13.4770000","DurationForVideoObject":"PT7M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"We mentioned earlier the possibility of measurement for an angle."},{"Start":"00:06.000 ","End":"00:09.870","Text":"The only thing we said was that if we have"},{"Start":"00:09.870 ","End":"00:15.120","Text":"an angle with an initial and terminal side and the rotation,"},{"Start":"00:15.120 ","End":"00:17.490","Text":"we just said that it will be positive if"},{"Start":"00:17.490 ","End":"00:21.645","Text":"the rotation is counterclockwise or negative if it\u0027s clockwise."},{"Start":"00:21.645 ","End":"00:23.670","Text":"If these 2 are the same line,"},{"Start":"00:23.670 ","End":"00:25.630","Text":"then measure is 0."},{"Start":"00:25.630 ","End":"00:28.740","Text":"But we didn\u0027t actually assign a number."},{"Start":"00:28.790 ","End":"00:32.810","Text":"There\u0027s 2 main ways of measuring angles."},{"Start":"00:32.810 ","End":"00:35.600","Text":"1 is in radians and 1 is in degrees."},{"Start":"00:35.600 ","End":"00:39.305","Text":"But we\u0027re not going to do an angle in general."},{"Start":"00:39.305 ","End":"00:42.620","Text":"We\u0027ll just do it for angles in standard position."},{"Start":"00:42.620 ","End":"00:48.050","Text":"You can always rotate it so the initial side is facing East,"},{"Start":"00:48.050 ","End":"00:54.260","Text":"I mean, in the direction of the positive x-axis with the vertex at the origin."},{"Start":"00:54.260 ","End":"00:57.130","Text":"It\u0027s no real restriction here."},{"Start":"00:57.130 ","End":"00:59.670","Text":"Here, this would be our angle."},{"Start":"00:59.670 ","End":"01:01.575","Text":"We know it\u0027s going to be positive."},{"Start":"01:01.575 ","End":"01:04.430","Text":"The question is what number do we give it?"},{"Start":"01:04.430 ","End":"01:08.234","Text":"Well, for radians, the way we do it is as follows."},{"Start":"01:08.234 ","End":"01:11.620","Text":"Let\u0027s say we have an angle and at first,"},{"Start":"01:11.620 ","End":"01:14.045","Text":"we\u0027ll take the positive case,"},{"Start":"01:14.045 ","End":"01:17.345","Text":"so we\u0027re going counterclockwise."},{"Start":"01:17.345 ","End":"01:22.970","Text":"Then we can draw a circle around the origin."},{"Start":"01:23.090 ","End":"01:26.310","Text":"Let\u0027s say the radius is r. The thing is,"},{"Start":"01:26.310 ","End":"01:28.875","Text":"it doesn\u0027t matter what r you choose."},{"Start":"01:28.875 ","End":"01:37.210","Text":"Then you measure the arc length just like I\u0027ve shaded here from"},{"Start":"01:37.210 ","End":"01:46.925","Text":"the initial side to the terminal side along the circle."},{"Start":"01:46.925 ","End":"01:50.265","Text":"Let\u0027s say this length is S,"},{"Start":"01:50.265 ","End":"01:55.520","Text":"then we can say that the measure of Theta in"},{"Start":"01:55.520 ","End":"02:03.490","Text":"radians is S/r radians."},{"Start":"02:03.490 ","End":"02:08.554","Text":"It doesn\u0027t matter what r is because it\u0027s all in the same proportion."},{"Start":"02:08.554 ","End":"02:11.015","Text":"I mean, if I double r, I\u0027ll double S,"},{"Start":"02:11.015 ","End":"02:13.745","Text":"and this ratio will be the same,"},{"Start":"02:13.745 ","End":"02:17.465","Text":"and this is the measure of the angle in radians."},{"Start":"02:17.465 ","End":"02:19.520","Text":"If it\u0027s clockwise, you do the same thing."},{"Start":"02:19.520 ","End":"02:21.770","Text":"You just put a minus in front."},{"Start":"02:21.770 ","End":"02:25.790","Text":"Again, you would take the arc length over the radius,"},{"Start":"02:25.790 ","End":"02:28.670","Text":"but you\u0027d make it negative."},{"Start":"02:28.670 ","End":"02:34.640","Text":"I\u0027ll give some important examples for special angles."},{"Start":"02:34.640 ","End":"02:39.680","Text":"First of all, I\u0027ll remind you from geometry that the circumference of"},{"Start":"02:39.680 ","End":"02:45.903","Text":"a circle circumference is 2(Pi) times the radius."},{"Start":"02:45.903 ","End":"02:48.270","Text":"For a whole circle,"},{"Start":"02:48.270 ","End":"02:58.525","Text":"the circumference is our S. This would be like our S and S/r would equal 2(Pi),"},{"Start":"02:58.525 ","End":"03:02.935","Text":"which means that all the way around the circle is 2(Pi)."},{"Start":"03:02.935 ","End":"03:05.610","Text":"We start off at 0,"},{"Start":"03:05.610 ","End":"03:08.320","Text":"we go all the way round, it\u0027s 2(Pi)."},{"Start":"03:08.320 ","End":"03:10.040","Text":"We only go halfway around,"},{"Start":"03:10.040 ","End":"03:12.205","Text":"it\u0027s only going to be Pi,"},{"Start":"03:12.205 ","End":"03:16.190","Text":"and if we go quarter of a circle up to here,"},{"Start":"03:16.190 ","End":"03:19.205","Text":"this angle would be Pi/2,"},{"Start":"03:19.205 ","End":"03:23.405","Text":"and 3/4 of a the circle is 3(pi)/2."},{"Start":"03:23.405 ","End":"03:26.335","Text":"These are important angles to remember."},{"Start":"03:26.335 ","End":"03:28.700","Text":"We\u0027ll get to degrees in a moment."},{"Start":"03:28.700 ","End":"03:31.790","Text":"We all know degrees, this would be like 90 degrees, 180,"},{"Start":"03:31.790 ","End":"03:36.290","Text":"270, but I\u0027ll revisit the concept of degrees."},{"Start":"03:36.290 ","End":"03:40.730","Text":"Note also that this formula can be written in"},{"Start":"03:40.730 ","End":"03:47.749","Text":"the form S=Theta times r or r times Theta."},{"Start":"03:47.749 ","End":"03:51.635","Text":"Which means that if we have the radius and we have the angle,"},{"Start":"03:51.635 ","End":"03:59.180","Text":"then we can compute the arc length along the circle for this angle."},{"Start":"03:59.180 ","End":"04:02.000","Text":"Now let\u0027s talk about degrees."},{"Start":"04:02.000 ","End":"04:03.470","Text":"Well, we know about degrees."},{"Start":"04:03.470 ","End":"04:09.320","Text":"We know that a full circle is 360 degrees."},{"Start":"04:09.320 ","End":"04:11.674","Text":"If we start off at 0 degrees,"},{"Start":"04:11.674 ","End":"04:14.915","Text":"then we know that a quarter circle is 90 degrees,"},{"Start":"04:14.915 ","End":"04:18.320","Text":"a half-circle is 180 degrees,"},{"Start":"04:18.320 ","End":"04:21.400","Text":"3/4 is 270 degrees."},{"Start":"04:21.400 ","End":"04:25.485","Text":"Notice that Pi radians is 180 degrees."},{"Start":"04:25.485 ","End":"04:32.195","Text":"Pi radians is 180 degrees."},{"Start":"04:32.195 ","End":"04:41.180","Text":"That gives us a formula that if we want to convert degrees to radians,"},{"Start":"04:41.180 ","End":"04:50.440","Text":"then we multiply by Pi/180."},{"Start":"04:50.440 ","End":"04:53.265","Text":"And if we want to do the other way round,"},{"Start":"04:53.265 ","End":"04:58.850","Text":"if we wanted to do from radians to degrees, again,"},{"Start":"04:58.850 ","End":"05:01.940","Text":"because this Pi radians is 180 degrees,"},{"Start":"05:01.940 ","End":"05:05.405","Text":"then each radian is 180/Pi."},{"Start":"05:05.405 ","End":"05:09.240","Text":"We multiply by 180/Pi,"},{"Start":"05:09.250 ","End":"05:12.215","Text":"which is like dividing by this."},{"Start":"05:12.215 ","End":"05:15.140","Text":"We can have a formula for going both ways."},{"Start":"05:15.140 ","End":"05:21.770","Text":"For example, we had 30 degrees."},{"Start":"05:21.770 ","End":"05:30.920","Text":"Then we would multiply that by Pi/180 and get Pi/6 radians."},{"Start":"05:30.920 ","End":"05:33.950","Text":"Let me just make a little table."},{"Start":"05:33.950 ","End":"05:35.720","Text":"Let\u0027s do a few of these."},{"Start":"05:35.720 ","End":"05:37.310","Text":"0 is 0."},{"Start":"05:37.310 ","End":"05:39.890","Text":"0 Degrees is 0 radians."},{"Start":"05:39.890 ","End":"05:43.435","Text":"Let\u0027s say at the top it\u0027s degrees,"},{"Start":"05:43.435 ","End":"05:47.525","Text":"at the bottom it\u0027s radians."},{"Start":"05:47.525 ","End":"05:53.145","Text":"Suppose I asked you, Pi/4 radians,"},{"Start":"05:53.145 ","End":"05:58.050","Text":"then multiply by 180/Pi and we get 180/4,"},{"Start":"05:58.050 ","End":"06:01.230","Text":"which is 45 degrees."},{"Start":"06:01.230 ","End":"06:03.480","Text":"I\u0027m going to write a few more,"},{"Start":"06:03.480 ","End":"06:10.920","Text":"60 degrees times Pi/180 is Pi/3."},{"Start":"06:10.920 ","End":"06:13.350","Text":"If I tell you Pi/2 radians,"},{"Start":"06:13.350 ","End":"06:17.560","Text":"well we\u0027ve had this before, that\u0027s 90 degrees."},{"Start":"06:18.140 ","End":"06:20.265","Text":"Let\u0027s do a few more."},{"Start":"06:20.265 ","End":"06:27.360","Text":"120 degrees times Pi/180 is 2/3 of Pi,"},{"Start":"06:27.360 ","End":"06:33.720","Text":"or 2(Pi)/3,135 degrees is 3"},{"Start":"06:33.720 ","End":"06:40.920","Text":"times 45, that\u0027s 3(Pi)/4."},{"Start":"06:40.920 ","End":"06:44.400","Text":"Let\u0027s see. I\u0027ll give you an example."},{"Start":"06:44.400 ","End":"06:48.945","Text":"What\u0027s 5(Pi)/6 radians?"},{"Start":"06:48.945 ","End":"06:57.930","Text":"I take 5(Pi)/6 multiply it by 180/Pi."},{"Start":"06:57.930 ","End":"07:03.950","Text":"That comes out to be 150 degrees. A couple more."},{"Start":"07:03.950 ","End":"07:10.230","Text":"180 degrees, we already said is Pi and will have room for 1 more."},{"Start":"07:10.230 ","End":"07:15.960","Text":"270 degrees is 3(Pi)/2,"},{"Start":"07:15.960 ","End":"07:18.089","Text":"2(Pi), 360 degrees,"},{"Start":"07:18.089 ","End":"07:22.735","Text":"and anyway, that will do for now."},{"Start":"07:22.735 ","End":"07:25.565","Text":"When we talk about an angle in degrees,"},{"Start":"07:25.565 ","End":"07:27.860","Text":"it doesn\u0027t always a whole number of degrees."},{"Start":"07:27.860 ","End":"07:29.510","Text":"It could be with a decimal."},{"Start":"07:29.510 ","End":"07:37.425","Text":"But there is an old-fashioned measure of angles,"},{"Start":"07:37.425 ","End":"07:40.790","Text":"where we use degrees,"},{"Start":"07:40.790 ","End":"07:43.405","Text":"minutes, and seconds."},{"Start":"07:43.405 ","End":"07:45.980","Text":"It\u0027s still used today."},{"Start":"07:45.980 ","End":"07:49.560","Text":"1 degree is divided in."}],"ID":10887},{"Watched":false,"Name":"Angles - Part 3","Duration":"5m 3s","ChapterTopicVideoID":10469,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10469.jpeg","UploadDate":"2021-06-28T15:50:33.1970000","DurationForVideoObject":"PT5M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"Our next topic is angles,"},{"Start":"00:01.950 ","End":"00:03.600","Text":"which you\u0027ve probably seen before."},{"Start":"00:03.600 ","End":"00:07.530","Text":"I\u0027m certain you have in geometry and in the Euclidean plane."},{"Start":"00:07.530 ","End":"00:12.090","Text":"Here we\u0027ll talk about angles in the Cartesian plane,"},{"Start":"00:12.090 ","End":"00:13.500","Text":"the x y plane,"},{"Start":"00:13.500 ","End":"00:19.545","Text":"where we have a y-axis and an x-axis."},{"Start":"00:19.545 ","End":"00:25.260","Text":"An angle consists of 2 rays."},{"Start":"00:25.260 ","End":"00:29.485","Text":"There will be an initial side and a terminal side."},{"Start":"00:29.485 ","End":"00:35.645","Text":"We rotate the initial side by a certain amount until we get to the terminal side."},{"Start":"00:35.645 ","End":"00:39.665","Text":"This is called trigonometric angle."},{"Start":"00:39.665 ","End":"00:44.420","Text":"If we turn this in a counterclockwise way,"},{"Start":"00:44.420 ","End":"00:47.075","Text":"it\u0027s called a positive angle."},{"Start":"00:47.075 ","End":"00:51.110","Text":"But we could have gotten from here to here clockwise,"},{"Start":"00:51.110 ","End":"00:55.115","Text":"and then it would\u0027ve been a negative angle."},{"Start":"00:55.115 ","End":"01:03.025","Text":"I could go all the way round and up to here."},{"Start":"01:03.025 ","End":"01:07.280","Text":"It could be there\u0027s more than 1 way of getting from"},{"Start":"01:07.280 ","End":"01:12.379","Text":"the initial side to the terminal side then we\u0027ll see more of this later."},{"Start":"01:12.940 ","End":"01:15.110","Text":"Let\u0027s put it at the side."},{"Start":"01:15.110 ","End":"01:17.255","Text":"This looks a bit messy."},{"Start":"01:17.255 ","End":"01:20.750","Text":"But the point is to remember that there\u0027s"},{"Start":"01:20.750 ","End":"01:25.475","Text":"more than 1 way of rotating the initial side to get to the terminal side."},{"Start":"01:25.475 ","End":"01:31.670","Text":"If it\u0027s counterclockwise, it\u0027s positive and clockwise it\u0027s considered negative."},{"Start":"01:31.670 ","End":"01:34.070","Text":"Now we actually give a measure to the angle,"},{"Start":"01:34.070 ","End":"01:35.270","Text":"but we\u0027ll see this later."},{"Start":"01:35.270 ","End":"01:38.035","Text":"It will be like degrees or radians."},{"Start":"01:38.035 ","End":"01:40.250","Text":"Perhaps I\u0027ll give 1 more example."},{"Start":"01:40.250 ","End":"01:43.700","Text":"Here\u0027s the initial,"},{"Start":"01:43.700 ","End":"01:46.460","Text":"and I put it in green, is where we start."},{"Start":"01:46.460 ","End":"01:49.430","Text":"Here\u0027s the terminal side,"},{"Start":"01:49.430 ","End":"01:51.695","Text":"it\u0027s in red where we stop."},{"Start":"01:51.695 ","End":"01:57.605","Text":"Here\u0027s the angle and we put a little arrow on it to show which way we\u0027re going."},{"Start":"01:57.605 ","End":"02:03.200","Text":"This would be a positive angle, a trigonometric angle."},{"Start":"02:03.200 ","End":"02:08.210","Text":"Mostly in this clip, I\u0027ll be using the Greek letter Theta for an angle,"},{"Start":"02:08.210 ","End":"02:10.130","Text":"but it could be anything."},{"Start":"02:10.130 ","End":"02:13.460","Text":"Could be using Alpha, Beta, even x,"},{"Start":"02:13.460 ","End":"02:17.540","Text":"anything, I just happen to like the Greek letter Theta."},{"Start":"02:17.540 ","End":"02:25.390","Text":"The next concept is an angle in standard position. I\u0027ll bring in a picture."},{"Start":"02:25.390 ","End":"02:30.320","Text":"The standard position means that 1 side is on the x-axis,"},{"Start":"02:30.320 ","End":"02:32.150","Text":"but not just on the x-axis,"},{"Start":"02:32.150 ","End":"02:36.335","Text":"but the vertex has to be at the origin."},{"Start":"02:36.335 ","End":"02:42.650","Text":"In fact, usually, we take it that the initial side"},{"Start":"02:42.650 ","End":"02:49.130","Text":"is the one on the x-axis and the terminal elsewhere."},{"Start":"02:49.130 ","End":"02:51.050","Text":"In this particular case,"},{"Start":"02:51.050 ","End":"02:55.190","Text":"it\u0027s a positive angle because I\u0027m indicating that this is my angle."},{"Start":"02:55.190 ","End":"02:57.277","Text":"I forgot to say earlier,"},{"Start":"02:57.277 ","End":"03:00.320","Text":"if the terminal side coincide with the initial side,"},{"Start":"03:00.320 ","End":"03:02.605","Text":"then it\u0027s called an angle of zero."},{"Start":"03:02.605 ","End":"03:04.670","Text":"It just looks like a ray."},{"Start":"03:04.670 ","End":"03:06.050","Text":"You don\u0027t see the angle."},{"Start":"03:06.050 ","End":"03:12.110","Text":"The next concept I\u0027m going to introduce is that of a quadrantal angle."},{"Start":"03:12.110 ","End":"03:13.940","Text":"Not sure how to pronounce that."},{"Start":"03:13.940 ","End":"03:17.195","Text":"Then we\u0027ll see what this second part is."},{"Start":"03:17.195 ","End":"03:20.510","Text":"Now I\u0027m still in angles in standard position,"},{"Start":"03:20.510 ","End":"03:23.510","Text":"which means that you actually don\u0027t have to sketch"},{"Start":"03:23.510 ","End":"03:29.930","Text":"the initial side because it\u0027s always the positive x-axis."},{"Start":"03:29.930 ","End":"03:31.925","Text":"We just have to show the terminal."},{"Start":"03:31.925 ","End":"03:35.176","Text":"Here I\u0027ve shown 4 different angles."},{"Start":"03:35.176 ","End":"03:42.030","Text":"A quadrantal angle is one which happens to lie on 1 of the axes."},{"Start":"03:42.030 ","End":"03:43.580","Text":"There\u0027s 4 ways that can be,"},{"Start":"03:43.580 ","End":"03:45.575","Text":"could be the positive x,"},{"Start":"03:45.575 ","End":"03:48.830","Text":"the positive y, the negative x, or the negative y."},{"Start":"03:48.830 ","End":"03:52.810","Text":"These are the 4 quadrantal angles."},{"Start":"03:52.810 ","End":"03:57.080","Text":"If it doesn\u0027t fall on top of 1 of the axes,"},{"Start":"03:57.080 ","End":"04:00.905","Text":"then it\u0027s got a fall into 1 of 4 quadrants,"},{"Start":"04:00.905 ","End":"04:03.050","Text":"where this is the first quadrant,"},{"Start":"04:03.050 ","End":"04:05.940","Text":"the second, the third, and the fourth."},{"Start":"04:06.530 ","End":"04:10.040","Text":"Here\u0027s the picture, but I should have explained,"},{"Start":"04:10.040 ","End":"04:11.690","Text":"when I say n here,"},{"Start":"04:11.690 ","End":"04:15.895","Text":"that\u0027s just my shortcut way of saying either 1,"},{"Start":"04:15.895 ","End":"04:20.530","Text":"2, 3 or 4,"},{"Start":"04:20.530 ","End":"04:26.075","Text":"and we just traditionally use Roman numerals for the 4 quadrants."},{"Start":"04:26.075 ","End":"04:31.045","Text":"This picture is an example of a quadrant 2 angle."},{"Start":"04:31.045 ","End":"04:38.930","Text":"Quadrant 2 angle means that the terminal side is inside quadrant 2."},{"Start":"04:38.930 ","End":"04:42.005","Text":"There\u0027s 4 possibilities."},{"Start":"04:42.005 ","End":"04:47.630","Text":"Here\u0027s another picture showing an examples of all 4 quadrant,"},{"Start":"04:47.630 ","End":"04:48.785","Text":"1 angle a quadrant,"},{"Start":"04:48.785 ","End":"04:49.850","Text":"2 angle a quadrant,"},{"Start":"04:49.850 ","End":"04:53.100","Text":"3 angle on a quadrant, 4 angle."},{"Start":"04:53.380 ","End":"05:00.371","Text":"I repeat, all these are angles in standard position."},{"Start":"05:00.371 ","End":"05:03.860","Text":"Let\u0027s continue."}],"ID":10888},{"Watched":false,"Name":"Angles - Part 4","Duration":"6m 39s","ChapterTopicVideoID":10470,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10470.jpeg","UploadDate":"2021-06-28T15:50:56.7700000","DurationForVideoObject":"PT6M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"Now we\u0027re going to talk about trigonometric functions of angles."},{"Start":"00:03.945 ","End":"00:08.010","Text":"This might seem like a strange topic."},{"Start":"00:08.010 ","End":"00:13.620","Text":"Because if you\u0027ve studied trigonometry in high school or college,"},{"Start":"00:13.620 ","End":"00:18.150","Text":"then you might wonder what else is there to take trigonometric functions of?"},{"Start":"00:18.150 ","End":"00:19.635","Text":"If I talk about,"},{"Start":"00:19.635 ","End":"00:21.060","Text":"I don\u0027t know, sine,"},{"Start":"00:21.060 ","End":"00:24.420","Text":"cosine and tangent, etc.,"},{"Start":"00:24.420 ","End":"00:27.705","Text":"where do we apply these two, if not angles?"},{"Start":"00:27.705 ","End":"00:30.270","Text":"But remember, in this course,"},{"Start":"00:30.270 ","End":"00:34.635","Text":"we took sine, cosine and tangent of numbers."},{"Start":"00:34.635 ","End":"00:39.790","Text":"There was that unit circle and going along from the 1, 0."},{"Start":"00:39.790 ","End":"00:47.240","Text":"So far we\u0027ve taken trigonometric function of numbers, not of angles."},{"Start":"00:47.240 ","End":"00:51.905","Text":"Now, I want to bring it back to angles,"},{"Start":"00:51.905 ","End":"00:57.200","Text":"and the answer is very simple."},{"Start":"00:57.200 ","End":"00:59.240","Text":"If we\u0027re in radians,"},{"Start":"00:59.240 ","End":"01:01.670","Text":"if I say, for example,"},{"Start":"01:01.670 ","End":"01:05.645","Text":"that Theta is an angle,"},{"Start":"01:05.645 ","End":"01:12.240","Text":"and let\u0027s say it\u0027s equal to Pi/6."},{"Start":"01:13.810 ","End":"01:18.530","Text":"I\u0027ll explicitly write radians, it\u0027s an angle."},{"Start":"01:18.530 ","End":"01:23.720","Text":"How do I find out what sine Theta equals,"},{"Start":"01:23.720 ","End":"01:28.025","Text":"cosine Theta equals, and so on."},{"Start":"01:28.025 ","End":"01:30.785","Text":"The sixth trigonometric functions in all."},{"Start":"01:30.785 ","End":"01:35.405","Text":"What we do is we convert from a radian to a number."},{"Start":"01:35.405 ","End":"01:39.320","Text":"We take the same Pi/6,"},{"Start":"01:39.320 ","End":"01:46.520","Text":"but we look at it this time as a number rather than as an angle."},{"Start":"01:46.520 ","End":"01:49.400","Text":"Then we know how to apply sine,"},{"Start":"01:49.400 ","End":"01:51.215","Text":"cosine and so on."},{"Start":"01:51.215 ","End":"01:55.475","Text":"We started off with an angle Theta equals Pi/6,"},{"Start":"01:55.475 ","End":"01:57.410","Text":"and then we move to a number,"},{"Start":"01:57.410 ","End":"02:01.235","Text":"let\u0027s say t equals Pi/6."},{"Start":"02:01.235 ","End":"02:07.650","Text":"Then we take sine of t is sine of Pi/6."},{"Start":"02:07.650 ","End":"02:09.435","Text":"If you look it up, it\u0027s a half,"},{"Start":"02:09.435 ","End":"02:15.510","Text":"and cosine of t is root 3/2 and so on."},{"Start":"02:15.510 ","End":"02:22.460","Text":"We would say that the sine of Theta and the cosine of Theta are the same thing."},{"Start":"02:22.460 ","End":"02:24.980","Text":"This is all very well for radians."},{"Start":"02:24.980 ","End":"02:27.170","Text":"Now if it\u0027s in degrees,"},{"Start":"02:27.170 ","End":"02:29.045","Text":"if I ask you,"},{"Start":"02:29.045 ","End":"02:34.800","Text":"what is sine of 30 degrees?"},{"Start":"02:34.810 ","End":"02:42.500","Text":"What is cosine 30 degrees and tangent and so on?"},{"Start":"02:42.500 ","End":"02:48.380","Text":"Then we convert the 30 degrees to radians."},{"Start":"02:48.380 ","End":"02:53.345","Text":"We say we have an angle of 30 degrees and that angle is equal to Pi/6,"},{"Start":"02:53.345 ","End":"02:57.875","Text":"and then from radians we just talk about instead of an angle,"},{"Start":"02:57.875 ","End":"02:59.840","Text":"we talked about a number,"},{"Start":"02:59.840 ","End":"03:03.680","Text":"t being Pi /6 from angle,"},{"Start":"03:03.680 ","End":"03:07.520","Text":"just repeating what I did above to number."},{"Start":"03:07.520 ","End":"03:13.230","Text":"Then we can say that sine of"},{"Start":"03:13.230 ","End":"03:18.810","Text":"30 degrees is sine of the number Pi/6,"},{"Start":"03:18.810 ","End":"03:22.380","Text":"which equals 1.5 and so on."},{"Start":"03:22.380 ","End":"03:27.440","Text":"To summarize, if it\u0027s in degrees, convert to radians."},{"Start":"03:27.440 ","End":"03:28.685","Text":"If it\u0027s in radians,"},{"Start":"03:28.685 ","End":"03:30.200","Text":"leave the number as is,"},{"Start":"03:30.200 ","End":"03:31.400","Text":"and then just apply sine,"},{"Start":"03:31.400 ","End":"03:36.090","Text":"cosine tangent to that quantity as a number."},{"Start":"03:36.550 ","End":"03:45.440","Text":"Next, I wanted to discuss another definition using the concept of a ratio."},{"Start":"03:45.440 ","End":"03:49.490","Text":"I\u0027ll draw a picture of an angle."},{"Start":"03:49.490 ","End":"03:53.480","Text":"Let\u0027s say this is the initial side,"},{"Start":"03:53.480 ","End":"03:59.210","Text":"and say this is the terminal side."},{"Start":"03:59.210 ","End":"04:01.610","Text":"This is a positive angle."},{"Start":"04:01.610 ","End":"04:03.605","Text":"We\u0027ll assume this is the angle."},{"Start":"04:03.605 ","End":"04:08.960","Text":"What we do is we choose any point on the terminal side,"},{"Start":"04:08.960 ","End":"04:10.775","Text":"but not the origin though."},{"Start":"04:10.775 ","End":"04:12.930","Text":"Let\u0027s say this point here."},{"Start":"04:12.930 ","End":"04:19.590","Text":"Let\u0027s say this is the point P whose coordinates are x, y."},{"Start":"04:21.080 ","End":"04:23.300","Text":"This is the origin."},{"Start":"04:23.300 ","End":"04:25.790","Text":"I didn\u0027t draw the axis in."},{"Start":"04:25.790 ","End":"04:28.100","Text":"What we do is,"},{"Start":"04:28.100 ","End":"04:32.840","Text":"first of all compute quantity r,"},{"Start":"04:32.840 ","End":"04:36.755","Text":"which is the distance from P to the origin,"},{"Start":"04:36.755 ","End":"04:43.405","Text":"which is the square root of x squared plus y squared."},{"Start":"04:43.405 ","End":"04:47.790","Text":"Then we define, I\u0027ll do three first."},{"Start":"04:47.790 ","End":"04:54.230","Text":"The sine is y/r,"},{"Start":"04:54.230 ","End":"04:57.095","Text":"the cosine is x/r,"},{"Start":"04:57.095 ","End":"05:01.380","Text":"and the tangent is y/x."},{"Start":"05:01.610 ","End":"05:06.735","Text":"Now the other three, cosecant Theta."},{"Start":"05:06.735 ","End":"05:09.060","Text":"We need secant Theta,"},{"Start":"05:09.060 ","End":"05:12.010","Text":"and we need cotangent Theta."},{"Start":"05:12.010 ","End":"05:14.270","Text":"These are the reciprocals of these,"},{"Start":"05:14.270 ","End":"05:22.790","Text":"r/y, r/x, x/y."},{"Start":"05:22.790 ","End":"05:25.745","Text":"Just to go for a concrete example,"},{"Start":"05:25.745 ","End":"05:32.790","Text":"let\u0027s suppose this was the 15, 8."},{"Start":"05:32.790 ","End":"05:36.375","Text":"Then it means that x is 15,"},{"Start":"05:36.375 ","End":"05:40.580","Text":"y is eight, and r equals."},{"Start":"05:40.580 ","End":"05:42.080","Text":"If you do the computation,"},{"Start":"05:42.080 ","End":"05:49.190","Text":"it will come out 17 because the square root of 15 squared is 225,"},{"Start":"05:49.190 ","End":"05:52.085","Text":"y squared is 64."},{"Start":"05:52.085 ","End":"05:58.695","Text":"That comes out to be root 289 and root of 299 is 17."},{"Start":"05:58.695 ","End":"06:00.900","Text":"So this is the x, the y,"},{"Start":"06:00.900 ","End":"06:05.220","Text":"and the r. Then we would have,"},{"Start":"06:05.220 ","End":"06:09.095","Text":"I have just wrote these copied from here, change the color."},{"Start":"06:09.095 ","End":"06:12.260","Text":"Sine Theta is y/r."},{"Start":"06:12.260 ","End":"06:17.180","Text":"It\u0027s just line substitution 8/17."},{"Start":"06:17.180 ","End":"06:22.190","Text":"Here we have 15/17,"},{"Start":"06:22.190 ","End":"06:25.640","Text":"y/x is 8/15,"},{"Start":"06:25.640 ","End":"06:30.445","Text":"and these are just the upside down of these 17/8,"},{"Start":"06:30.445 ","End":"06:36.830","Text":"17/15 and 15/8."},{"Start":"06:36.830 ","End":"06:40.170","Text":"So that\u0027s the definition as a ratio."}],"ID":10889},{"Watched":false,"Name":"Angles - Part 5","Duration":"7m 24s","ChapterTopicVideoID":10471,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10471.jpeg","UploadDate":"2021-06-28T15:51:27.1530000","DurationForVideoObject":"PT7M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.125","Text":"Now let\u0027s talk about another way of defining trigonometric functions for acute angles."},{"Start":"00:07.125 ","End":"00:16.320","Text":"We have a right angle triangle and therefore Theta is an acute angle."},{"Start":"00:16.320 ","End":"00:20.310","Text":"We label the 3 sides opposite,"},{"Start":"00:20.310 ","End":"00:23.050","Text":"adjacent, and hypotenuse."},{"Start":"00:23.050 ","End":"00:27.530","Text":"Now what I did is rotated it because we have"},{"Start":"00:27.530 ","End":"00:32.536","Text":"a definition for triangles in standard position."},{"Start":"00:32.536 ","End":"00:36.080","Text":"Everything\u0027s the same. The hypotenuse,"},{"Start":"00:36.080 ","End":"00:40.620","Text":"is r and the opposite is y and adjacent as length x and"},{"Start":"00:40.620 ","End":"00:45.755","Text":"I copy pasted the definitions that we had with y, r, and x."},{"Start":"00:45.755 ","End":"00:49.310","Text":"But now let\u0027s rewrite these, we\u0027ll abbreviate."},{"Start":"00:49.310 ","End":"00:51.809","Text":"Where I see hypotenuse,"},{"Start":"00:51.809 ","End":"00:53.370","Text":"we\u0027ll put hyp, here,"},{"Start":"00:53.370 ","End":"00:56.790","Text":"we\u0027ll just call it opp and here adj."},{"Start":"00:56.790 ","End":"01:02.520","Text":"Now we can get different definitions."},{"Start":"01:02.650 ","End":"01:05.960","Text":"There are more mnemonic."},{"Start":"01:05.960 ","End":"01:07.150","Text":"You can remember them."},{"Start":"01:07.150 ","End":"01:08.855","Text":"You don\u0027t know what y and x are,"},{"Start":"01:08.855 ","End":"01:10.220","Text":"but you could say that,"},{"Start":"01:10.220 ","End":"01:12.170","Text":"for example, the tangent,"},{"Start":"01:12.170 ","End":"01:13.715","Text":"that\u0027s how I remember it,"},{"Start":"01:13.715 ","End":"01:18.690","Text":"is opposite over adjacent."},{"Start":"01:18.690 ","End":"01:25.490","Text":"Sine is opposite over hypotenuse."},{"Start":"01:25.490 ","End":"01:31.640","Text":"Cosine is adjacent over hypotenuse."},{"Start":"01:31.640 ","End":"01:33.290","Text":"These are the 3 important ones."},{"Start":"01:33.290 ","End":"01:38.725","Text":"The other ones, just flip these, invert them."},{"Start":"01:38.725 ","End":"01:44.870","Text":"That\u0027s for the right angle triangles and with opposite,"},{"Start":"01:44.870 ","End":"01:47.580","Text":"adjacent, and hypotenuse."},{"Start":"01:47.730 ","End":"01:52.270","Text":"Now if we know the trigonometric functions of acute angles,"},{"Start":"01:52.270 ","End":"01:58.105","Text":"we can talk about any angle using the concept of a reference angle."},{"Start":"01:58.105 ","End":"02:00.695","Text":"I\u0027ll show you what I mean."},{"Start":"02:00.695 ","End":"02:08.005","Text":"I brought a diagram that we had earlier showing the different quadrants."},{"Start":"02:08.005 ","End":"02:12.313","Text":"These are 4 examples of angles."},{"Start":"02:12.313 ","End":"02:15.805","Text":"In the first quadrant this would be Theta,"},{"Start":"02:15.805 ","End":"02:18.475","Text":"and in the second quadrant,"},{"Start":"02:18.475 ","End":"02:20.967","Text":"this would be my Theta."},{"Start":"02:20.967 ","End":"02:22.330","Text":"I don\u0027t want to clutter it up."},{"Start":"02:22.330 ","End":"02:25.630","Text":"I want to explain what a reference angle is."},{"Start":"02:25.630 ","End":"02:31.275","Text":"It\u0027s the angle with the x-axis that makes it acute."},{"Start":"02:31.275 ","End":"02:34.835","Text":"In quadrant 1 for this angle,"},{"Start":"02:34.835 ","End":"02:38.570","Text":"this thing itself would be the reference angle."},{"Start":"02:38.570 ","End":"02:42.025","Text":"For this angle in quadrant 2,"},{"Start":"02:42.025 ","End":"02:43.130","Text":"this is the angle,"},{"Start":"02:43.130 ","End":"02:45.665","Text":"but the reference angle would be this."},{"Start":"02:45.665 ","End":"02:48.305","Text":"It\u0027s the angle with the x-axis,"},{"Start":"02:48.305 ","End":"02:50.175","Text":"but not this 1,"},{"Start":"02:50.175 ","End":"02:51.895","Text":"the 1 that makes it acute."},{"Start":"02:51.895 ","End":"02:54.005","Text":"So in the third quadrant,"},{"Start":"02:54.005 ","End":"02:56.869","Text":"this would be the reference angle,"},{"Start":"02:56.869 ","End":"02:58.429","Text":"and in the fourth quadrant,"},{"Start":"02:58.429 ","End":"03:01.645","Text":"this would be the reference angle."},{"Start":"03:01.645 ","End":"03:06.110","Text":"It turns out that these trigonometric functions of"},{"Start":"03:06.110 ","End":"03:09.275","Text":"the reference angle are the"},{"Start":"03:09.275 ","End":"03:13.790","Text":"same as for the original angle except for the matter of the sine."},{"Start":"03:13.790 ","End":"03:16.910","Text":"So what we do, let\u0027s say in the second quadrant,"},{"Start":"03:16.910 ","End":"03:18.478","Text":"to find the sine, cosine,"},{"Start":"03:18.478 ","End":"03:20.990","Text":"and tangent is to find the sine, cosine,"},{"Start":"03:20.990 ","End":"03:25.130","Text":"or tangent of this 1 and then fix it with a plus or minus."},{"Start":"03:25.130 ","End":"03:28.085","Text":"Now how do we know if it\u0027s plus or minus?"},{"Start":"03:28.085 ","End":"03:30.270","Text":"There\u0027s a mnemonic."},{"Start":"03:30.270 ","End":"03:32.925","Text":"If you remember the phrase,"},{"Start":"03:32.925 ","End":"03:38.350","Text":"all students"},{"Start":"03:38.600 ","End":"03:44.500","Text":"take calculus."},{"Start":"03:44.560 ","End":"03:49.535","Text":"It\u0027s a mnemonic that here I have the letter a here,"},{"Start":"03:49.535 ","End":"03:51.275","Text":"s, here t,"},{"Start":"03:51.275 ","End":"03:53.150","Text":"and here c,"},{"Start":"03:53.150 ","End":"03:55.040","Text":"and the a stands for all,"},{"Start":"03:55.040 ","End":"03:56.180","Text":"just like the word all,"},{"Start":"03:56.180 ","End":"04:00.620","Text":"means all the trigonometric functions are positive here."},{"Start":"04:00.620 ","End":"04:05.195","Text":"For the moment, we\u0027ll just talk about the 3 main ones: the sine,"},{"Start":"04:05.195 ","End":"04:07.895","Text":"cosine, and tangent."},{"Start":"04:07.895 ","End":"04:10.020","Text":"The other 3 we can get from these."},{"Start":"04:10.020 ","End":"04:12.529","Text":"Here, the s stands for sine."},{"Start":"04:12.529 ","End":"04:15.275","Text":"The sine is positive and the other 2 are negative."},{"Start":"04:15.275 ","End":"04:20.135","Text":"Here the tangent is positive in the third quadrant and in the fourth quadrant,"},{"Start":"04:20.135 ","End":"04:23.000","Text":"the cosine is positive."},{"Start":"04:23.000 ","End":"04:26.705","Text":"Here\u0027s a picture I found on the web."},{"Start":"04:26.705 ","End":"04:29.150","Text":"Yeah, all students take calculus."},{"Start":"04:29.150 ","End":"04:30.620","Text":"I\u0027ve seen others,"},{"Start":"04:30.620 ","End":"04:32.825","Text":"add sugar to coffee,"},{"Start":"04:32.825 ","End":"04:35.180","Text":"all sinners take care."},{"Start":"04:35.180 ","End":"04:40.050","Text":"Anyway, the ASTC mnemonic."},{"Start":"04:40.050 ","End":"04:43.270","Text":"ASTC for the quadrants."},{"Start":"04:43.270 ","End":"04:45.260","Text":"This is just for sine,"},{"Start":"04:45.260 ","End":"04:46.610","Text":"cosine, and tangent."},{"Start":"04:46.610 ","End":"04:51.650","Text":"The other 3 are reciprocals of these and have the same sign."},{"Start":"04:51.650 ","End":"04:54.530","Text":"So sine has the same sign,"},{"Start":"04:54.530 ","End":"04:56.105","Text":"sign is ign,"},{"Start":"04:56.105 ","End":"05:01.865","Text":"as cosecant and cosine has the same sign plus minus"},{"Start":"05:01.865 ","End":"05:08.840","Text":"as the secant and tangent has the same sine as cotangent."},{"Start":"05:08.840 ","End":"05:12.790","Text":"Let\u0027s take an example in the second quadrant."},{"Start":"05:12.790 ","End":"05:14.630","Text":"In the second quadrant,"},{"Start":"05:14.630 ","End":"05:17.135","Text":"let\u0027s say that this was,"},{"Start":"05:17.135 ","End":"05:21.950","Text":"I don\u0027t know, 2 Pi over 3, like 120 degrees."},{"Start":"05:21.950 ","End":"05:26.525","Text":"The reference angle here would be 180,"},{"Start":"05:26.525 ","End":"05:29.600","Text":"I mean Pi of course, minus this."},{"Start":"05:29.600 ","End":"05:33.890","Text":"So the reference angle is Pi-2 Pi over 3."},{"Start":"05:33.890 ","End":"05:37.340","Text":"So that\u0027s Pi over 3."},{"Start":"05:37.340 ","End":"05:40.315","Text":"Then if we want the sine, cosine,"},{"Start":"05:40.315 ","End":"05:46.725","Text":"and tangent of 2 Pi over 3, these we know."},{"Start":"05:46.725 ","End":"05:52.250","Text":"These are the reference angles and we are in the second quadrant,"},{"Start":"05:52.250 ","End":"05:53.870","Text":"which has an s on it,"},{"Start":"05:53.870 ","End":"05:59.705","Text":"which means that in the second quadrant the sine would be plus,"},{"Start":"05:59.705 ","End":"06:02.825","Text":"this would be minus and this would be minus."},{"Start":"06:02.825 ","End":"06:08.480","Text":"Now I can say that the sine of 2 Pi over 3, the original angle,"},{"Start":"06:08.480 ","End":"06:10.145","Text":"not the reference angle,"},{"Start":"06:10.145 ","End":"06:15.260","Text":"is plus root 3 over 2."},{"Start":"06:15.260 ","End":"06:22.220","Text":"But the cosine of 2 Pi over 3 is -1/2."},{"Start":"06:22.220 ","End":"06:24.515","Text":"The minus here is important."},{"Start":"06:24.515 ","End":"06:31.710","Text":"The tangent of 2 Pi over 3 is minus root 3."},{"Start":"06:31.710 ","End":"06:33.945","Text":"So we take the sine, cosine,"},{"Start":"06:33.945 ","End":"06:38.600","Text":"tangent of the reference angle or whichever 1 we need,"},{"Start":"06:38.600 ","End":"06:40.325","Text":"we might just need the cosine,"},{"Start":"06:40.325 ","End":"06:44.910","Text":"and then we add the sign according to this."},{"Start":"06:46.000 ","End":"06:48.800","Text":"If I really want to complete the picture,"},{"Start":"06:48.800 ","End":"06:51.065","Text":"I could do the cosecant,"},{"Start":"06:51.065 ","End":"06:56.075","Text":"the secant, and the cotangent,"},{"Start":"06:56.075 ","End":"06:59.615","Text":"and the signs would be the same."},{"Start":"06:59.615 ","End":"07:03.450","Text":"You would just flip these."},{"Start":"07:04.250 ","End":"07:08.280","Text":"Cosecant would be 2 over root 3,"},{"Start":"07:08.280 ","End":"07:12.180","Text":"secant would be -2,"},{"Start":"07:12.180 ","End":"07:17.995","Text":"and the cotangent would be -1 over root 3."},{"Start":"07:17.995 ","End":"07:24.720","Text":"With this, we\u0027re done with the section on angles."}],"ID":10890},{"Watched":false,"Name":"Exercise 1","Duration":"2m 21s","ChapterTopicVideoID":10377,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10377.jpeg","UploadDate":"2017-11-02T15:53:22.0870000","DurationForVideoObject":"PT2M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"This exercise is here to make sure you understand what the term"},{"Start":"00:04.500 ","End":"00:08.790","Text":"coterminal means and it consists of 2 parts,"},{"Start":"00:08.790 ","End":"00:10.950","Text":"1 in degrees and 1 in radians,"},{"Start":"00:10.950 ","End":"00:15.780","Text":"and we just have to list all the angles coterminal with the given angle in Part A,"},{"Start":"00:15.780 ","End":"00:21.120","Text":"it\u0027s 50 degrees, and in Part B it\u0027s 3 pi over 4 radians."},{"Start":"00:21.120 ","End":"00:26.489","Text":"Now the main idea is to take the angle."},{"Start":"00:26.489 ","End":"00:29.370","Text":"I can, the first part 50 degrees,"},{"Start":"00:29.370 ","End":"00:32.985","Text":"and just add multiples of 360."},{"Start":"00:32.985 ","End":"00:36.120","Text":"If I add 360,"},{"Start":"00:36.120 ","End":"00:43.235","Text":"I will get 410 and if I take that and add 360,"},{"Start":"00:43.235 ","End":"00:49.250","Text":"then I\u0027ll get 770."},{"Start":"00:49.250 ","End":"00:53.330","Text":"You could also subtract 360 and"},{"Start":"00:53.330 ","End":"01:01.790","Text":"get minus 310 and"},{"Start":"01:01.790 ","End":"01:04.535","Text":"subtract another 360 and so on."},{"Start":"01:04.535 ","End":"01:10.190","Text":"Now, there are infinitely many so we can in general say that the solution is,"},{"Start":"01:10.190 ","End":"01:15.320","Text":"50 degrees plus some multiple n could be positive or negative,"},{"Start":"01:15.320 ","End":"01:20.810","Text":"or zero times 360 degrees."},{"Start":"01:20.810 ","End":"01:23.875","Text":"That will be the answer for Part A."},{"Start":"01:23.875 ","End":"01:30.190","Text":"In Part B, we are working in radians on the equivalent to 360 is 2 pi;"},{"Start":"01:30.190 ","End":"01:39.725","Text":"1 example of a coterminal angle with 3 pi over 4 would be to add 2 pi."},{"Start":"01:39.725 ","End":"01:42.430","Text":"I could simplify this; doesn\u0027t matter."},{"Start":"01:42.430 ","End":"01:50.835","Text":"I could add 3 pi over 4 plus 4 pi."},{"Start":"01:50.835 ","End":"01:57.550","Text":"Or we could even subtract 3 pi over 4 minus 2 pi and so on."},{"Start":"01:57.550 ","End":"01:58.990","Text":"These are just examples."},{"Start":"01:58.990 ","End":"02:06.030","Text":"The general expression would be 3 pi over 4 plus some whole multiple."},{"Start":"02:06.030 ","End":"02:13.355","Text":"I\u0027ll write the 2 pi first and then times n, where again,"},{"Start":"02:13.355 ","End":"02:17.145","Text":"n is an integer,"},{"Start":"02:17.145 ","End":"02:22.180","Text":"zero, or positive or negative. That\u0027s it."}],"ID":10737},{"Watched":false,"Name":"Exercise 2","Duration":"3m 28s","ChapterTopicVideoID":10378,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10378.jpeg","UploadDate":"2017-11-02T15:53:35.5430000","DurationForVideoObject":"PT3M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"In this exercise, we want to find all 6 trigonometric functions."},{"Start":"00:05.565 ","End":"00:11.160","Text":"First of 720 degrees and then of minus a 180 degrees."},{"Start":"00:11.160 ","End":"00:13.080","Text":"There\u0027s more than 1 way to do this,"},{"Start":"00:13.080 ","End":"00:14.745","Text":"and here\u0027s what I recommend."},{"Start":"00:14.745 ","End":"00:20.940","Text":"I recommend finding a coterminal angle which is somewhere between 0 and 360,"},{"Start":"00:20.940 ","End":"00:22.705","Text":"then it\u0027s easier to deal with."},{"Start":"00:22.705 ","End":"00:29.315","Text":"If I take the 720 degrees and I subtract twice 360,"},{"Start":"00:29.315 ","End":"00:33.320","Text":"this would be 0 degrees."},{"Start":"00:33.320 ","End":"00:34.490","Text":"If its coterminal,"},{"Start":"00:34.490 ","End":"00:38.675","Text":"it has the same trigonometric functions."},{"Start":"00:38.675 ","End":"00:43.325","Text":"Usually, I also convert this to radians into a more familiar."},{"Start":"00:43.325 ","End":"00:47.850","Text":"It\u0027s like 0 radians."},{"Start":"00:49.060 ","End":"00:51.700","Text":"Let\u0027s see, sin(0),"},{"Start":"00:51.700 ","End":"00:56.470","Text":"we know this very well is 0."},{"Start":"00:57.470 ","End":"01:04.920","Text":"Cosine(0) is 1, tan(0) is 0."},{"Start":"01:04.920 ","End":"01:12.260","Text":"Then we have the cosecant(0) is undefined."},{"Start":"01:12.260 ","End":"01:16.520","Text":"I\u0027ll just write the letter u for undefined,"},{"Start":"01:16.520 ","End":"01:18.950","Text":"it\u0027s actually 1/0,"},{"Start":"01:18.950 ","End":"01:21.110","Text":"cosecant is 1 over sine,"},{"Start":"01:21.110 ","End":"01:24.330","Text":"secant is 1 over cosine,"},{"Start":"01:24.380 ","End":"01:34.020","Text":"it\u0027s 1 and the cotangent of 0 once again is not defined."},{"Start":"01:34.390 ","End":"01:40.165","Text":"That\u0027s part a. In part b,"},{"Start":"01:40.165 ","End":"01:44.055","Text":"we have minus 180 degrees."},{"Start":"01:44.055 ","End":"01:48.350","Text":"Like I said, 1 method that I suggest is just to add or subtract"},{"Start":"01:48.350 ","End":"01:53.685","Text":"multiples of 360 to bring it to between 0 and 360."},{"Start":"01:53.685 ","End":"01:55.665","Text":"If I add 360,"},{"Start":"01:55.665 ","End":"01:58.125","Text":"it\u0027s like a 180 degrees,"},{"Start":"01:58.125 ","End":"02:02.265","Text":"then if I put it in radians, that\u0027s pi."},{"Start":"02:02.265 ","End":"02:10.115","Text":"We are in familiar territory because we know that sin(pi) is 0."},{"Start":"02:10.115 ","End":"02:15.420","Text":"We know that cosine(pi) is minus 1,"},{"Start":"02:18.800 ","End":"02:22.795","Text":"sine over cosine is 1 way, if you\u0027ve forgotten,"},{"Start":"02:22.795 ","End":"02:25.449","Text":"is 0 over minus 1 is 0,"},{"Start":"02:25.449 ","End":"02:28.450","Text":"then the cosecant is 1 over the sine,"},{"Start":"02:28.450 ","End":"02:30.385","Text":"I can use the reciprocal."},{"Start":"02:30.385 ","End":"02:33.010","Text":"You have forgot to write the pi."},{"Start":"02:33.010 ","End":"02:38.650","Text":"Then we want secant of pi and we want cotangent of pi."},{"Start":"02:38.650 ","End":"02:43.225","Text":"I wrote them this way so that these are the reciprocals of these, so 1/0,"},{"Start":"02:43.225 ","End":"02:45.880","Text":"that\u0027s undefined, 1 over minus 1,"},{"Start":"02:45.880 ","End":"02:47.470","Text":"that\u0027s minus 1,"},{"Start":"02:47.470 ","End":"02:50.540","Text":"1/0, once again undefined."},{"Start":"02:51.290 ","End":"02:53.470","Text":"That would be the answer."},{"Start":"02:53.470 ","End":"02:58.865","Text":"Instead of pi, we would put back the original angle minus 180 degrees."},{"Start":"02:58.865 ","End":"03:06.515","Text":"I would say like sin(-180) degrees is 0 and so on for the other 5."},{"Start":"03:06.515 ","End":"03:16.070","Text":"Perhaps here also, I would maybe go back and rewrite these as sin(720) degrees is 0,"},{"Start":"03:16.070 ","End":"03:18.980","Text":"cosine(720) degrees is 1,"},{"Start":"03:18.980 ","End":"03:21.750","Text":"and so on for the others."},{"Start":"03:21.950 ","End":"03:28.720","Text":"I\u0027m not going to do that, but you could. That\u0027s it."}],"ID":10738},{"Watched":false,"Name":"Exercise 3","Duration":"1m 49s","ChapterTopicVideoID":10379,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10379.jpeg","UploadDate":"2017-11-02T15:53:42.3430000","DurationForVideoObject":"PT1M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This exercise is mainly to go over"},{"Start":"00:02.820 ","End":"00:11.625","Text":"the term supplementary angle and use the Alpha naught Theta or any letter will do."},{"Start":"00:11.625 ","End":"00:14.745","Text":"There\u0027s no big deal here."},{"Start":"00:14.745 ","End":"00:19.110","Text":"Just have to remember the supplementary in degrees means 180"},{"Start":"00:19.110 ","End":"00:23.760","Text":"minus the angle and in radians,"},{"Start":"00:23.760 ","End":"00:26.400","Text":"180 degrees is Pi minus this."},{"Start":"00:26.400 ","End":"00:31.995","Text":"Here it\u0027s radians if it\u0027s not given any units, it\u0027s radians."},{"Start":"00:31.995 ","End":"00:38.130","Text":"We want to say Pi minus Pi/6 and that\u0027s the answer,"},{"Start":"00:38.130 ","End":"00:39.405","Text":"but we want to simplify it,"},{"Start":"00:39.405 ","End":"00:41.865","Text":"so 1 minus 5/6."},{"Start":"00:41.865 ","End":"00:47.145","Text":"The answer is 5/6 Pi or 5 Pi/6."},{"Start":"00:47.145 ","End":"00:49.755","Text":"In part b,"},{"Start":"00:49.755 ","End":"00:53.490","Text":"where in degrees and minutes."},{"Start":"00:53.490 ","End":"00:56.650","Text":"I\u0027m going to subtract"},{"Start":"01:02.770 ","End":"01:11.515","Text":"180 degrees minus 42 degrees and 25 minutes."},{"Start":"01:11.515 ","End":"01:15.680","Text":"If you\u0027re not sure how to do this without a calculator,"},{"Start":"01:15.680 ","End":"01:20.150","Text":"one way would be to break up one of the degrees into minutes."},{"Start":"01:20.150 ","End":"01:27.495","Text":"I could say 179 degrees and 60 minutes minus 42,"},{"Start":"01:27.495 ","End":"01:33.840","Text":"25 and that would equal 179 minus"},{"Start":"01:33.840 ","End":"01:42.800","Text":"42 is 137 degrees and 60 minus 25 is 35,"},{"Start":"01:42.800 ","End":"01:46.625","Text":"so that would be the answer."},{"Start":"01:46.625 ","End":"01:50.160","Text":"That\u0027s all there is to this exercise."}],"ID":10739},{"Watched":false,"Name":"Exercise 4","Duration":"3m 26s","ChapterTopicVideoID":10380,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10380.jpeg","UploadDate":"2017-11-02T15:53:53.1730000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.270","Text":"In this exercise, we\u0027re going to convert 4 radians to degrees,"},{"Start":"00:06.270 ","End":"00:08.100","Text":"minutes, and seconds."},{"Start":"00:08.100 ","End":"00:12.240","Text":"Now if it was just to convert to degrees, that\u0027s straightforward."},{"Start":"00:12.240 ","End":"00:16.875","Text":"What we could say is that we would take the 4,"},{"Start":"00:16.875 ","End":"00:22.785","Text":"and when we convert radians to degrees,"},{"Start":"00:22.785 ","End":"00:30.990","Text":"we multiply by 180 over Pi."},{"Start":"00:30.990 ","End":"00:35.110","Text":"That\u0027 basically because a 180 degrees is Pi radians."},{"Start":"00:35.110 ","End":"00:38.960","Text":"If we do this from a simple calculator,"},{"Start":"00:38.960 ","End":"00:45.075","Text":"what we get, let\u0027s see 229."},{"Start":"00:45.075 ","End":"00:48.375","Text":"I\u0027m just going to use 6 decimal places,"},{"Start":"00:48.375 ","End":"00:53.985","Text":"183,118, I\u0027m copying from the calculator."},{"Start":"00:53.985 ","End":"00:57.405","Text":"Now that\u0027s not degrees in minutes."},{"Start":"00:57.405 ","End":"01:00.395","Text":"Most modern calculators do have degrees,"},{"Start":"01:00.395 ","End":"01:01.685","Text":"minutes, and seconds."},{"Start":"01:01.685 ","End":"01:03.875","Text":"If you have such a calculator,"},{"Start":"01:03.875 ","End":"01:06.875","Text":"then there\u0027s no need to follow the rest."},{"Start":"01:06.875 ","End":"01:09.095","Text":"But it might be instructive."},{"Start":"01:09.095 ","End":"01:13.850","Text":"Supposing you have just a very basic calculator that doesn\u0027t have these conversions,"},{"Start":"01:13.850 ","End":"01:15.320","Text":"what you would do is say, okay,"},{"Start":"01:15.320 ","End":"01:17.075","Text":"we have 229 degrees,"},{"Start":"01:17.075 ","End":"01:22.235","Text":"now what is this decimal fraction in minutes and seconds?"},{"Start":"01:22.235 ","End":"01:24.260","Text":"First of all, let\u0027s take care of the minutes."},{"Start":"01:24.260 ","End":"01:34.725","Text":"Now this decimal is 183,118 over a million,"},{"Start":"01:34.725 ","End":"01:38.594","Text":"6 zeros because it\u0027s 6 decimal places."},{"Start":"01:38.594 ","End":"01:41.590","Text":"If I do this."},{"Start":"01:43.490 ","End":"01:45.660","Text":"This much of a degree,"},{"Start":"01:45.660 ","End":"01:50.835","Text":"so I multiply by 60 to get it in minutes."},{"Start":"01:50.835 ","End":"01:58.830","Text":"This comes out to be, 10.987."},{"Start":"01:58.830 ","End":"02:02.150","Text":"I think 4 decimal places will be sufficient this time."},{"Start":"02:02.150 ","End":"02:05.300","Text":"What I actually did was I didn\u0027t rewrite this."},{"Start":"02:05.300 ","End":"02:07.505","Text":"I kept this in the calculator,"},{"Start":"02:07.505 ","End":"02:11.420","Text":"subtracted 229 to get the fraction part and"},{"Start":"02:11.420 ","End":"02:15.695","Text":"then multiply the answer by 60 and not lose any accuracy."},{"Start":"02:15.695 ","End":"02:17.240","Text":"Now that I have this,"},{"Start":"02:17.240 ","End":"02:24.675","Text":"now we know that it\u0027s 229 degrees and 10 minutes."},{"Start":"02:24.675 ","End":"02:28.174","Text":"Now this part, just like we converted this into degrees,"},{"Start":"02:28.174 ","End":"02:32.205","Text":"we want to convert this into minutes."},{"Start":"02:32.205 ","End":"02:34.305","Text":"We convert this into seconds."},{"Start":"02:34.305 ","End":"02:37.220","Text":"Once again, we just multiply by 60,"},{"Start":"02:37.220 ","End":"02:42.515","Text":"take off the 10 and multiply it by 60 or if you like, I mean,"},{"Start":"02:42.515 ","End":"02:49.370","Text":"officially it\u0027s a formerly 9,870 and this time we need 4 zeros and a 1,"},{"Start":"02:49.370 ","End":"02:58.185","Text":"so that\u0027s over 10,000 and then multiply that by 60 and what we get 59.2 something."},{"Start":"02:58.185 ","End":"03:01.665","Text":"Already we know that we rounded off to 59,"},{"Start":"03:01.665 ","End":"03:06.990","Text":"so that will be 229 degrees,"},{"Start":"03:06.990 ","End":"03:10.080","Text":"10 minutes and 59."},{"Start":"03:10.080 ","End":"03:14.880","Text":"Sorry. Yeah, 1 tick for a minute,"},{"Start":"03:14.880 ","End":"03:16.890","Text":"2 ticks for seconds."},{"Start":"03:16.890 ","End":"03:18.845","Text":"That\u0027s the answer but like I said,"},{"Start":"03:18.845 ","End":"03:22.100","Text":"many calculators already do this conversion for"},{"Start":"03:22.100 ","End":"03:26.520","Text":"you so you don\u0027t need to do all this. That\u0027s it."}],"ID":10740},{"Watched":false,"Name":"Exercise 5","Duration":"1m 16s","ChapterTopicVideoID":10381,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10381.jpeg","UploadDate":"2017-11-02T15:53:57.0430000","DurationForVideoObject":"PT1M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"In this exercise,"},{"Start":"00:01.950 ","End":"00:08.520","Text":"we want to convert from degrees minute seconds to radians."},{"Start":"00:08.520 ","End":"00:10.110","Text":"We\u0027ll do it in two steps."},{"Start":"00:10.110 ","End":"00:15.585","Text":"First of all, we\u0027ll convert it to degrees as a decimal and then two radians."},{"Start":"00:15.585 ","End":"00:18.930","Text":"Now, this as a fraction of degrees is"},{"Start":"00:18.930 ","End":"00:28.410","Text":"563 plus each minute is 160th of a degree but a second is a 60th of a minute,"},{"Start":"00:28.410 ","End":"00:32.055","Text":"so 60*60 is 3600,"},{"Start":"00:32.055 ","End":"00:35.115","Text":"so this is the fraction we have to compute."},{"Start":"00:35.115 ","End":"00:45.260","Text":"On the calculator it comes out to be exactly 563.415 degrees."},{"Start":"00:45.260 ","End":"00:47.135","Text":"Now as a decimal,"},{"Start":"00:47.135 ","End":"00:49.585","Text":"I\u0027ll just copy that here,"},{"Start":"00:49.585 ","End":"00:53.990","Text":"563.415, and as usual,"},{"Start":"00:53.990 ","End":"00:56.540","Text":"from degrees to radians,"},{"Start":"00:56.540 ","End":"01:02.185","Text":"we multiply by Pi/180,"},{"Start":"01:02.185 ","End":"01:10.775","Text":"and that will give us 9.8334 something."},{"Start":"01:10.775 ","End":"01:12.625","Text":"I\u0027ll stop there."},{"Start":"01:12.625 ","End":"01:16.240","Text":"That\u0027s all there is to it. That\u0027s the answer."}],"ID":10741},{"Watched":false,"Name":"Exercise 6","Duration":"2m 56s","ChapterTopicVideoID":10382,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10382.jpeg","UploadDate":"2017-11-02T15:54:08.9400000","DurationForVideoObject":"PT2M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.375","Text":"In this exercise, we\u0027re given an angle theta in standard position."},{"Start":"00:06.375 ","End":"00:11.385","Text":"We also know that the point (12,"},{"Start":"00:11.385 ","End":"00:14.610","Text":"-35) is on its terminal side."},{"Start":"00:14.610 ","End":"00:19.530","Text":"We have to compute the six trigonometric functions of this angle theta."},{"Start":"00:19.530 ","End":"00:26.820","Text":"This 12 is the x and the -35 is the y. x is positive, y is negative."},{"Start":"00:26.820 ","End":"00:28.450","Text":"We\u0027re in the fourth quadrant."},{"Start":"00:28.450 ","End":"00:29.630","Text":"We could draw a sketch."},{"Start":"00:29.630 ","End":"00:31.295","Text":"There is no need to."},{"Start":"00:31.295 ","End":"00:35.390","Text":"All we have to do is compute also r,"},{"Start":"00:35.390 ","End":"00:38.806","Text":"which is the distance from the origin."},{"Start":"00:38.806 ","End":"00:40.430","Text":"So r which is, in general,"},{"Start":"00:40.430 ","End":"00:45.230","Text":"the square root of x^2 plus y^2."},{"Start":"00:45.230 ","End":"00:50.605","Text":"In this case comes out to be the square root of,"},{"Start":"00:50.605 ","End":"00:56.210","Text":"let\u0027s see.12^2 is 144,"},{"Start":"00:56.210 ","End":"01:07.430","Text":"and 35^2 squared is 2025."},{"Start":"01:07.430 ","End":"01:11.770","Text":"That makes it the square root of"},{"Start":"01:11.770 ","End":"01:19.640","Text":"2169,"},{"Start":"01:19.640 ","End":"01:26.790","Text":"and this comes out to be 37."},{"Start":"01:27.070 ","End":"01:31.970","Text":"Now, we just have to blindly substitute in the formulas."},{"Start":"01:31.970 ","End":"01:35.720","Text":"Like sine theta is y/r,"},{"Start":"01:35.720 ","End":"01:42.920","Text":"which comes out to be -35/37."},{"Start":"01:42.920 ","End":"01:50.405","Text":"Cosine theta is x/r, which is 12/37."},{"Start":"01:50.405 ","End":"01:55.850","Text":"Tangent theta is y/x,"},{"Start":"01:55.850 ","End":"02:00.050","Text":"so that comes out to be -35,"},{"Start":"02:00.050 ","End":"02:03.095","Text":"that\u0027s the y, and over 12."},{"Start":"02:03.095 ","End":"02:07.760","Text":"The other three, I\u0027ll write them as cosecant,"},{"Start":"02:07.760 ","End":"02:11.915","Text":"secant, and cotangent."},{"Start":"02:11.915 ","End":"02:15.990","Text":"Also same angle theta."},{"Start":"02:16.450 ","End":"02:20.699","Text":"These are the reciprocals of these."},{"Start":"02:21.790 ","End":"02:29.030","Text":"r/y, this would be r/x and this x/y,"},{"Start":"02:29.030 ","End":"02:30.890","Text":"which comes out to be,"},{"Start":"02:30.890 ","End":"02:31.970","Text":"since we have everything,"},{"Start":"02:31.970 ","End":"02:38.750","Text":"r is 37, y is -35."},{"Start":"02:38.750 ","End":"02:42.920","Text":"Here we have 37/12,"},{"Start":"02:42.920 ","End":"02:47.730","Text":"which is x, and x/y is -12/35."},{"Start":"02:50.020 ","End":"02:57.240","Text":"Quite a technical exercise and that\u0027s it. We\u0027re done."}],"ID":10742},{"Watched":false,"Name":"Exercise 7","Duration":"5m 22s","ChapterTopicVideoID":10383,"CourseChapterTopicPlaylistID":257202,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10383.jpeg","UploadDate":"2017-11-02T15:54:30.5730000","DurationForVideoObject":"PT5M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise we have to compute"},{"Start":"00:03.630 ","End":"00:09.060","Text":"6 trigonometric functions for each of 3 angles: 45 degrees,"},{"Start":"00:09.060 ","End":"00:10.800","Text":"30 degrees, and 60 degrees,"},{"Start":"00:10.800 ","End":"00:13.214","Text":"and these are special angles."},{"Start":"00:13.214 ","End":"00:15.720","Text":"Also note that they\u0027re all acute angles."},{"Start":"00:15.720 ","End":"00:19.553","Text":"Now for the first 1 I have a diagram,"},{"Start":"00:19.553 ","End":"00:22.460","Text":"and here is a triangle with 45,"},{"Start":"00:22.460 ","End":"00:25.345","Text":"45, and 90 degrees."},{"Start":"00:25.345 ","End":"00:28.590","Text":"Let\u0027s just pick 1 of these;"},{"Start":"00:28.590 ","End":"00:29.790","Text":"let\u0027s just take this 1;"},{"Start":"00:29.790 ","End":"00:32.830","Text":"the 45 degrees here."},{"Start":"00:32.990 ","End":"00:37.535","Text":"Then this 1 becomes the side that\u0027s opposite,"},{"Start":"00:37.535 ","End":"00:40.865","Text":"this is the side that\u0027s adjacent,"},{"Start":"00:40.865 ","End":"00:43.565","Text":"and this is the hypotenuse."},{"Start":"00:43.565 ","End":"00:46.400","Text":"For acute angles there are formulas involving opposite,"},{"Start":"00:46.400 ","End":"00:48.565","Text":"adjacent,, and hypotenuse."},{"Start":"00:48.565 ","End":"00:57.215","Text":"For example, sine is the opposite over the hypotenuse."},{"Start":"00:57.215 ","End":"01:00.230","Text":"I won\u0027t write that; opposite over hypotenuse,"},{"Start":"01:00.230 ","End":"01:03.420","Text":"1 over the square root of 2."},{"Start":"01:03.470 ","End":"01:10.580","Text":"The cosine is the adjacent over the hypotenuse."},{"Start":"01:10.580 ","End":"01:14.810","Text":"Once again it\u0027s 1 over the square root of 2,"},{"Start":"01:14.810 ","End":"01:22.025","Text":"and the tangent is the side opposite over the side adjacent to the angle,"},{"Start":"01:22.025 ","End":"01:24.565","Text":"1 over 1,"},{"Start":"01:24.565 ","End":"01:27.640","Text":"which is just 1."},{"Start":"01:27.640 ","End":"01:32.525","Text":"Then the other 3: the cosecant,"},{"Start":"01:32.525 ","End":"01:41.625","Text":"the secant, and the cotangent all at 45 degrees."},{"Start":"01:41.625 ","End":"01:46.685","Text":"These 3 are the reciprocals of these 3."},{"Start":"01:46.685 ","End":"01:52.970","Text":"If this is opposite over hypotenuse and this 1 is hypotenuse over opposite,"},{"Start":"01:52.970 ","End":"01:56.495","Text":"I can just reverse these 4 reciprocal."},{"Start":"01:56.495 ","End":"01:59.585","Text":"This 1 is going to be square root of 2, 1 over this."},{"Start":"01:59.585 ","End":"02:06.055","Text":"This 1 is also going to be square root of 2 and reciprocal of 1 is just 1."},{"Start":"02:06.055 ","End":"02:08.745","Text":"That\u0027s the 45 degrees."},{"Start":"02:08.745 ","End":"02:12.280","Text":"What about 30 and 60?"},{"Start":"02:13.330 ","End":"02:20.900","Text":"The diagram here is if we take an equilateral triangle and most convenient take it;"},{"Start":"02:20.900 ","End":"02:21.950","Text":"2, 2,"},{"Start":"02:21.950 ","End":"02:26.870","Text":"2, and we drop a perpendicular here which is"},{"Start":"02:26.870 ","End":"02:32.355","Text":"also the angle bisector and also the median so we get 2 triangles."},{"Start":"02:32.355 ","End":"02:38.785","Text":"Let\u0027s go with the right-hand 1 which are 30, 60, 90."},{"Start":"02:38.785 ","End":"02:43.655","Text":"If I want to let\u0027s take the 60 degrees first."},{"Start":"02:43.655 ","End":"02:49.575","Text":"If we take the 60, then this becomes the opposite,"},{"Start":"02:49.575 ","End":"02:52.625","Text":"this becomes the adjacent,"},{"Start":"02:52.625 ","End":"02:56.100","Text":"and this is the hypotenuse."},{"Start":"02:56.150 ","End":"03:01.230","Text":"You know what? I\u0027ll just write all the function sine,"},{"Start":"03:01.230 ","End":"03:03.420","Text":"cosine, tangent,"},{"Start":"03:03.420 ","End":"03:06.140","Text":"and then we\u0027ll have cosecant,"},{"Start":"03:06.140 ","End":"03:10.710","Text":"secant, and cotangent,"},{"Start":"03:10.710 ","End":"03:16.140","Text":"and it\u0027s going to be 60 degrees,"},{"Start":"03:16.140 ","End":"03:18.510","Text":"and all the rest of them."},{"Start":"03:18.510 ","End":"03:28.393","Text":"Now the sine as we said is opposite over hypotenuse so that would be root 3 over 2."},{"Start":"03:28.393 ","End":"03:34.160","Text":"The cosine is adjacent over hypotenuse, so it\u0027s 1/2."},{"Start":"03:34.160 ","End":"03:40.030","Text":"Tangent opposite over adjacent root 3."},{"Start":"03:40.330 ","End":"03:43.370","Text":"I\u0027ll remind you the opposite is root 3,"},{"Start":"03:43.370 ","End":"03:44.450","Text":"the adjacent is 1,"},{"Start":"03:44.450 ","End":"03:46.595","Text":"hypotenuse is 2 just like written here."},{"Start":"03:46.595 ","End":"03:55.385","Text":"The cosecant is just the inverse of the reciprocal of the sin so that\u0027s 2 over root 3,"},{"Start":"03:55.385 ","End":"03:59.625","Text":"and I have the inverse of this which is 2,"},{"Start":"03:59.625 ","End":"04:07.885","Text":"and the inverse meaning reciprocal of this is 1 over root 3."},{"Start":"04:07.885 ","End":"04:12.545","Text":"Now we want to do the same thing for 30 degrees."},{"Start":"04:12.545 ","End":"04:14.700","Text":"You now what? I\u0027ll go in the other triangle,"},{"Start":"04:14.700 ","End":"04:17.965","Text":"so I won\u0027t have to erase."},{"Start":"04:17.965 ","End":"04:21.210","Text":"Let\u0027s say we take this 30 degrees,"},{"Start":"04:21.210 ","End":"04:24.570","Text":"so this becomes the side opposite,"},{"Start":"04:24.570 ","End":"04:28.890","Text":"this side becomes the side adjacent,"},{"Start":"04:28.890 ","End":"04:32.750","Text":"and this is the hypotenuse."},{"Start":"04:32.750 ","End":"04:41.675","Text":"Here I just wrote them all."},{"Start":"04:41.675 ","End":"04:48.455","Text":"Once again, sine is opposite over hypotenuse;"},{"Start":"04:48.455 ","End":"04:52.985","Text":"1/2, cosine adjacent over hypotenuse;"},{"Start":"04:52.985 ","End":"04:55.565","Text":"root 3 over 2."},{"Start":"04:55.565 ","End":"04:59.645","Text":"Tangent is opposite over adjacent,"},{"Start":"04:59.645 ","End":"05:02.120","Text":"1 over root 3."},{"Start":"05:02.120 ","End":"05:04.885","Text":"Then I have the 3 reciprocals of these,"},{"Start":"05:04.885 ","End":"05:08.790","Text":"so 1/2 means 2 root 3 over 2,"},{"Start":"05:08.790 ","End":"05:11.910","Text":"so 2 over root 3,"},{"Start":"05:11.910 ","End":"05:17.140","Text":"and the inverse of this is just root 3."},{"Start":"05:17.990 ","End":"05:21.910","Text":"That\u0027s it."}],"ID":10743}],"Thumbnail":null,"ID":257202},{"Name":"Trig Identities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Trigonometric Identities","Duration":"3m ","ChapterTopicVideoID":10480,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10480.jpeg","UploadDate":"2021-06-29T13:21:07.8070000","DurationForVideoObject":"PT3M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"We\u0027re returning to trigonometric identities."},{"Start":"00:03.570 ","End":"00:12.300","Text":"I gave these identities in the first chapter on trigonometric functions,"},{"Start":"00:12.300 ","End":"00:16.470","Text":"and now we\u0027re just going to show how we can use these plus"},{"Start":"00:16.470 ","End":"00:20.880","Text":"a bit of algebra to prove a whole lot of identities."},{"Start":"00:20.880 ","End":"00:26.085","Text":"I mean, there\u0027s endless numbers of identities 1 could make up as examples,"},{"Start":"00:26.085 ","End":"00:29.340","Text":"and I\u0027ll start with 1."},{"Start":"00:29.340 ","End":"00:39.330","Text":"We have to prove that cosecant of Theta minus the sine of Theta is"},{"Start":"00:39.330 ","End":"00:49.310","Text":"equal to cosine of Theta times cotangent Theta and to say it\u0027s an identity,"},{"Start":"00:49.310 ","End":"00:51.400","Text":"it means that for all Theta."},{"Start":"00:51.400 ","End":"00:53.990","Text":"If you find even 1 counter example,"},{"Start":"00:53.990 ","End":"00:56.375","Text":"then this is not an identity."},{"Start":"00:56.375 ","End":"00:58.900","Text":"This is fairly typical,"},{"Start":"00:58.900 ","End":"01:02.735","Text":"and we often want to convert everything to sine and cosine."},{"Start":"01:02.735 ","End":"01:05.554","Text":"It\u0027s not always, but it\u0027s 1 technique."},{"Start":"01:05.554 ","End":"01:07.745","Text":"We\u0027ll start with the left-hand side."},{"Start":"01:07.745 ","End":"01:12.195","Text":"Cosecant Theta minus sine Theta is equal to."},{"Start":"01:12.195 ","End":"01:15.435","Text":"Now cosecant is 1 over sine,"},{"Start":"01:15.435 ","End":"01:20.130","Text":"that\u0027s from the reciprocal identities,"},{"Start":"01:20.130 ","End":"01:23.620","Text":"minus sine Theta,"},{"Start":"01:23.620 ","End":"01:29.990","Text":"and now we can do a bit of algebra and put a common denominator,"},{"Start":"01:29.990 ","End":"01:34.358","Text":"put everything over sine Theta."},{"Start":"01:34.358 ","End":"01:38.900","Text":"Here we have 1 and here we have minus this times this,"},{"Start":"01:38.900 ","End":"01:43.860","Text":"so it\u0027s sine Theta or sine squared Theta."},{"Start":"01:44.020 ","End":"01:49.670","Text":"Now here we can use 1 of the Pythagorean identities."},{"Start":"01:49.670 ","End":"01:53.180","Text":"If we look at this one here,"},{"Start":"01:53.180 ","End":"01:56.570","Text":"1 minus sine squared is cosine"},{"Start":"01:56.570 ","End":"02:04.960","Text":"squared Theta over sine Theta."},{"Start":"02:05.360 ","End":"02:12.585","Text":"Now I can write the cosine squared as cosine times cosine."},{"Start":"02:12.585 ","End":"02:18.710","Text":"This is cosine Theta times cosine Theta."},{"Start":"02:18.710 ","End":"02:21.435","Text":"I\u0027ll just put brackets for emphasis,"},{"Start":"02:21.435 ","End":"02:27.580","Text":"divided by sine Theta."},{"Start":"02:27.580 ","End":"02:36.965","Text":"Now, what I can do is take the cosine Theta from here and this quotient cosine"},{"Start":"02:36.965 ","End":"02:46.340","Text":"Theta over sine Theta is equal to cotangent Theta and that\u0027s basically it."},{"Start":"02:46.340 ","End":"02:49.520","Text":"We started from the left-hand side and reach the right-hand side,"},{"Start":"02:49.520 ","End":"02:51.400","Text":"and so we\u0027re done,"},{"Start":"02:51.400 ","End":"02:54.170","Text":"and I won\u0027t do anymore."},{"Start":"02:54.170 ","End":"03:00.060","Text":"There\u0027s plenty of solved examples after the tutorial."}],"ID":10900},{"Watched":false,"Name":"Exercise 1","Duration":"1m 25s","ChapterTopicVideoID":10384,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10384.jpeg","UploadDate":"2017-11-02T15:59:37.8630000","DurationForVideoObject":"PT1M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.815","Text":"In this exercise, we have a trigonometric identity to prove."},{"Start":"00:04.815 ","End":"00:07.320","Text":"In this case, as in most cases,"},{"Start":"00:07.320 ","End":"00:09.000","Text":"will start from one side,"},{"Start":"00:09.000 ","End":"00:11.880","Text":"the left-hand side, and reach the right-hand side."},{"Start":"00:11.880 ","End":"00:13.140","Text":"On the left-hand side,"},{"Start":"00:13.140 ","End":"00:14.460","Text":"first of all, I\u0027m just going to copy it."},{"Start":"00:14.460 ","End":"00:18.660","Text":"It\u0027s secant t minus cosine t. Now,"},{"Start":"00:18.660 ","End":"00:24.150","Text":"remember that secant is one over cosine is one of the reciprocal identities."},{"Start":"00:24.150 ","End":"00:29.660","Text":"It\u0027s 1 over cosine t minus cosine t. Now use a bit of"},{"Start":"00:29.660 ","End":"00:36.205","Text":"algebra to put a common denominator of cosine t. Here we have 1,"},{"Start":"00:36.205 ","End":"00:41.630","Text":"and here we\u0027ll get cosine^2 t. Now"},{"Start":"00:41.630 ","End":"00:47.150","Text":"the Pythagorean identity which follows from cosine squared plus sine^2 is 1,"},{"Start":"00:47.150 ","End":"00:52.745","Text":"that 1 minus cosine^2 is equal to"},{"Start":"00:52.745 ","End":"00:59.450","Text":"sine^2 t over cosine t. A bit of algebra,"},{"Start":"00:59.450 ","End":"01:07.220","Text":"sine^2 I can separate into sine times sine and the cosine."},{"Start":"01:07.220 ","End":"01:11.310","Text":"I\u0027ll put over the first one under."},{"Start":"01:11.780 ","End":"01:17.540","Text":"Now, sine t over cosine t is equal to tangent t and"},{"Start":"01:17.540 ","End":"01:22.430","Text":"here I\u0027ll just copy the sine t. Low and behold,"},{"Start":"01:22.430 ","End":"01:25.980","Text":"this is what we had to prove and so we are done."}],"ID":10744},{"Watched":false,"Name":"Exercise 2","Duration":"1m 36s","ChapterTopicVideoID":10385,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10385.jpeg","UploadDate":"2017-11-02T15:59:44.0830000","DurationForVideoObject":"PT1M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.715","Text":"In this exercise, we have to prove a trigonometric identity that"},{"Start":"00:05.715 ","End":"00:12.315","Text":"cos^4 minus sin^4 is the same as cos^2 minus sin^2."},{"Start":"00:12.315 ","End":"00:18.120","Text":"What I\u0027m going to do here is rely on the algebraic formula that"},{"Start":"00:18.120 ","End":"00:25.485","Text":"a^2 minus b^2 equals a minus b, a plus b."},{"Start":"00:25.485 ","End":"00:28.664","Text":"Sometimes called the difference of squares formula."},{"Start":"00:28.664 ","End":"00:31.650","Text":"Back here, I\u0027m going to rewrite this,"},{"Start":"00:31.650 ","End":"00:34.995","Text":"well, the left-hand side,"},{"Start":"00:34.995 ","End":"00:39.990","Text":"as cos^4 is cos^2^2,"},{"Start":"00:39.990 ","End":"00:43.185","Text":"so it\u0027s cos^2 Alpha^2."},{"Start":"00:43.185 ","End":"00:46.005","Text":"Then similarly here,"},{"Start":"00:46.005 ","End":"00:48.030","Text":"^4 is ^2 ^2,"},{"Start":"00:48.030 ","End":"00:52.215","Text":"so it\u0027s sin^2 Alpha^2."},{"Start":"00:52.215 ","End":"00:55.130","Text":"I\u0027m starting from the left-hand side and I\u0027m trying to reach the right-hand side."},{"Start":"00:55.130 ","End":"00:59.420","Text":"Now, using this formula with a as cos^2 and b as sin^2,"},{"Start":"00:59.420 ","End":"01:08.100","Text":"we get that this is cos^2 Alpha minus sin^2 Alpha."},{"Start":"01:08.100 ","End":"01:09.600","Text":"That\u0027s the first bracket."},{"Start":"01:09.600 ","End":"01:17.970","Text":"The second bracket is the same thing just with a plus here, sin^2 Alpha."},{"Start":"01:19.150 ","End":"01:25.430","Text":"Well, this is equal to 1 by one of the Pythagorean identities,"},{"Start":"01:25.430 ","End":"01:26.855","Text":"the most basic one."},{"Start":"01:26.855 ","End":"01:33.185","Text":"We\u0027re just left with cos^2 Alpha minus sin^2 Alpha."},{"Start":"01:33.185 ","End":"01:35.700","Text":"That\u0027s all there is to it."}],"ID":10745},{"Watched":false,"Name":"Exercise 3","Duration":"2m 9s","ChapterTopicVideoID":10386,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10386.jpeg","UploadDate":"2017-11-02T15:59:51.1500000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.700","Text":"This exercise is another trigonometric identity to prove and as I do in most cases,"},{"Start":"00:08.700 ","End":"00:13.455","Text":"I start from the left-hand side and work my way to the right-hand side."},{"Start":"00:13.455 ","End":"00:16.140","Text":"I won\u0027t copy it,"},{"Start":"00:16.140 ","End":"00:18.000","Text":"we\u0027ll just take it straight from here."},{"Start":"00:18.000 ","End":"00:22.350","Text":"I\u0027m going to put the common denominator for this expression."},{"Start":"00:22.350 ","End":"00:27.300","Text":"The common denominator is just the product of these 2 so I\u0027ll"},{"Start":"00:27.300 ","End":"00:35.355","Text":"take (1 +Sine Theta) (1 - sine Theta)."},{"Start":"00:35.355 ","End":"00:39.030","Text":"What I\u0027m going to do is just cross multiply,"},{"Start":"00:39.030 ","End":"00:44.430","Text":"I mean this times this is (1-sine Theta)."},{"Start":"00:44.430 ","End":"00:46.305","Text":"The plus from here,"},{"Start":"00:46.305 ","End":"00:51.465","Text":"this times this is (1+ sine"},{"Start":"00:51.465 ","End":"00:57.485","Text":"Theta) and so this is equal to this,"},{"Start":"00:57.485 ","End":"00:59.105","Text":"and this is equal too."},{"Start":"00:59.105 ","End":"01:00.665","Text":"Now the numerator,"},{"Start":"01:00.665 ","End":"01:01.880","Text":"if I open the brackets,"},{"Start":"01:01.880 ","End":"01:08.090","Text":"the sine Theta- sine Theta cancels so I get 2 and when I get on the denominator,"},{"Start":"01:08.090 ","End":"01:11.075","Text":"is a difference of squares formula."},{"Start":"01:11.075 ","End":"01:19.430","Text":"To remind you (a+ b ) times (a- b)= a^2- b^2,"},{"Start":"01:19.430 ","End":"01:24.620","Text":"so here I get 1- sine^2Theta."},{"Start":"01:24.620 ","End":"01:31.290","Text":"Well 1^2, that\u0027s 1. We\u0027re getting close."},{"Start":"01:31.540 ","End":"01:42.570","Text":"1- sine^2 by one of the Pythagorean identities is cosine^2 and"},{"Start":"01:42.570 ","End":"01:48.110","Text":"this is equal to twice now I can write"},{"Start":"01:48.110 ","End":"01:54.725","Text":"the square as (1/cosine Theta)^2."},{"Start":"01:54.725 ","End":"02:01.310","Text":"The reason I\u0027m doing this is because another 1/cosine is secant so this is just"},{"Start":"02:01.310 ","End":"02:09.540","Text":"2 secant Theta^2 or secant^2 Theta and we are done."}],"ID":10746},{"Watched":false,"Name":"Exercise 4","Duration":"2m 21s","ChapterTopicVideoID":10387,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10387.jpeg","UploadDate":"2017-11-02T15:59:59.1730000","DurationForVideoObject":"PT2M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.090","Text":"In this exercise, we have the following trigonometric identity to prove, verify."},{"Start":"00:07.220 ","End":"00:11.020","Text":"We\u0027re going to start from the left-hand side and reach the right-hand side,"},{"Start":"00:11.020 ","End":"00:14.940","Text":"and at first, it\u0027s not quite clear what should be done."},{"Start":"00:14.940 ","End":"00:18.480","Text":"The trick I\u0027m going to do is to multiply top and bottom by"},{"Start":"00:18.480 ","End":"00:21.720","Text":"cosine and then I\u0027ll have cosine^2 here,"},{"Start":"00:21.720 ","End":"00:23.670","Text":"and then I\u0027ll use one of the Pythagorean."},{"Start":"00:23.670 ","End":"00:31.260","Text":"So let me just copy 1 plus sine x over cosine x,"},{"Start":"00:31.260 ","End":"00:37.855","Text":"and this is equal to x here,"},{"Start":"00:37.855 ","End":"00:43.834","Text":"1 plus sine x times cosine x."},{"Start":"00:43.834 ","End":"00:48.880","Text":"Like I said, I\u0027m going to multiply top and bottom by cosine,"},{"Start":"00:48.880 ","End":"00:51.050","Text":"that gives me cosine^2."},{"Start":"00:51.050 ","End":"00:54.215","Text":"Now, what I can do with that,"},{"Start":"00:54.215 ","End":"01:05.010","Text":"let me just copy the numerator is to write this denominator as 1 minus sine^2 x."},{"Start":"01:05.170 ","End":"01:09.905","Text":"Like to remind you of the algebraic,"},{"Start":"01:09.905 ","End":"01:13.895","Text":"a^2 minus b^2 is a minus b,"},{"Start":"01:13.895 ","End":"01:17.135","Text":"a plus b difference of squares formula."},{"Start":"01:17.135 ","End":"01:27.995","Text":"So that I can write the denominator as cosine^2 x."},{"Start":"01:27.995 ","End":"01:37.885","Text":"Now, one of the cosines will cancel and we\u0027re left with 1 plus sine x."},{"Start":"01:37.885 ","End":"01:41.430","Text":"This cosine cancels with the squared,"},{"Start":"01:41.430 ","End":"01:44.835","Text":"so it\u0027s just cosine x,"},{"Start":"01:44.835 ","End":"01:50.400","Text":"and so I can"},{"Start":"01:50.400 ","End":"01:56.760","Text":"rewrite 1 minus sine^2 as I\u0027ll take the plus first,"},{"Start":"01:56.760 ","End":"02:00.870","Text":"then the minus, so 1 plus sine x,"},{"Start":"02:00.870 ","End":"02:04.170","Text":"1 minus sine x,"},{"Start":"02:04.170 ","End":"02:08.190","Text":"and this first term cancels,"},{"Start":"02:08.190 ","End":"02:17.370","Text":"so we\u0027re left with cosine x over 1 minus sine x because this goes with this,"},{"Start":"02:17.370 ","End":"02:20.010","Text":"and that\u0027s exactly the right-hand side,"},{"Start":"02:20.010 ","End":"02:22.480","Text":"and so we are done."}],"ID":10747},{"Watched":false,"Name":"Exercise 5","Duration":"1m 54s","ChapterTopicVideoID":10388,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10388.jpeg","UploadDate":"2017-11-02T16:00:05.4030000","DurationForVideoObject":"PT1M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:08.800","Text":"In this exercise, we have to prove that this here is a trigonometric identity."},{"Start":"00:08.870 ","End":"00:11.400","Text":"As we often do, we start from"},{"Start":"00:11.400 ","End":"00:15.420","Text":"the left-hand side and see if we can reach the right-hand side."},{"Start":"00:15.420 ","End":"00:21.330","Text":"Now, cotangent can be expressed in terms of sine and cosine. Let\u0027s do that."},{"Start":"00:21.330 ","End":"00:25.320","Text":"I\u0027ll just remind you that cotangent of, say,"},{"Start":"00:25.320 ","End":"00:32.115","Text":"Alpha is equal to cosine of Alpha/sine of Alpha."},{"Start":"00:32.115 ","End":"00:34.859","Text":"If I apply that here,"},{"Start":"00:34.859 ","End":"00:45.110","Text":"then we get cotangent of u. Cosine u/sine u plus same thing for v,"},{"Start":"00:45.110 ","End":"00:53.150","Text":"cosine v/sine v, and then over cotangent of u again,"},{"Start":"00:53.150 ","End":"00:57.230","Text":"cosine u/sine u. Cotangent v"},{"Start":"00:57.230 ","End":"01:04.450","Text":"is cosine v/sine v-1."},{"Start":"01:04.610 ","End":"01:06.620","Text":"What do we do with this?"},{"Start":"01:06.620 ","End":"01:11.300","Text":"Well, we\u0027ll multiply top and bottom by sine u sine v,"},{"Start":"01:11.300 ","End":"01:13.970","Text":"and then we can get rid of all of the fractions."},{"Start":"01:13.970 ","End":"01:16.760","Text":"If we do that here,"},{"Start":"01:16.760 ","End":"01:21.170","Text":"we\u0027ll get cosine u times sine v,"},{"Start":"01:21.170 ","End":"01:30.650","Text":"like cross multiplication plus this times this: sine u cosine v/sine u sine v,"},{"Start":"01:30.650 ","End":"01:32.800","Text":"but that\u0027s what we\u0027re multiplying by."},{"Start":"01:32.800 ","End":"01:35.000","Text":"On the denominator, the sine u,"},{"Start":"01:35.000 ","End":"01:36.470","Text":"sine v will disappear."},{"Start":"01:36.470 ","End":"01:41.300","Text":"We get cosine u cosine v minus."},{"Start":"01:41.300 ","End":"01:44.240","Text":"Here we have multiplied by sine u,"},{"Start":"01:44.240 ","End":"01:51.710","Text":"sine v. Looks like we\u0027ve done it already in 1 step."},{"Start":"01:51.710 ","End":"01:54.540","Text":"I guess that\u0027s it.c"}],"ID":10748},{"Watched":false,"Name":"Exercise 6","Duration":"1m ","ChapterTopicVideoID":10389,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10389.jpeg","UploadDate":"2017-11-02T16:00:08.4600000","DurationForVideoObject":"PT1M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.794","Text":"In this exercise, we have to verify or prove"},{"Start":"00:03.794 ","End":"00:08.205","Text":"this trigonometric identity as written here. I won\u0027t read it out."},{"Start":"00:08.205 ","End":"00:12.060","Text":"What we\u0027re going to do, as we most often do,"},{"Start":"00:12.060 ","End":"00:16.095","Text":"is start from the left hand side and reach the right hand side."},{"Start":"00:16.095 ","End":"00:19.020","Text":"Now, the obvious thing to do is to put"},{"Start":"00:19.020 ","End":"00:23.895","Text":"a common denominator here and cosine Alpha it will be."},{"Start":"00:23.895 ","End":"00:27.390","Text":"The left hand side is equal to,"},{"Start":"00:27.390 ","End":"00:30.810","Text":"I\u0027ll put it over cosine Alpha."},{"Start":"00:30.810 ","End":"00:36.095","Text":"Now here I have 1 and here I must multiply top and bottom by cosine Alpha,"},{"Start":"00:36.095 ","End":"00:41.435","Text":"so I get cosine^2 Alpha that\u0027s what this equals."},{"Start":"00:41.435 ","End":"00:43.055","Text":"What does this equal?"},{"Start":"00:43.055 ","End":"00:49.605","Text":"Well, one of the Pythagorean identities is that 1 minus cosine^2 is sine^2 Alpha,"},{"Start":"00:49.605 ","End":"00:51.900","Text":"because sine^2 plus cosine^2 is 1,"},{"Start":"00:51.900 ","End":"00:56.955","Text":"so we have sine^2 Alpha over cosine Alpha and boy,"},{"Start":"00:56.955 ","End":"00:59.860","Text":"that\u0027s it already. That was short."}],"ID":10749},{"Watched":false,"Name":"Exercise 7","Duration":"2m 37s","ChapterTopicVideoID":10390,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10390.jpeg","UploadDate":"2017-11-02T16:00:17.5870000","DurationForVideoObject":"PT2M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"In this exercise, we have"},{"Start":"00:01.650 ","End":"00:06.330","Text":"yet another trigonometric identity to prove and as written here,"},{"Start":"00:06.330 ","End":"00:07.980","Text":"I won\u0027t read it out."},{"Start":"00:07.980 ","End":"00:10.950","Text":"What we\u0027ll do is we\u0027ll start from the left-hand side and try"},{"Start":"00:10.950 ","End":"00:14.010","Text":"and simplify it until we get to the right-hand side."},{"Start":"00:14.010 ","End":"00:17.940","Text":"What I suggest is a common denominator,"},{"Start":"00:17.940 ","End":"00:21.060","Text":"so this left-hand side will equal."},{"Start":"00:21.060 ","End":"00:25.410","Text":"Now a denominator will be this times this, in other words,"},{"Start":"00:25.410 ","End":"00:31.560","Text":"sine x times 1 plus cosine x and here I get the cross-product."},{"Start":"00:31.560 ","End":"00:35.380","Text":"Just multiply 1 plus cosine x,"},{"Start":"00:35.380 ","End":"00:37.339","Text":"I have to put the brackets here,"},{"Start":"00:37.339 ","End":"00:42.720","Text":"times 1 plus cosine x. I can instead of writing it twice,"},{"Start":"00:42.720 ","End":"00:47.119","Text":"just put a square here and then plus this times this,"},{"Start":"00:47.119 ","End":"00:51.900","Text":"sine x times sine x, so sine^2x."},{"Start":"00:53.590 ","End":"00:56.195","Text":"Now let\u0027s do a bit of algebra,"},{"Start":"00:56.195 ","End":"00:59.660","Text":"(1+cosx)^2 using the binomial expansion,"},{"Start":"00:59.660 ","End":"01:08.400","Text":"a plus b^2 is a^2 which is 1^2 plus twice 1 times cosine x plus cosine(x)^2,"},{"Start":"01:08.400 ","End":"01:12.830","Text":"which is cosine^2(x) and then the sine^2x was here"},{"Start":"01:12.830 ","End":"01:17.690","Text":"already and the denominator doesn\u0027t change,"},{"Start":"01:17.690 ","End":"01:23.735","Text":"sine x times 1 plus cosine x."},{"Start":"01:23.735 ","End":"01:28.375","Text":"Now look, this and this together is equal to 1."},{"Start":"01:28.375 ","End":"01:35.340","Text":"I can say this is 1 plus 1 and then on the next line since it\u0027s 2 plus 2 cosine x,"},{"Start":"01:35.340 ","End":"01:42.200","Text":"I can take 2 outside the brackets and I get 1 plus cosine x."},{"Start":"01:42.200 ","End":"01:43.490","Text":"That shouldn\u0027t be hard to follow,"},{"Start":"01:43.490 ","End":"01:48.695","Text":"1 plus 1 is 2 take the 2 out of the bracket 2 plus 2 cosine x is this."},{"Start":"01:48.695 ","End":"01:50.919","Text":"On the denominator,"},{"Start":"01:50.919 ","End":"01:56.640","Text":"sine x times 1 plus cosine x."},{"Start":"01:56.640 ","End":"02:00.675","Text":"Now, we can cancel top and bottom."},{"Start":"02:00.675 ","End":"02:05.340","Text":"Here we have 1 plus cosine x and here 1 plus cosine x."},{"Start":"02:05.340 ","End":"02:08.170","Text":"By the way, in all the exercises"},{"Start":"02:08.170 ","End":"02:13.280","Text":"the identity is only supposed to hold for where it\u0027s defined or make sense."},{"Start":"02:13.280 ","End":"02:18.155","Text":"1 plus cosine x is not going to be 0 and this won\u0027t make sense."},{"Start":"02:18.155 ","End":"02:22.560","Text":"What we\u0027re left with here is 2 over"},{"Start":"02:23.870 ","End":"02:31.310","Text":"sine x but 1 over sine x by a reciprocal identity is the cosecant of x."},{"Start":"02:31.310 ","End":"02:37.710","Text":"This is just twice cosecant of x and we are done."}],"ID":10750},{"Watched":false,"Name":"Exercise 8","Duration":"4m 45s","ChapterTopicVideoID":10391,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10391.jpeg","UploadDate":"2017-11-02T16:00:40.7930000","DurationForVideoObject":"PT4M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.280","Text":"In this exercise, we have a trigonometric identity to prove or verify."},{"Start":"00:06.710 ","End":"00:10.560","Text":"This time we have the letter Phi,"},{"Start":"00:10.560 ","End":"00:13.215","Text":"just in case you don\u0027t know how to pronounce it."},{"Start":"00:13.215 ","End":"00:17.880","Text":"This is the Greek letter Phi or little Phi,"},{"Start":"00:17.880 ","End":"00:19.965","Text":"because there\u0027s also a capital Phi,"},{"Start":"00:19.965 ","End":"00:22.170","Text":"which is more like this."},{"Start":"00:22.170 ","End":"00:26.430","Text":"The little Phi is sometimes written like this and sometimes"},{"Start":"00:26.430 ","End":"00:31.570","Text":"it\u0027s just written as a smaller one of these. That\u0027s an aside."},{"Start":"00:31.580 ","End":"00:37.500","Text":"As usual, we start with the left-hand sides and try to simplify it,"},{"Start":"00:37.500 ","End":"00:41.505","Text":"and use other identities to get to the right-hand side."},{"Start":"00:41.505 ","End":"00:46.220","Text":"Let\u0027s first of all square this using binomial expansion. You know what?"},{"Start":"00:46.220 ","End":"00:47.420","Text":"I\u0027ll just copy this again."},{"Start":"00:47.420 ","End":"00:52.920","Text":"Cosecant Phi plus cotangent Phi^2 is equal to."},{"Start":"00:53.270 ","End":"00:57.870","Text":"I\u0027m going to use the famous binomial expansion,"},{"Start":"00:57.870 ","End":"01:07.120","Text":"a plus b^2 is equal to a^2 plus 2ab plus b^2 from algebra."},{"Start":"01:07.640 ","End":"01:17.220","Text":"We get cosecant^2 Phi plus twice cosecant"},{"Start":"01:17.220 ","End":"01:23.775","Text":"Phi times cotangent Phi"},{"Start":"01:23.775 ","End":"01:31.290","Text":"plus cotangent^2 of Phi."},{"Start":"01:31.870 ","End":"01:34.820","Text":"Just look at what we\u0027re aiming for."},{"Start":"01:34.820 ","End":"01:39.680","Text":"We\u0027re aiming for this which contains cosine only and not these."},{"Start":"01:39.680 ","End":"01:42.650","Text":"There is a standard thing you can always do,"},{"Start":"01:42.650 ","End":"01:48.425","Text":"is if all 6 trigonometric functions can be written in terms of just sine and cosine,"},{"Start":"01:48.425 ","End":"01:51.140","Text":"I mean sine and cosine themselves of course."},{"Start":"01:51.140 ","End":"01:56.580","Text":"Then, for example, cosecant of whatever, in this case,"},{"Start":"01:56.580 ","End":"02:03.485","Text":"Phi is 1 over sine Phi and the other one we need is the cotangent."},{"Start":"02:03.485 ","End":"02:06.515","Text":"Cotangent is cosine over sine."},{"Start":"02:06.515 ","End":"02:09.450","Text":"In this case, Phi."},{"Start":"02:09.710 ","End":"02:13.880","Text":"Plug these in here, we\u0027ll get everything in terms of sine and cosine."},{"Start":"02:13.880 ","End":"02:18.840","Text":"Hopefully, later we\u0027ll be able to get rid of this sine and be left with this."},{"Start":"02:18.950 ","End":"02:21.870","Text":"We get cosecant is 1 over sine,"},{"Start":"02:21.870 ","End":"02:25.170","Text":"so this is 1 over sine^2 Phi."},{"Start":"02:25.170 ","End":"02:28.335","Text":"Then we have 2."},{"Start":"02:28.335 ","End":"02:31.905","Text":"Cosecant is 1 over sine."},{"Start":"02:31.905 ","End":"02:41.610","Text":"I can put a sine Phi here and cotangent is cosine over sine."},{"Start":"02:43.120 ","End":"02:47.450","Text":"I could write this as sine^2Phi and I will in a moment,"},{"Start":"02:47.450 ","End":"02:51.730","Text":"then plus cotangent, which is cosine over sine squared."},{"Start":"02:51.730 ","End":"02:58.140","Text":"We\u0027ve got cosine^2 Phi over sine^2 Phi."},{"Start":"02:58.140 ","End":"03:01.180","Text":"Like I said, this is sine^2."},{"Start":"03:01.180 ","End":"03:06.695","Text":"In fact, everything can be written over sine^2Phi."},{"Start":"03:06.695 ","End":"03:07.830","Text":"Let\u0027s see what we have left."},{"Start":"03:07.830 ","End":"03:09.540","Text":"We have 1 from here,"},{"Start":"03:09.540 ","End":"03:13.335","Text":"plus 2 cosine Phi."},{"Start":"03:13.335 ","End":"03:16.930","Text":"Then cosine^2Phi."},{"Start":"03:21.950 ","End":"03:26.045","Text":"I\u0027m going to be lazy."},{"Start":"03:26.045 ","End":"03:28.000","Text":"I don\u0027t want to copy everything again,"},{"Start":"03:28.000 ","End":"03:33.265","Text":"but sine^2 is 1 minus cosine^2."},{"Start":"03:33.265 ","End":"03:35.250","Text":"We can do this."},{"Start":"03:35.250 ","End":"03:37.470","Text":"Now already we just have cosine."},{"Start":"03:37.470 ","End":"03:41.095","Text":"Then the bit more algebra maybe we can get to this."},{"Start":"03:41.095 ","End":"03:44.845","Text":"For the numerator, I\u0027m going to use this rule here."},{"Start":"03:44.845 ","End":"03:46.120","Text":"For the denominator."},{"Start":"03:46.120 ","End":"03:52.120","Text":"I\u0027m going to use another rule which is the difference of squares formula from algebra,"},{"Start":"03:52.120 ","End":"03:59.740","Text":"which is a^2 minus b^2 is a plus b times a minus b."},{"Start":"03:59.740 ","End":"04:02.645","Text":"If we apply that here,"},{"Start":"04:02.645 ","End":"04:07.395","Text":"then we get on the numerator, this is 1^2."},{"Start":"04:07.395 ","End":"04:14.250","Text":"We get 1 plus cosine Phi^2."},{"Start":"04:14.250 ","End":"04:20.955","Text":"This first formula over 1 minus cosine^2 using this would be"},{"Start":"04:20.955 ","End":"04:29.130","Text":"1 plus cosine Phi times 1 minus cosine Phi."},{"Start":"04:29.130 ","End":"04:34.760","Text":"Now if I cancel 1 plus cosine Phi from here and here it\u0027s squared,"},{"Start":"04:34.760 ","End":"04:37.640","Text":"so I just cancel the two."},{"Start":"04:38.060 ","End":"04:42.780","Text":"What we\u0027re left with it is this."},{"Start":"04:42.780 ","End":"04:44.345","Text":"I\u0027m not going to copy it again."},{"Start":"04:44.345 ","End":"04:46.680","Text":"Check we\u0027re done."}],"ID":10751},{"Watched":false,"Name":"Exercise 9","Duration":"2m 25s","ChapterTopicVideoID":10392,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10392.jpeg","UploadDate":"2017-11-02T16:00:56.9000000","DurationForVideoObject":"PT2M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.264","Text":"Here we have another exercise with a trigonometric identity to prove."},{"Start":"00:05.264 ","End":"00:08.580","Text":"We have to prove this equals this and I\u0027ll start"},{"Start":"00:08.580 ","End":"00:12.180","Text":"from the left-hand side and try and reach the right-hand side."},{"Start":"00:12.180 ","End":"00:14.430","Text":"Now, of course, when I solve it,"},{"Start":"00:14.430 ","End":"00:20.820","Text":"I\u0027ve tried various combinations on pencil and paper is trial and error,"},{"Start":"00:20.820 ","End":"00:24.170","Text":"but there\u0027s one thing that often works is"},{"Start":"00:24.170 ","End":"00:28.490","Text":"to convert everything in terms of sine and cosine."},{"Start":"00:28.490 ","End":"00:31.045","Text":"That will often do."},{"Start":"00:31.045 ","End":"00:33.000","Text":"The left-hand side,"},{"Start":"00:33.000 ","End":"00:36.000","Text":"I could write the cosecant is"},{"Start":"00:36.000 ","End":"00:41.945","Text":"1/sine and I\u0027m hoping at the end to get to sine/cosine, which is tangent."},{"Start":"00:41.945 ","End":"00:47.000","Text":"Let\u0027s do that. This left hand side is equal to cosine and this"},{"Start":"00:47.000 ","End":"00:52.670","Text":"is the Greek letter Beta, of course you know that."},{"Start":"00:52.670 ","End":"01:01.590","Text":"We have cosine Beta/1/sine Beta"},{"Start":"01:01.590 ","End":"01:05.280","Text":"minus sine Beta."},{"Start":"01:05.280 ","End":"01:08.970","Text":"Now, algebraic common denominator,"},{"Start":"01:08.970 ","End":"01:12.580","Text":"I\u0027ll just multiply top and bottom by sine Beta,"},{"Start":"01:12.580 ","End":"01:16.100","Text":"that will get rid of this fraction in the denominator."},{"Start":"01:16.100 ","End":"01:20.540","Text":"If I take sine Beta, multiply the top,"},{"Start":"01:20.540 ","End":"01:25.175","Text":"and then on the bottom the denominator multiplied by sine Beta,"},{"Start":"01:25.175 ","End":"01:28.190","Text":"I get 1 minus sine Beta."},{"Start":"01:28.190 ","End":"01:33.840","Text":"Sine Beta, which is sine^2 Beta."},{"Start":"01:35.230 ","End":"01:40.595","Text":"Next thing is to use the famous Pythagorean identity,"},{"Start":"01:40.595 ","End":"01:45.235","Text":"easily remembered as cosine^2 plus sine^2 is 1,"},{"Start":"01:45.235 ","End":"01:49.580","Text":"and so therefore 1 minus sine^2 is cosine^2."},{"Start":"01:49.580 ","End":"01:54.560","Text":"We have sine Beta cosine better than numerator,"},{"Start":"01:54.560 ","End":"01:55.895","Text":"I\u0027m leaving as is,"},{"Start":"01:55.895 ","End":"02:00.975","Text":"and here, cosine^2 of Beta."},{"Start":"02:00.975 ","End":"02:05.660","Text":"We can cancel cosine Beta from the top and from the bottom on the bottom,"},{"Start":"02:05.660 ","End":"02:15.035","Text":"it just means we remove the two and so what we\u0027re left with is sine Beta/cosine Beta."},{"Start":"02:15.035 ","End":"02:20.700","Text":"That\u0027s another quotient identity that sine/cosine is tangent,"},{"Start":"02:20.700 ","End":"02:26.010","Text":"so this is equal to tangent Beta and we are done."}],"ID":10752},{"Watched":false,"Name":"Exercise 10","Duration":"4m 10s","ChapterTopicVideoID":10393,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10393.jpeg","UploadDate":"2017-11-02T16:01:15.4470000","DurationForVideoObject":"PT4M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"Here we have another trigonometric identity to prove or"},{"Start":"00:04.650 ","End":"00:08.970","Text":"verify as written here, I won\u0027t read it out."},{"Start":"00:08.970 ","End":"00:12.344","Text":"They\u0027re going to use the common trick."},{"Start":"00:12.344 ","End":"00:14.460","Text":"It\u0027s not really a trick, it\u0027s a technique,"},{"Start":"00:14.460 ","End":"00:18.900","Text":"of writing everything in terms of sine and cosine. What do I have here?"},{"Start":"00:18.900 ","End":"00:21.630","Text":"Altogether I have a cosecant and a cotangent."},{"Start":"00:21.630 ","End":"00:25.080","Text":"I just have to remember that cosecant,"},{"Start":"00:25.080 ","End":"00:26.400","Text":"I won\u0027t even bother writing the x."},{"Start":"00:26.400 ","End":"00:34.600","Text":"In general, cosecant is 1/sin and cotangent is equal to cos/sin."},{"Start":"00:35.330 ","End":"00:40.390","Text":"All the 4 other ones can be written in terms of sine and cosine."},{"Start":"00:40.390 ","End":"00:42.965","Text":"If I plug that in here,"},{"Start":"00:42.965 ","End":"00:52.290","Text":"then we can get that the left-hand side is equal to cosecant^2 is 1/sin^2 squared,"},{"Start":"00:52.290 ","End":"00:57.640","Text":"and the argument is x minus,"},{"Start":"00:58.310 ","End":"01:01.840","Text":"then here I have"},{"Start":"01:04.130 ","End":"01:15.105","Text":"(1+cotangent is cosx/sinx) ^2."},{"Start":"01:15.105 ","End":"01:16.955","Text":"That\u0027s the left-hand side."},{"Start":"01:16.955 ","End":"01:19.980","Text":"Let\u0027s see, what do we get?"},{"Start":"01:20.560 ","End":"01:31.100","Text":"Don\u0027t forget from algebra that (a+b)^2 is a^2+2ab+ b^2."},{"Start":"01:31.100 ","End":"01:32.200","Text":"I know you know it by heart,"},{"Start":"01:32.200 ","End":"01:35.030","Text":"but sometimes people just forget this."},{"Start":"01:35.550 ","End":"01:41.200","Text":"The first term as is sin^2x minus."},{"Start":"01:41.200 ","End":"01:46.585","Text":"Now, there\u0027s brackets here,"},{"Start":"01:46.585 ","End":"01:49.060","Text":"I\u0027ll just put everything with a minus sign."},{"Start":"01:49.060 ","End":"01:50.695","Text":"I\u0027ll open the minus out."},{"Start":"01:50.695 ","End":"01:52.900","Text":"This plus this squared is this squared,"},{"Start":"01:52.900 ","End":"01:54.745","Text":"which is 1^2, which is 1,"},{"Start":"01:54.745 ","End":"01:56.995","Text":"then twice this times this,"},{"Start":"01:56.995 ","End":"01:59.125","Text":"but it\u0027s with a minus,"},{"Start":"01:59.125 ","End":"02:05.765","Text":"so it\u0027s twice cosx/sinx."},{"Start":"02:05.765 ","End":"02:07.185","Text":"Then this thing squared,"},{"Start":"02:07.185 ","End":"02:14.830","Text":"but still with the -cos^2x/sin^2x."},{"Start":"02:16.070 ","End":"02:19.880","Text":"Still working on the left-hand side."},{"Start":"02:19.880 ","End":"02:26.465","Text":"Now this can be written in terms of common denominator."},{"Start":"02:26.465 ","End":"02:31.490","Text":"I could put everything over sin^2x. Let\u0027s do that."},{"Start":"02:31.490 ","End":"02:36.650","Text":"I\u0027ll put a dividing line and here sin^2x."},{"Start":"02:36.650 ","End":"02:38.485","Text":"Here I have 1."},{"Start":"02:38.485 ","End":"02:42.960","Text":"Here I have -sin^2x."},{"Start":"02:43.330 ","End":"02:47.455","Text":"Then here I still have to complete another sign,"},{"Start":"02:47.455 ","End":"02:53.570","Text":"x. It\u0027s 2cosx sinx."},{"Start":"02:53.570 ","End":"02:56.165","Text":"It\u0027s a bit longer."},{"Start":"02:56.165 ","End":"03:04.860","Text":"Then -cos^2x. Let\u0027s see."},{"Start":"03:05.120 ","End":"03:10.545","Text":"Look, I have a 1 here and here I have a sin^2 and a cos^2."},{"Start":"03:10.545 ","End":"03:18.045","Text":"If we remember that sin^2 of whatever plus cos^2 of whatever,"},{"Start":"03:18.045 ","End":"03:19.995","Text":"same thing is equal to 1,"},{"Start":"03:19.995 ","End":"03:23.625","Text":"then 1-sin^2-cos^2 is 0."},{"Start":"03:23.625 ","End":"03:25.705","Text":"It\u0027s like 1,"},{"Start":"03:25.705 ","End":"03:30.170","Text":"and then it cancels with the -sin^2x and the -cos^2x."},{"Start":"03:30.170 ","End":"03:32.000","Text":"After I\u0027ve done that,"},{"Start":"03:32.000 ","End":"03:35.809","Text":"then I can cancel the sine from the top and the bottom"},{"Start":"03:35.809 ","End":"03:39.510","Text":"and take this sinx with one of these."},{"Start":"03:39.510 ","End":"03:42.390","Text":"That\u0027s like crossing it with that."},{"Start":"03:42.390 ","End":"03:44.310","Text":"What we\u0027re left with,"},{"Start":"03:44.310 ","End":"03:49.330","Text":"let\u0027s see, is -2cosx/sinx."},{"Start":"03:54.700 ","End":"03:57.965","Text":"The only thing now to remember is that,"},{"Start":"03:57.965 ","End":"03:59.150","Text":"well, we don\u0027t have to remember it."},{"Start":"03:59.150 ","End":"04:03.050","Text":"I wrote it here that cotangent is cos/sin."},{"Start":"04:03.050 ","End":"04:07.715","Text":"This is just equal to -2cotangent x,"},{"Start":"04:07.715 ","End":"04:10.200","Text":"and we are done."}],"ID":10753},{"Watched":false,"Name":"Exercise 11","Duration":"1m 36s","ChapterTopicVideoID":10394,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10394.jpeg","UploadDate":"2017-11-02T16:01:20.7970000","DurationForVideoObject":"PT1M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"Here we have another trigonometric identity that we need to prove,"},{"Start":"00:05.175 ","End":"00:08.865","Text":"verify, demonstrate, show whatever."},{"Start":"00:08.865 ","End":"00:14.850","Text":"I\u0027m going to use the technique of converting everything to sines and cosine."},{"Start":"00:14.850 ","End":"00:21.920","Text":"Recall that tangent of something could"},{"Start":"00:21.920 ","End":"00:28.475","Text":"be u is equal to sine u over cosine u,"},{"Start":"00:28.475 ","End":"00:34.770","Text":"and we\u0027ll start with the left-hand side and see if we can reach the right-hand side."},{"Start":"00:35.330 ","End":"00:40.680","Text":"1 over tangent u would be the upside-down of this."},{"Start":"00:40.680 ","End":"00:46.110","Text":"We would have cosine u over sine u,"},{"Start":"00:46.110 ","End":"00:48.690","Text":"that\u0027s this part Plus"},{"Start":"00:48.690 ","End":"00:56.475","Text":"this part is just sine u over cosine u,"},{"Start":"00:56.475 ","End":"01:00.065","Text":"and then we\u0027ll work on the left-hand side and see if we can reach the right."},{"Start":"01:00.065 ","End":"01:02.585","Text":"Now it put a common denominator."},{"Start":"01:02.585 ","End":"01:07.595","Text":"The common denominator would be just sine u times"},{"Start":"01:07.595 ","End":"01:14.600","Text":"cosine u and cross-multiply cosine times cosine is cos^u,"},{"Start":"01:14.600 ","End":"01:20.120","Text":"sine times sine is sin^u."},{"Start":"01:20.120 ","End":"01:22.970","Text":"Then all we have to do is remember that there\u0027s"},{"Start":"01:22.970 ","End":"01:26.120","Text":"a Pythagorean identity that cosine squared"},{"Start":"01:26.120 ","End":"01:32.045","Text":"plus sine squared u is equal to 1 and copy the denominator,"},{"Start":"01:32.045 ","End":"01:34.340","Text":"and this is the right-hand side."},{"Start":"01:34.340 ","End":"01:36.630","Text":"We are done."}],"ID":10754},{"Watched":false,"Name":"Exercise 12","Duration":"1m 46s","ChapterTopicVideoID":10395,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10395.jpeg","UploadDate":"2017-11-02T16:01:27.1570000","DurationForVideoObject":"PT1M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we have another trigonometric identity to"},{"Start":"00:04.950 ","End":"00:09.705","Text":"verify or prove. This is what it is."},{"Start":"00:09.705 ","End":"00:11.775","Text":"We only have sines and cosines."},{"Start":"00:11.775 ","End":"00:19.110","Text":"But this expression in the parentheses really reminds one of"},{"Start":"00:19.110 ","End":"00:28.965","Text":"the algebraic identity where (a-b)^2 is a^2 minus 2ab plus b^2."},{"Start":"00:28.965 ","End":"00:30.630","Text":"I think it\u0027s pretty straightforward."},{"Start":"00:30.630 ","End":"00:39.525","Text":"If we let a=1 and b is cosine^2(x),"},{"Start":"00:39.525 ","End":"00:41.980","Text":"then that\u0027s what we should get."},{"Start":"00:43.550 ","End":"00:46.640","Text":"I won\u0027t copy, I\u0027ll just work straight from here."},{"Start":"00:46.640 ","End":"00:48.905","Text":"Sine x as is,"},{"Start":"00:48.905 ","End":"00:57.635","Text":"and this will be (1-cosine^2 x)^2 from this formula here,"},{"Start":"00:57.635 ","End":"01:01.580","Text":"where a is 1 and b is the cosine^2"},{"Start":"01:01.580 ","End":"01:08.350","Text":"x. Cosine^(2)^2 is cosine^4."},{"Start":"01:08.350 ","End":"01:10.700","Text":"Now this is equal to,"},{"Start":"01:10.700 ","End":"01:16.210","Text":"using the Pythagorean identity that 1 minus cosine^2 is sine^2,"},{"Start":"01:16.210 ","End":"01:21.630","Text":"so we have sine x times sine^2"},{"Start":"01:21.630 ","End":"01:27.180","Text":"x^2."},{"Start":"01:27.180 ","End":"01:29.970","Text":"It\u0027s pretty straightforward that this is sine^5."},{"Start":"01:29.970 ","End":"01:34.750","Text":"I mean, sine^(2)^2 is sine^4."},{"Start":"01:34.750 ","End":"01:39.290","Text":"I don\u0027t think there\u0027s any need to write an intermediate step,"},{"Start":"01:39.290 ","End":"01:43.775","Text":"sine^(2)^2 sine^4 sine^24 times sine is sine^5 x."},{"Start":"01:43.775 ","End":"01:46.500","Text":"We are done."}],"ID":10755},{"Watched":false,"Name":"Exercise 13","Duration":"2m 32s","ChapterTopicVideoID":10396,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10396.jpeg","UploadDate":"2017-11-02T16:01:35.6300000","DurationForVideoObject":"PT2M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:10.125","Text":"In this exercise, we have a trigonometric identity to verify, show, prove, whatever."},{"Start":"00:10.125 ","End":"00:13.770","Text":"As usual, we\u0027ll start from"},{"Start":"00:13.770 ","End":"00:18.630","Text":"the left hand side and see if we can work our way towards the right hand side."},{"Start":"00:18.630 ","End":"00:21.950","Text":"But, there is a bit of algebra here,"},{"Start":"00:21.950 ","End":"00:24.960","Text":"we have something cubed minus something cubed."},{"Start":"00:24.960 ","End":"00:27.870","Text":"There is a formula in algebra, in fact,"},{"Start":"00:27.870 ","End":"00:32.565","Text":"there\u0027s a general formula for a^n minus b^n."},{"Start":"00:32.565 ","End":"00:34.590","Text":"But I won\u0027t use the general one,"},{"Start":"00:34.590 ","End":"00:42.110","Text":"we\u0027ll just use the one for n equals 3. a^3 minus b^3 is"},{"Start":"00:42.110 ","End":"00:51.425","Text":"equal to a minus b times a^2 plus ab plus b^2."},{"Start":"00:51.425 ","End":"00:56.525","Text":"Of course, you could just multiply this out and see that this is what you get,"},{"Start":"00:56.525 ","End":"01:01.055","Text":"or we could just remember it."},{"Start":"01:01.055 ","End":"01:03.800","Text":"Go look it up in an algebra book."},{"Start":"01:03.800 ","End":"01:12.000","Text":"Let\u0027s apply that here with a being cos(t) and b as sin(t)."},{"Start":"01:12.690 ","End":"01:16.510","Text":"We get using algebra on the top here."},{"Start":"01:16.510 ","End":"01:23.530","Text":"We have cos(t) minus sin(t) and"},{"Start":"01:23.530 ","End":"01:29.230","Text":"then cos^2(t) plus cos(t)"},{"Start":"01:29.230 ","End":"01:35.615","Text":", sin(t) plus sin^2(t)."},{"Start":"01:35.615 ","End":"01:45.600","Text":"All this is the numerator over cos(t) minus sin(t)."},{"Start":"01:45.600 ","End":"01:51.165","Text":"Now look, cos(t) minus sin(t) on the top and the bottom."},{"Start":"01:51.165 ","End":"01:54.630","Text":"This will cancel with this."},{"Start":"01:54.630 ","End":"01:56.779","Text":"There\u0027s a 1 on the denominator,"},{"Start":"01:56.779 ","End":"01:59.700","Text":"but it\u0027s like no denominator now."},{"Start":"02:00.520 ","End":"02:05.460","Text":"The other thing I could do is combine these two."},{"Start":"02:06.010 ","End":"02:11.195","Text":"What we\u0027re left with is this plus this equals 1."},{"Start":"02:11.195 ","End":"02:15.125","Text":"From Pythagorean identity, the most basic one,"},{"Start":"02:15.125 ","End":"02:16.775","Text":"cos^2 plus sin^2 is 1."},{"Start":"02:16.775 ","End":"02:24.510","Text":"We have 1 plus cos(t) sin(t)."},{"Start":"02:24.510 ","End":"02:27.570","Text":"That\u0027s it because this is canceled already denominator\u0027s 1,"},{"Start":"02:27.570 ","End":"02:29.780","Text":"that\u0027s all there is, and that\u0027s equal to this."},{"Start":"02:29.780 ","End":"02:32.970","Text":"So yes, we are done."}],"ID":10756},{"Watched":false,"Name":"Exercise 14","Duration":"2m 17s","ChapterTopicVideoID":10397,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10397.jpeg","UploadDate":"2017-11-02T16:01:43.4900000","DurationForVideoObject":"PT2M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.485","Text":"In this exercise, we have to prove the following trigonometric identity."},{"Start":"00:08.485 ","End":"00:11.420","Text":"We\u0027ll start from the left-hand side,"},{"Start":"00:11.420 ","End":"00:14.575","Text":"and work our way to the right-hand side."},{"Start":"00:14.575 ","End":"00:19.470","Text":"One thing that\u0027s commonly done is to write everything in terms of sine and cosine."},{"Start":"00:19.470 ","End":"00:28.020","Text":"For example, the cotangent of whatever is equal to the cosine"},{"Start":"00:28.020 ","End":"00:37.850","Text":"over the sine of that same thing and the tangent is sine over cosine."},{"Start":"00:37.850 ","End":"00:41.575","Text":"If I use that here on the left,"},{"Start":"00:41.575 ","End":"00:48.425","Text":"then we shall get cosine x over sine x"},{"Start":"00:48.425 ","End":"00:55.755","Text":"minus sine x over cosine x over."},{"Start":"00:55.755 ","End":"01:00.600","Text":"Then cosine x over sine x as above,"},{"Start":"01:00.600 ","End":"01:06.425","Text":"this time with a plus for the sine x over the cosine x."},{"Start":"01:06.425 ","End":"01:08.420","Text":"Now, we want to do is simplify this,"},{"Start":"01:08.420 ","End":"01:11.450","Text":"get rid of the fractions within the fractions."},{"Start":"01:11.450 ","End":"01:19.235","Text":"What I suggest is multiplying top and bottom of the big fraction by sine x cosine x,"},{"Start":"01:19.235 ","End":"01:22.045","Text":"that will get rid of all the fractions."},{"Start":"01:22.045 ","End":"01:26.510","Text":"If we multiply the top by sine x cosine x,"},{"Start":"01:26.510 ","End":"01:30.200","Text":"we just get the sum of the cross-product, no the difference."},{"Start":"01:30.200 ","End":"01:31.955","Text":"We get cosine, cosine,"},{"Start":"01:31.955 ","End":"01:38.269","Text":"which is cosine squared x minus sine squared x."},{"Start":"01:38.269 ","End":"01:40.010","Text":"On the denominator,"},{"Start":"01:40.010 ","End":"01:41.825","Text":"we get the same thing with a plus."},{"Start":"01:41.825 ","End":"01:47.555","Text":"Cosine squared x plus sine squared x."},{"Start":"01:47.555 ","End":"01:58.620","Text":"Now, what we have to do is to remember that cosine squared x plus sine squared x is 1."},{"Start":"02:00.190 ","End":"02:06.200","Text":"This on the Pythagorean identities is equal to 1."},{"Start":"02:06.200 ","End":"02:09.050","Text":"Let\u0027s say we just have the numerator,"},{"Start":"02:09.050 ","End":"02:12.380","Text":"which is cosine squared x minus sine squared x."},{"Start":"02:12.380 ","End":"02:14.780","Text":"This isn\u0027t it exactly what\u0027s written here."},{"Start":"02:14.780 ","End":"02:17.460","Text":"Yeah, we are done."}],"ID":10757},{"Watched":false,"Name":"Exercise 15","Duration":"1m 49s","ChapterTopicVideoID":10398,"CourseChapterTopicPlaylistID":257203,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10398.jpeg","UploadDate":"2017-11-02T16:01:49.6830000","DurationForVideoObject":"PT1M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.585","Text":"In this exercise, we have to show that this is an identity."},{"Start":"00:06.585 ","End":"00:08.460","Text":"It\u0027s not purely trigonometric,"},{"Start":"00:08.460 ","End":"00:10.740","Text":"it has some logarithms in it."},{"Start":"00:10.740 ","End":"00:15.165","Text":"But let\u0027s see how we do this."},{"Start":"00:15.165 ","End":"00:18.330","Text":"Of course, this means that it\u0027s an identity wherever it\u0027s"},{"Start":"00:18.330 ","End":"00:23.445","Text":"defined for both the left-hand and right-hand sides."},{"Start":"00:23.445 ","End":"00:26.490","Text":"Let\u0027s not worry about that."},{"Start":"00:26.490 ","End":"00:28.995","Text":"Now, secant and cosine,"},{"Start":"00:28.995 ","End":"00:31.080","Text":"that\u0027s one of the differences between left and right."},{"Start":"00:31.080 ","End":"00:34.690","Text":"What is the relationship between secant and cosine?"},{"Start":"00:34.690 ","End":"00:39.320","Text":"The key thing here is that secant, in this case,"},{"Start":"00:39.320 ","End":"00:43.760","Text":"Alpha is the reciprocal of the cosine."},{"Start":"00:43.760 ","End":"00:47.310","Text":"This is the key to solving this."},{"Start":"00:47.660 ","End":"00:52.425","Text":"The left-hand side is equal to"},{"Start":"00:52.425 ","End":"01:01.085","Text":"natural log of 1/cosine Alpha."},{"Start":"01:01.085 ","End":"01:04.670","Text":"Now by the properties of the logarithm,"},{"Start":"01:04.670 ","End":"01:09.200","Text":"I can either do it as a quotient and say it\u0027s natural log of"},{"Start":"01:09.200 ","End":"01:15.930","Text":"1 minus natural log of cosine Alpha."},{"Start":"01:15.930 ","End":"01:23.115","Text":"The other way to do it would be to say that this is cosine Alpha^ minus 1."},{"Start":"01:23.115 ","End":"01:25.700","Text":"It gets to the same thing in the end because"},{"Start":"01:25.700 ","End":"01:28.895","Text":"natural log of 1 is 0 so either way we would get"},{"Start":"01:28.895 ","End":"01:34.800","Text":"to minus natural log of cosine Alpha."},{"Start":"01:34.800 ","End":"01:40.640","Text":"Probably best to put brackets here because this natural log of 1 is 0."},{"Start":"01:40.640 ","End":"01:45.680","Text":"That\u0027s it. It was easier than you might have thought."},{"Start":"01:45.680 ","End":"01:50.710","Text":"But here\u0027s the key. Done."}],"ID":10758}],"Thumbnail":null,"ID":257203},{"Name":"Advanced Trigonometric Identities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Explanation and Examples - Part 1","Duration":"8m 15s","ChapterTopicVideoID":13609,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13609.jpeg","UploadDate":"2021-06-28T14:51:11.8330000","DurationForVideoObject":"PT8M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"Continuing with trigonometric identities."},{"Start":"00:02.850 ","End":"00:08.670","Text":"Next, the double-angle identities that tells us the sine,"},{"Start":"00:08.670 ","End":"00:11.175","Text":"cosine, and tangent of twice an angle,"},{"Start":"00:11.175 ","End":"00:13.095","Text":"sine 2 Alpha,"},{"Start":"00:13.095 ","End":"00:18.150","Text":"we have to know both sine Alpha and cosine Alpha for this and it\u0027s this formula,"},{"Start":"00:18.150 ","End":"00:19.470","Text":"for cosine 2 Alpha,"},{"Start":"00:19.470 ","End":"00:24.825","Text":"we have a choice of 3 formulas that we usually use one of these two,"},{"Start":"00:24.825 ","End":"00:29.220","Text":"because you either have the cosine of Alpha or you have the sine of Alpha,"},{"Start":"00:29.220 ","End":"00:31.980","Text":"and from that we compute cosine 2 Alpha,"},{"Start":"00:31.980 ","End":"00:38.130","Text":"and then there\u0027s also a tangent of 2 Alpha computed from tangent Alpha."},{"Start":"00:38.130 ","End":"00:41.060","Text":"I\u0027ll just do one example because there are"},{"Start":"00:41.060 ","End":"00:45.889","Text":"examples that are solved in the exercises following the tutorial."},{"Start":"00:45.889 ","End":"00:51.105","Text":"Let\u0027s say we\u0027re given that sine of,"},{"Start":"00:51.105 ","End":"00:52.510","Text":"let\u0027s say Theta,"},{"Start":"00:52.510 ","End":"00:53.720","Text":"use a different letter,"},{"Start":"00:53.720 ","End":"00:56.720","Text":"is equal to 1/3,"},{"Start":"00:56.720 ","End":"01:03.025","Text":"we have to find cosine of 2 Theta."},{"Start":"01:03.025 ","End":"01:06.770","Text":"We see that the formula that will help us is this one"},{"Start":"01:06.770 ","End":"01:10.745","Text":"that gives us cosine of twice the angle in terms of the sine,"},{"Start":"01:10.745 ","End":"01:19.205","Text":"and we say, cosine 2 Theta is 1 minus 2 sine squared Theta."},{"Start":"01:19.205 ","End":"01:23.245","Text":"In our case this is equal to 1 minus 2,"},{"Start":"01:23.245 ","End":"01:27.165","Text":"and sine Theta was given to be a 1/3^2,"},{"Start":"01:27.165 ","End":"01:29.585","Text":"a 1/3^2 is a 1/9,"},{"Start":"01:29.585 ","End":"01:32.535","Text":"twice 1/9 is 2/9,"},{"Start":"01:32.535 ","End":"01:35.490","Text":"1 minus 2/9 is 7/9."},{"Start":"01:35.490 ","End":"01:39.240","Text":"I\u0027ll settle for that,"},{"Start":"01:39.240 ","End":"01:41.652","Text":"you have these written."},{"Start":"01:41.652 ","End":"01:44.565","Text":"Let\u0027s move on to the next."},{"Start":"01:44.565 ","End":"01:48.960","Text":"Next, we have the half angle identities,"},{"Start":"01:48.960 ","End":"01:52.580","Text":"sometimes these are called the half angle identities,"},{"Start":"01:52.580 ","End":"01:55.115","Text":"and these are the half angle formulas."},{"Start":"01:55.115 ","End":"01:56.825","Text":"Anyway, there\u0027s 2 sets,"},{"Start":"01:56.825 ","End":"02:04.775","Text":"and the relation is that if you let Theta equals 2x,"},{"Start":"02:04.775 ","End":"02:12.130","Text":"then x is Theta over 2 and these two become these two,"},{"Start":"02:12.130 ","End":"02:14.930","Text":"and there\u0027s also a formula for the tangent of"},{"Start":"02:14.930 ","End":"02:20.235","Text":"a half angle in terms of the sine and cosine, there\u0027s 2 variations."},{"Start":"02:20.235 ","End":"02:27.410","Text":"A thing to watch out for in these are these 2 formulas because there\u0027s a plus or minus,"},{"Start":"02:27.410 ","End":"02:31.610","Text":"and that\u0027s the only real difficulty other than that it\u0027s just substituting,"},{"Start":"02:31.610 ","End":"02:33.590","Text":"is to decide the plus or minus,"},{"Start":"02:33.590 ","End":"02:40.370","Text":"and this usually we do because we know the quadrant in which Theta lies."},{"Start":"02:40.370 ","End":"02:43.680","Text":"I\u0027ll just give 1 example,"},{"Start":"02:46.430 ","End":"02:55.035","Text":"suppose we are given that cosine of Theta is equal to"},{"Start":"02:55.035 ","End":"03:03.015","Text":"1/3 and that Theta is in the fourth quadrant,"},{"Start":"03:03.015 ","End":"03:09.000","Text":"quadrant IV, Latin."},{"Start":"03:09.000 ","End":"03:15.290","Text":"The question is, find the cosine"},{"Start":"03:15.290 ","End":"03:22.280","Text":"of Theta over 2 and sine Theta over 2,"},{"Start":"03:22.280 ","End":"03:24.460","Text":"let\u0027s find them both."},{"Start":"03:24.460 ","End":"03:27.410","Text":"Now in the solution, like I said,"},{"Start":"03:27.410 ","End":"03:31.415","Text":"the most important thing is to find out the sine."},{"Start":"03:31.415 ","End":"03:34.370","Text":"What is quadrant 4?"},{"Start":"03:34.370 ","End":"03:43.160","Text":"It means that Theta is between 270 degrees and 360 degrees."},{"Start":"03:43.160 ","End":"03:48.765","Text":"In other words, 2 Pi and 3 Pi over 2,"},{"Start":"03:48.765 ","End":"03:55.125","Text":"and this means that Theta over 2 is between Pi,"},{"Start":"03:55.125 ","End":"04:00.610","Text":"or rather between 3 Pi over 4 and Pi."},{"Start":"04:00.610 ","End":"04:02.510","Text":"Now this is a 135 degrees,"},{"Start":"04:02.510 ","End":"04:04.040","Text":"this is 180 degrees,"},{"Start":"04:04.040 ","End":"04:07.530","Text":"but it\u0027s all in second quadrant,"},{"Start":"04:07.530 ","End":"04:13.320","Text":"so Theta over 2 is in quadrant number 2,"},{"Start":"04:13.320 ","End":"04:18.135","Text":"and so sine is"},{"Start":"04:18.135 ","End":"04:25.590","Text":"positive and the cosine in quadrant 2 is negative."},{"Start":"04:25.590 ","End":"04:30.170","Text":"Now we get to substituting in these 2 formulas,"},{"Start":"04:30.170 ","End":"04:35.690","Text":"sine of Theta over 2 is equal to, because it\u0027s positive,"},{"Start":"04:35.690 ","End":"04:45.450","Text":"it\u0027s just the plus square root of 1 minus cosine Theta is a 1/3 over 2,"},{"Start":"04:45.450 ","End":"04:48.660","Text":"and this equals 1 minus a 1/3 is 2/3,"},{"Start":"04:48.660 ","End":"04:50.805","Text":"2/3 over 2 is a 1/3,"},{"Start":"04:50.805 ","End":"04:58.430","Text":"that\u0027s the square root of 1/3 and the cosine of Theta over 2 is going to be,"},{"Start":"04:58.430 ","End":"05:07.785","Text":"because it\u0027s negative minus the square root of 1 plus a 1/3 over 2,"},{"Start":"05:07.785 ","End":"05:10.365","Text":"which is minus the square root,"},{"Start":"05:10.365 ","End":"05:12.150","Text":"1 and 1/3 is 4/3,"},{"Start":"05:12.150 ","End":"05:15.585","Text":"so here we have 2/3."},{"Start":"05:15.585 ","End":"05:18.535","Text":"Let\u0027s move on to the next."},{"Start":"05:18.535 ","End":"05:23.180","Text":"Next we have the product to sum identities,"},{"Start":"05:23.180 ","End":"05:25.370","Text":"which in my experience are not often used,"},{"Start":"05:25.370 ","End":"05:28.250","Text":"but they are used and you should know them,"},{"Start":"05:28.250 ","End":"05:29.795","Text":"and there\u0027s 4 of them,"},{"Start":"05:29.795 ","End":"05:31.805","Text":"sine times the cosine,"},{"Start":"05:31.805 ","End":"05:35.720","Text":"cosine times the sine or sine sine or cosine cosine,"},{"Start":"05:35.720 ","End":"05:40.040","Text":"it converts a product of trigonometric functions into a sum,"},{"Start":"05:40.040 ","End":"05:43.745","Text":"and the sum will either be sines or cosines."},{"Start":"05:43.745 ","End":"05:46.865","Text":"Let\u0027s just do an example,"},{"Start":"05:46.865 ","End":"05:48.995","Text":"not with numbers, but let\u0027s say,"},{"Start":"05:48.995 ","End":"05:53.565","Text":"I ask you to take cosine 5x,"},{"Start":"05:53.565 ","End":"06:00.800","Text":"sine 4x and write it as a sum rather than a product."},{"Start":"06:00.800 ","End":"06:04.889","Text":"What we would do is locate the right formula."},{"Start":"06:04.889 ","End":"06:09.640","Text":"We have cosine times sine, this one."},{"Start":"06:09.710 ","End":"06:16.700","Text":"We get 1/2 of sine."},{"Start":"06:16.700 ","End":"06:20.375","Text":"Now alpha, this is the Alpha and this is the Beta,"},{"Start":"06:20.375 ","End":"06:24.900","Text":"Alpha plus Beta is 9x,"},{"Start":"06:25.130 ","End":"06:32.825","Text":"and then we have minus sine of Alpha minus Beta,"},{"Start":"06:32.825 ","End":"06:34.850","Text":"which is x,"},{"Start":"06:34.850 ","End":"06:40.965","Text":"1/2 sine 9x minus"},{"Start":"06:40.965 ","End":"06:46.705","Text":"1/2 sine x and that\u0027s it."},{"Start":"06:46.705 ","End":"06:55.190","Text":"These are the sum to product identities that express the sum of 2 sines or"},{"Start":"06:55.190 ","End":"07:04.250","Text":"the difference in terms of a product of sine and cosine and sometimes cosine,"},{"Start":"07:04.250 ","End":"07:05.420","Text":"cosine or sine and sine,"},{"Start":"07:05.420 ","End":"07:09.080","Text":"but it converts a sum or difference to a product."},{"Start":"07:09.080 ","End":"07:12.140","Text":"Let\u0027s just take one example,"},{"Start":"07:12.140 ","End":"07:17.510","Text":"let\u0027s convert cosine 9x plus"},{"Start":"07:17.510 ","End":"07:23.910","Text":"cosine 5x into a product."},{"Start":"07:24.170 ","End":"07:28.470","Text":"We\u0027re going to use this formula,"},{"Start":"07:28.470 ","End":"07:32.580","Text":"we get 2 cosine,"},{"Start":"07:32.580 ","End":"07:36.090","Text":"now this is Alpha and this is Beta,"},{"Start":"07:36.090 ","End":"07:42.840","Text":"so Alpha plus Beta over 2 is 9x plus 5x over"},{"Start":"07:42.840 ","End":"07:52.065","Text":"2 and then cosine of 9x minus 5x over 2."},{"Start":"07:52.065 ","End":"07:59.560","Text":"This equals to cosine 9 plus 5 is 14 over 2 is 7x,"},{"Start":"08:00.080 ","End":"08:05.610","Text":"cosine 9 minus 5 is 4 over 2 is 2,"},{"Start":"08:05.610 ","End":"08:11.365","Text":"so the answer is 2 cosine 7x cosine 2x and that\u0027s it."},{"Start":"08:11.365 ","End":"08:15.900","Text":"That concludes the trigonometric identities."}],"ID":14328},{"Watched":false,"Name":"Explanation and Examples - Part 2","Duration":"8m 5s","ChapterTopicVideoID":13610,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13610.jpeg","UploadDate":"2021-06-28T14:51:46.1930000","DurationForVideoObject":"PT8M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"We\u0027re continuing with trigonometric identities."},{"Start":"00:03.675 ","End":"00:06.810","Text":"There\u0027s quite a lot of them and you usually get"},{"Start":"00:06.810 ","End":"00:10.515","Text":"a formula sheet because it\u0027s just too many to memorize."},{"Start":"00:10.515 ","End":"00:16.965","Text":"The next group is the angle sum or angle difference identities."},{"Start":"00:16.965 ","End":"00:18.300","Text":"The sine of a sum,"},{"Start":"00:18.300 ","End":"00:20.910","Text":"the sine of a difference cosine of sum and difference,"},{"Start":"00:20.910 ","End":"00:25.455","Text":"tangent of sum and difference. There they are."},{"Start":"00:25.455 ","End":"00:31.590","Text":"I\u0027ll do an example and the example will be as follows."},{"Start":"00:31.590 ","End":"00:38.280","Text":"We\u0027re given that cosine of Alpha is 3/5,"},{"Start":"00:38.280 ","End":"00:46.455","Text":"and we\u0027re given cosine Beta is 8 over 17."},{"Start":"00:46.455 ","End":"00:51.630","Text":"We\u0027re told that Alpha and Beta"},{"Start":"00:51.630 ","End":"00:58.060","Text":"are in quadrant number 1 and A,"},{"Start":"00:58.060 ","End":"01:06.095","Text":"we have to find what is sine of Alpha plus Beta and b,"},{"Start":"01:06.095 ","End":"01:13.885","Text":"is to find the cosine of Alpha minus Beta."},{"Start":"01:13.885 ","End":"01:19.960","Text":"In a, we can use the formula for sine of Alpha plus Beta."},{"Start":"01:19.960 ","End":"01:21.070","Text":"We look it up here,"},{"Start":"01:21.070 ","End":"01:24.385","Text":"it\u0027s the first 1 and this is equal to"},{"Start":"01:24.385 ","End":"01:33.505","Text":"sine Alpha cosine Beta plus cosine Alpha sine Beta."},{"Start":"01:33.505 ","End":"01:36.785","Text":"Now the thing is that we have cosine Beta,"},{"Start":"01:36.785 ","End":"01:41.160","Text":"this is 8 over 17."},{"Start":"01:41.160 ","End":"01:45.285","Text":"And we have cosine Alpha, which is 3/5."},{"Start":"01:45.285 ","End":"01:46.815","Text":"What we don\u0027t have,"},{"Start":"01:46.815 ","End":"01:49.900","Text":"a sine Alpha, and sine Beta."},{"Start":"01:49.900 ","End":"01:53.840","Text":"For this, we can use a different identity."},{"Start":"01:53.840 ","End":"01:58.375","Text":"The fact, remember that sine squared plus cosine squared is 1."},{"Start":"01:58.375 ","End":"02:08.910","Text":"So sine Alpha is the square root of 1 minus cosine squared Alpha."},{"Start":"02:08.910 ","End":"02:12.030","Text":"Only thing is it has to be plus or minus,"},{"Start":"02:12.030 ","End":"02:17.630","Text":"and we\u0027ll know the plus or minus because of the quadrant information."},{"Start":"02:17.630 ","End":"02:19.400","Text":"Okay, Let\u0027s continue."},{"Start":"02:19.400 ","End":"02:28.250","Text":"This is equal to plus or minus the square root of 1 minus."},{"Start":"02:28.250 ","End":"02:31.565","Text":"Now, we have 3/5 squared,"},{"Start":"02:31.565 ","End":"02:36.920","Text":"which is 9 over 25."},{"Start":"02:36.920 ","End":"02:41.975","Text":"That\u0027s equal to plus or minus the square root"},{"Start":"02:41.975 ","End":"02:48.015","Text":"of 225 minus 9 is 16 over 25."},{"Start":"02:48.015 ","End":"02:51.860","Text":"And this is 4/5."},{"Start":"02:51.860 ","End":"02:54.515","Text":"Do we take the plus or the minus?"},{"Start":"02:54.515 ","End":"02:57.590","Text":"Well, we\u0027re told that Alpha and Beta are in the first quadrant,"},{"Start":"02:57.590 ","End":"02:59.720","Text":"so the answer is just 4/5."},{"Start":"02:59.720 ","End":"03:02.420","Text":"I\u0027ll just emphasize it that we took the plus."},{"Start":"03:02.420 ","End":"03:07.200","Text":"Now let\u0027s find sine of Beta."},{"Start":"03:07.200 ","End":"03:10.040","Text":"We also know that it\u0027s going to be positive,"},{"Start":"03:10.040 ","End":"03:12.335","Text":"so I don\u0027t need the plus or minus,"},{"Start":"03:12.335 ","End":"03:18.510","Text":"but it will be the square root of 1 minus cosine squared Beta,"},{"Start":"03:18.510 ","End":"03:26.295","Text":"which is the square root of 1 minus 8 over 17 squared,"},{"Start":"03:26.295 ","End":"03:31.150","Text":"is 64 over 289."},{"Start":"03:31.150 ","End":"03:37.295","Text":"We get 225 over 289."},{"Start":"03:37.295 ","End":"03:41.525","Text":"This comes out to be 15 in the numerator,"},{"Start":"03:41.525 ","End":"03:45.130","Text":"and 17 on the denominator."},{"Start":"03:45.130 ","End":"03:47.420","Text":"That\u0027s sine Alpha sine Beta,"},{"Start":"03:47.420 ","End":"03:49.010","Text":"but we\u0027re not done."},{"Start":"03:49.010 ","End":"03:51.875","Text":"We have to now substitute in here."},{"Start":"03:51.875 ","End":"03:57.630","Text":"We get that sine of Alpha plus Beta."},{"Start":"03:57.630 ","End":"03:58.710","Text":"We now have everything."},{"Start":"03:58.710 ","End":"04:02.460","Text":"Sine Alpha is 4/5."},{"Start":"04:02.460 ","End":"04:06.375","Text":"Cosine Beta 8 over 17,"},{"Start":"04:06.375 ","End":"04:17.285","Text":"and then 3/5 and then sine of Beta is 15 over 17."},{"Start":"04:17.285 ","End":"04:18.740","Text":"What does this come out?"},{"Start":"04:18.740 ","End":"04:20.300","Text":"Same denominator on both."},{"Start":"04:20.300 ","End":"04:24.515","Text":"5 times 17 is 85."},{"Start":"04:24.515 ","End":"04:28.680","Text":"Here we get 4 times 8 is 32,"},{"Start":"04:28.680 ","End":"04:32.520","Text":"3 times 15 is 45,"},{"Start":"04:32.520 ","End":"04:41.115","Text":"and so 77 over 85."},{"Start":"04:41.115 ","End":"04:43.185","Text":"That\u0027s just part a."},{"Start":"04:43.185 ","End":"04:49.170","Text":"Part b, cosine of Alpha minus Beta."},{"Start":"04:49.170 ","End":"04:51.290","Text":"This is the formula for Part 1."},{"Start":"04:51.290 ","End":"04:54.935","Text":"We use this formula for Part 2, this formula."},{"Start":"04:54.935 ","End":"05:00.030","Text":"We have cosine Alpha,"},{"Start":"05:00.030 ","End":"05:08.560","Text":"cosine Beta plus sine Alpha, sine Beta."},{"Start":"05:08.560 ","End":"05:13.175","Text":"But we can reuse the results from part a."},{"Start":"05:13.175 ","End":"05:19.940","Text":"We had the sine Alpha was 4/5,"},{"Start":"05:19.940 ","End":"05:27.345","Text":"and sine Beta was 15 over 17."},{"Start":"05:27.345 ","End":"05:33.875","Text":"That doesn\u0027t change so now we just have to substitute again and we get this continuing,"},{"Start":"05:33.875 ","End":"05:39.470","Text":"cosine Alpha is 3/5,"},{"Start":"05:39.470 ","End":"05:44.180","Text":"cosine Beta 8/17 plus"},{"Start":"05:44.180 ","End":"05:51.640","Text":"sine Alpha is 4/5 sine Beta 15 over 17."},{"Start":"05:51.640 ","End":"05:57.165","Text":"Just as in part a, it\u0027s over 85."},{"Start":"05:57.165 ","End":"06:00.180","Text":"Here we have 3 times 8 is 24,"},{"Start":"06:00.180 ","End":"06:05.070","Text":"plus 4 times 15 is 60 so the answer"},{"Start":"06:05.070 ","End":"06:10.950","Text":"is 84 over 85."},{"Start":"06:10.950 ","End":"06:15.680","Text":"The next set of identities or the co-function identities."},{"Start":"06:15.680 ","End":"06:22.070","Text":"You notice that the trigonometric functions come in pairs with a co and without a co."},{"Start":"06:22.070 ","End":"06:23.750","Text":"There\u0027s sine and there\u0027s cosine,"},{"Start":"06:23.750 ","End":"06:28.225","Text":"tangent, cotangent, secant, cosecant."},{"Start":"06:28.225 ","End":"06:31.395","Text":"Sine, tangent, or secant of an angle,"},{"Start":"06:31.395 ","End":"06:36.320","Text":"it\u0027s the same as the cofunction of the complement of the angle."},{"Start":"06:36.320 ","End":"06:38.630","Text":"If we have an angle Theta,"},{"Start":"06:38.630 ","End":"06:44.840","Text":"then the complement is pi over 2 minus Theta,"},{"Start":"06:44.840 ","End":"06:49.765","Text":"or if you like, 90 degrees minus Theta."},{"Start":"06:49.765 ","End":"06:55.550","Text":"Just for example, suppose you know that sine 30 degrees,"},{"Start":"06:55.550 ","End":"06:57.440","Text":"I\u0027ll do it in degrees,"},{"Start":"06:57.440 ","End":"07:01.705","Text":"is equal to 1/2."},{"Start":"07:01.705 ","End":"07:12.635","Text":"Then we can say that cosine of 60 degrees is sine of the complement of 90 minus 60,"},{"Start":"07:12.635 ","End":"07:16.235","Text":"which is sine of 30 degrees."},{"Start":"07:16.235 ","End":"07:18.665","Text":"That\u0027s also going be a half."},{"Start":"07:18.665 ","End":"07:20.485","Text":"Do another example,"},{"Start":"07:20.485 ","End":"07:25.550","Text":"if I know that tangent and this time I\u0027ll do it in radians,"},{"Start":"07:25.550 ","End":"07:32.110","Text":"of pi over 3 is the square root of 3."},{"Start":"07:32.110 ","End":"07:40.350","Text":"Then I can figure that the cotangent of Pi over"},{"Start":"07:40.350 ","End":"07:50.615","Text":"6 is equal to the tangent of pi over 2 minus pi over 6."},{"Start":"07:50.615 ","End":"07:53.425","Text":"Pi over 2 minus pi over 6."},{"Start":"07:53.425 ","End":"07:59.840","Text":"This is like pi over 3 and that\u0027s also equal to square root of 3."},{"Start":"07:59.840 ","End":"08:01.310","Text":"There\u0027s nothing much more to it."},{"Start":"08:01.310 ","End":"08:06.750","Text":"Let\u0027s take a break now and then continue with more trigonometric identities."}],"ID":14329},{"Watched":false,"Name":"Exercise 1","Duration":"3m 29s","ChapterTopicVideoID":13611,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13611.jpeg","UploadDate":"2018-10-21T13:37:31.5330000","DurationForVideoObject":"PT3M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.820","Text":"In this exercise, well there\u0027s 8 parts we have to say which quadrant"},{"Start":"00:05.820 ","End":"00:09.030","Text":"the angle theta lies in and we\u0027re given 2 hints about"},{"Start":"00:09.030 ","End":"00:12.750","Text":"the sine of 2 trigonometric functions like in the first one,"},{"Start":"00:12.750 ","End":"00:15.570","Text":"we\u0027re told that the tangent and the cosine of negative,"},{"Start":"00:15.570 ","End":"00:17.400","Text":"which quadrant is theta in."},{"Start":"00:17.400 ","End":"00:21.420","Text":"I brought in a mnemonic that will help"},{"Start":"00:21.420 ","End":"00:29.715","Text":"us all students take calculus all sine,"},{"Start":"00:29.715 ","End":"00:34.380","Text":"tangent, cosine here all the 3 functions"},{"Start":"00:34.380 ","End":"00:39.040","Text":"are positive here just a sine here there\u0027s a tangent and here the cosine."},{"Start":"00:39.350 ","End":"00:45.455","Text":"We go through these but looking ahead I noticed that there\u0027s also a cotangent."},{"Start":"00:45.455 ","End":"00:49.175","Text":"The cotangent is 1 over the tangent,"},{"Start":"00:49.175 ","End":"00:54.435","Text":"so it has the same sine these are pairs if you have"},{"Start":"00:54.435 ","End":"01:00.555","Text":"a secant then it\u0027s like the cosine that\u0027s 1 over cosine so it has the same sine,"},{"Start":"01:00.555 ","End":"01:04.080","Text":"and cosecant is paired with sine."},{"Start":"01:04.080 ","End":"01:08.640","Text":"We really just need the 3 let\u0027s see,"},{"Start":"01:08.640 ","End":"01:10.725","Text":"it\u0027s questionable elimination,"},{"Start":"01:10.725 ","End":"01:13.250","Text":"tangent and cosine are negative,"},{"Start":"01:13.250 ","End":"01:16.430","Text":"so we need to find where tangent and cosine are"},{"Start":"01:16.430 ","End":"01:20.180","Text":"both negative and we find that here tangent and cosine are"},{"Start":"01:20.180 ","End":"01:23.790","Text":"negative so it\u0027s in"},{"Start":"01:23.790 ","End":"01:27.930","Text":"the second quadrant in"},{"Start":"01:27.930 ","End":"01:34.985","Text":"B we need tangent positive and cosine negative."},{"Start":"01:34.985 ","End":"01:40.100","Text":"Tangent positive is either here or here and"},{"Start":"01:40.100 ","End":"01:45.480","Text":"cosine negative makes it this one so that is quadrant 3."},{"Start":"01:47.540 ","End":"01:54.650","Text":"Part C sine and cosine both negative so we just look around and"},{"Start":"01:54.650 ","End":"02:01.375","Text":"see I have sine and cosine negative that is also the 3rd quadrant,"},{"Start":"02:01.375 ","End":"02:06.755","Text":"D we want cosine positive but sine negative."},{"Start":"02:06.755 ","End":"02:09.440","Text":"Cosine is positive in these 2,"},{"Start":"02:09.440 ","End":"02:13.580","Text":"sine is negative in these 2 here is where both happen"},{"Start":"02:13.580 ","End":"02:19.520","Text":"cosine positive sign negative quadrant number 4."},{"Start":"02:19.520 ","End":"02:25.734","Text":"Then we have d, e, f,"},{"Start":"02:25.734 ","End":"02:33.230","Text":"and g. No, I miss count it\u0027s only"},{"Start":"02:33.230 ","End":"02:38.255","Text":"7 in E we want cosine negative"},{"Start":"02:38.255 ","End":"02:43.790","Text":"and tangent positive because cotangent and tangent like I said,"},{"Start":"02:43.790 ","End":"02:45.470","Text":"so cosine negative tangent,"},{"Start":"02:45.470 ","End":"02:48.710","Text":"positive cosine is negative in these 2,"},{"Start":"02:48.710 ","End":"02:55.440","Text":"tangent is positive in these 2 so here it is in the 3rd quadrant."},{"Start":"02:57.800 ","End":"03:02.420","Text":"Let\u0027s see, cotangent negative so we\u0027re looking for"},{"Start":"03:02.420 ","End":"03:06.515","Text":"like tangent negative and sine negative."},{"Start":"03:06.515 ","End":"03:09.860","Text":"Tangent and sine are both negative in quadrant"},{"Start":"03:09.860 ","End":"03:15.740","Text":"4 and sine and tangent negative just by traveling and"},{"Start":"03:15.740 ","End":"03:20.640","Text":"looking also sine and tangent are negative is"},{"Start":"03:20.640 ","End":"03:26.390","Text":"quadrant 4 which is actually really the same as the previous problem tangent and sine."},{"Start":"03:26.390 ","End":"03:29.820","Text":"Yeah. Okay, that\u0027s it."}],"ID":14330},{"Watched":false,"Name":"Exercise 2","Duration":"7m 47s","ChapterTopicVideoID":13612,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13612.jpeg","UploadDate":"2018-10-21T13:38:58.5570000","DurationForVideoObject":"PT7M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.054","Text":"In this exercise, we\u0027re given a point on the terminal side of an angle."},{"Start":"00:05.054 ","End":"00:11.550","Text":"We have to find the value of the trigonometric functions given,"},{"Start":"00:11.550 ","End":"00:14.775","Text":"like in the first case, sine and cosine."},{"Start":"00:14.775 ","End":"00:17.355","Text":"The point is (3,1)."},{"Start":"00:17.355 ","End":"00:23.565","Text":"All we have to do is find out which quadrant this point lies in."},{"Start":"00:23.565 ","End":"00:27.000","Text":"The initial side is always the positive x-axis."},{"Start":"00:27.000 ","End":"00:30.360","Text":"Really, just the question is whether this point is in the first,"},{"Start":"00:30.360 ","End":"00:32.189","Text":"second, third, or fourth quadrant."},{"Start":"00:32.189 ","End":"00:40.510","Text":"In a, the x is positive and the y is positive so it\u0027s the first quadrant."},{"Start":"00:43.330 ","End":"00:49.610","Text":"We know that the sine will be positive and the cosine will be positive here."},{"Start":"00:49.610 ","End":"00:50.990","Text":"But how do we find it?"},{"Start":"00:50.990 ","End":"00:52.760","Text":"When you have the coordinates,"},{"Start":"00:52.760 ","End":"00:55.430","Text":"easiest is to find the tangent of the angle."},{"Start":"00:55.430 ","End":"00:57.605","Text":"This is the rise and the run."},{"Start":"00:57.605 ","End":"01:02.610","Text":"I know that tangent Theta is 1 over 3."},{"Start":"01:03.050 ","End":"01:06.965","Text":"Then we use trigonometric identities to find the other."},{"Start":"01:06.965 ","End":"01:12.500","Text":"For example, I know that there\u0027s a trigonometric identity that"},{"Start":"01:12.500 ","End":"01:22.235","Text":"1 + tangent^2 Theta equals secant^2 Theta."},{"Start":"01:22.235 ","End":"01:25.460","Text":"Or We could write this as 1 over cosine^2."},{"Start":"01:25.460 ","End":"01:26.870","Text":"I guess we\u0027ll do that in a moment."},{"Start":"01:26.870 ","End":"01:32.750","Text":"What we have is that 1 + tangent^2"},{"Start":"01:32.750 ","End":"01:41.190","Text":"Theta would be 1 over 9 = secant^2 Theta,"},{"Start":"01:41.230 ","End":"01:48.200","Text":"so secant^2 Theta ="},{"Start":"01:48.200 ","End":"01:57.270","Text":"10 over 9,"},{"Start":"01:57.270 ","End":"02:01.900","Text":"so cosine^2 Theta is 1 over that is 9 over 10,"},{"Start":"02:02.810 ","End":"02:13.150","Text":"and so cosine Theta is 3 over the square root of 10."},{"Start":"02:13.820 ","End":"02:16.480","Text":"Now we have the cosine."},{"Start":"02:16.480 ","End":"02:18.850","Text":"Now we need the sine."},{"Start":"02:18.850 ","End":"02:23.770","Text":"Probably the easiest thing is because we know"},{"Start":"02:23.770 ","End":"02:30.280","Text":"that tangent Theta = sine Theta over cosine Theta."},{"Start":"02:30.280 ","End":"02:39.665","Text":"We can also say that sine Theta = tangent Theta times cosine Theta."},{"Start":"02:39.665 ","End":"02:43.700","Text":"In this case, we know tangent Theta = 1 over 3 and"},{"Start":"02:43.700 ","End":"02:50.820","Text":"cosine Theta = 3 over the square root of 10."},{"Start":"02:50.820 ","End":"02:57.790","Text":"This is equal to 1 over the square root of 10."},{"Start":"02:57.860 ","End":"03:02.120","Text":"We found the sine and we found the cosine."},{"Start":"03:02.120 ","End":"03:07.460","Text":"Notice that when I took the square root from here to here,"},{"Start":"03:07.460 ","End":"03:09.530","Text":"I knew that the cosine was positive,"},{"Start":"03:09.530 ","End":"03:12.020","Text":"otherwise I would have a plus or minus here."},{"Start":"03:12.020 ","End":"03:19.550","Text":"Let me just put a plus here to show that I didn\u0027t forget about the sign. That was Part A."},{"Start":"03:19.550 ","End":"03:21.620","Text":"Before we move on to part b,"},{"Start":"03:21.620 ","End":"03:26.520","Text":"I think a diagram and mnemonic would help."},{"Start":"03:26.930 ","End":"03:30.313","Text":"In Part B, minus 2,"},{"Start":"03:30.313 ","End":"03:35.275","Text":"minus 1 it\u0027s in the third quadrant."},{"Start":"03:35.275 ","End":"03:39.380","Text":"That means that we want the tangent."},{"Start":"03:39.380 ","End":"03:42.360","Text":"The tangent is positive."},{"Start":"03:42.580 ","End":"03:48.020","Text":"All we have to do, we know the formula for tangent of Theta."},{"Start":"03:48.020 ","End":"03:49.835","Text":"It\u0027s the y over the x,"},{"Start":"03:49.835 ","End":"03:58.540","Text":"it\u0027s minus 1 over minus 2 so the answer is 1/2 and that\u0027s part B."},{"Start":"03:58.540 ","End":"04:02.900","Text":"In Part C and D, I noticed that we have secant, cosecant,"},{"Start":"04:02.900 ","End":"04:06.680","Text":"and cotangent, which are not in the diagram,"},{"Start":"04:06.680 ","End":"04:11.000","Text":"but we don\u0027t need them because the secant is the inverse of the cosine."},{"Start":"04:11.000 ","End":"04:13.220","Text":"The cosecant is the inverse,"},{"Start":"04:13.220 ","End":"04:17.300","Text":"meaning 1 over the reciprocal of the sine and"},{"Start":"04:17.300 ","End":"04:23.600","Text":"the cotangent is the reciprocal of the tangent so they\u0027re going to have the same signs."},{"Start":"04:23.600 ","End":"04:28.485","Text":"In C, 4 - 5 is somewhere,"},{"Start":"04:28.485 ","End":"04:33.780","Text":"x is positive, y is negative so that brings us to the fourth quadrant."},{"Start":"04:33.780 ","End":"04:43.160","Text":"In the fourth quadrant here the cosine is positive and so the secant is also positive."},{"Start":"04:43.160 ","End":"04:48.109","Text":"Let\u0027s see, we find the tangent first."},{"Start":"04:48.109 ","End":"04:53.015","Text":"Tangent Theta = minus 5 over 4."},{"Start":"04:53.015 ","End":"04:58.730","Text":"Then we can get the secant from the identity"},{"Start":"04:58.730 ","End":"05:07.015","Text":"that tangent^2 Theta + 1 = secant^2 Theta."},{"Start":"05:07.015 ","End":"05:13.170","Text":"Tangent^2 is 25 over 16 +"},{"Start":"05:13.170 ","End":"05:20.345","Text":"1 = secant^2 Theta."},{"Start":"05:20.345 ","End":"05:28.505","Text":"We have that secant Theta and we know what\u0027s positive is the square root,"},{"Start":"05:28.505 ","End":"05:33.815","Text":"25 + 16 is 41 over 16,"},{"Start":"05:33.815 ","End":"05:42.860","Text":"which is the square root of 41 over 4, and from here,"},{"Start":"05:42.860 ","End":"05:45.410","Text":"we get that the cosine Theta,"},{"Start":"05:45.410 ","End":"05:49.955","Text":"which is the reciprocal of secant Theta,"},{"Start":"05:49.955 ","End":"05:54.880","Text":"is going to be 4 over the square root of 41."},{"Start":"05:54.880 ","End":"06:01.150","Text":"That\u0027s part C. Finally, Part D,"},{"Start":"06:01.150 ","End":"06:08.360","Text":"where (minus 3,5) is in the second quadrant."},{"Start":"06:08.360 ","End":"06:10.010","Text":"In the second quadrant,"},{"Start":"06:10.010 ","End":"06:17.070","Text":"the tangent is negative and the sine is positive."},{"Start":"06:17.360 ","End":"06:28.140","Text":"Let\u0027s see, the tangent of Theta is the y over the x is 5 over minus 3."},{"Start":"06:28.140 ","End":"06:35.330","Text":"I\u0027ll just put the minus in front and so the cotangent of Theta is the reciprocal,"},{"Start":"06:35.330 ","End":"06:39.455","Text":"is minus 3 over 5."},{"Start":"06:39.455 ","End":"06:43.280","Text":"Now we have the cotangent."},{"Start":"06:43.280 ","End":"06:45.455","Text":"What about the cosecant?"},{"Start":"06:45.455 ","End":"06:50.105","Text":"Well, there\u0027s an identity that says that"},{"Start":"06:50.105 ","End":"06:58.730","Text":"1 + cotangent^2 = cosecant^2."},{"Start":"06:58.730 ","End":"07:02.735","Text":"We have the cotangent, so we have 1,"},{"Start":"07:02.735 ","End":"07:05.480","Text":"and then when we square it,"},{"Start":"07:05.480 ","End":"07:09.670","Text":"we\u0027ll get plus 9/5."},{"Start":"07:09.670 ","End":"07:15.080","Text":"I\u0027ll write it a cosecant."},{"Start":"07:15.080 ","End":"07:19.085","Text":"Sometimes it\u0027s written like this, cosecant^2 Theta."},{"Start":"07:19.085 ","End":"07:23.745","Text":"Cosecant of Theta,"},{"Start":"07:23.745 ","End":"07:25.520","Text":"we know it\u0027s positive,"},{"Start":"07:25.520 ","End":"07:29.480","Text":"so it\u0027s plus the square root of this,"},{"Start":"07:29.480 ","End":"07:35.829","Text":"which is 5 + 9 is 14 over"},{"Start":"07:35.829 ","End":"07:42.080","Text":"5 and that\u0027s it."},{"Start":"07:42.080 ","End":"07:44.540","Text":"We found the tangent, we found the cosecant,"},{"Start":"07:44.540 ","End":"07:47.790","Text":"and that\u0027s the last part so we are done."}],"ID":14331},{"Watched":false,"Name":"Exercise 3","Duration":"7m 7s","ChapterTopicVideoID":13613,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13613.jpeg","UploadDate":"2018-10-21T13:40:20.6800000","DurationForVideoObject":"PT7M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, we\u0027re given some hints about an angle"},{"Start":"00:04.110 ","End":"00:09.540","Text":"Theta and we have to find the specified trigonometric functions."},{"Start":"00:09.540 ","End":"00:14.535","Text":"Like in part a, we\u0027re given the sine of Theta and the quadrant that Theta is in,"},{"Start":"00:14.535 ","End":"00:21.580","Text":"and we have to find the cosine and the tangent and we\u0027ll use a mnemonic."},{"Start":"00:21.830 ","End":"00:26.240","Text":"What I suggest that just go over these first and place the signs,"},{"Start":"00:26.240 ","End":"00:30.500","Text":"in quadrant 1, everything, all positive."},{"Start":"00:30.500 ","End":"00:33.575","Text":"That\u0027s plus and that\u0027s plus."},{"Start":"00:33.575 ","End":"00:35.855","Text":"In quadrant 2,"},{"Start":"00:35.855 ","End":"00:38.105","Text":"the sine is positive,"},{"Start":"00:38.105 ","End":"00:40.700","Text":"but the cotangent is like the tangent,"},{"Start":"00:40.700 ","End":"00:42.095","Text":"the reciprocal."},{"Start":"00:42.095 ","End":"00:44.120","Text":"It\u0027s also negative."},{"Start":"00:44.120 ","End":"00:45.570","Text":"In quadrant 4,"},{"Start":"00:45.570 ","End":"00:46.870","Text":"which is this one,"},{"Start":"00:46.870 ","End":"00:49.210","Text":"the cosine is positive."},{"Start":"00:49.210 ","End":"00:54.155","Text":"The secant is its reciprocal will also be positive."},{"Start":"00:54.155 ","End":"00:57.395","Text":"In the third quadrant here,"},{"Start":"00:57.395 ","End":"01:00.215","Text":"the cosine is negative,"},{"Start":"01:00.215 ","End":"01:03.690","Text":"but the tangent is positive."},{"Start":"01:03.830 ","End":"01:07.789","Text":"In part a, we have the sine."},{"Start":"01:07.789 ","End":"01:10.295","Text":"We want the cosine and the tangent."},{"Start":"01:10.295 ","End":"01:18.505","Text":"The easiest is to find the cosine because we know that sine^2 plus cosine ^2 is 1,"},{"Start":"01:18.505 ","End":"01:19.790","Text":"or in other words,"},{"Start":"01:19.790 ","End":"01:27.930","Text":"cosine of Theta is plus or minus the square root of 1 minus sine^2 Theta."},{"Start":"01:27.930 ","End":"01:33.960","Text":"But we know that cosine is positive."},{"Start":"01:33.960 ","End":"01:37.770","Text":"We don\u0027t need the plus or minus."},{"Start":"01:37.770 ","End":"01:39.290","Text":"We know it\u0027ll be plus."},{"Start":"01:39.290 ","End":"01:44.120","Text":"This is equal to the square root of 1 minus"},{"Start":"01:44.120 ","End":"01:48.070","Text":"7/25 squared"},{"Start":"01:48.070 ","End":"01:55.970","Text":"is 49/625."},{"Start":"01:55.970 ","End":"01:59.135","Text":"That is equal to the square root."},{"Start":"01:59.135 ","End":"02:05.250","Text":"625 minus 49 is 576 over"},{"Start":"02:05.250 ","End":"02:13.680","Text":"625 and that comes out to be exactly 24/25."},{"Start":"02:13.680 ","End":"02:15.940","Text":"That\u0027s the cosine."},{"Start":"02:15.940 ","End":"02:21.335","Text":"Then the tangent is easy once we have the sine and the cosine,"},{"Start":"02:21.335 ","End":"02:27.560","Text":"because the tangent is the sine over the cosine."},{"Start":"02:27.560 ","End":"02:30.990","Text":"This will equal. Sine,"},{"Start":"02:30.990 ","End":"02:33.860","Text":"we\u0027re given as 7/25."},{"Start":"02:33.860 ","End":"02:40.310","Text":"Cosine we found is 24/25 and"},{"Start":"02:40.310 ","End":"02:47.210","Text":"so the answer is 7/24."},{"Start":"02:47.210 ","End":"02:51.920","Text":"That\u0027s part a. For part b,"},{"Start":"02:51.920 ","End":"02:55.685","Text":"we have the cosine, you want the sine and the cotangent."},{"Start":"02:55.685 ","End":"02:59.375","Text":"We\u0027ll find the sine first similar to part a,"},{"Start":"02:59.375 ","End":"03:03.660","Text":"and we know the sign is going to be positive."},{"Start":"03:03.660 ","End":"03:13.960","Text":"We have that sine Theta is plus square root of 1 minus cosine^2 Theta,"},{"Start":"03:13.960 ","End":"03:23.980","Text":"which is the square root of 1 minus 4/7 squared is 16/49."},{"Start":"03:23.980 ","End":"03:34.720","Text":"That comes out to be 49 minus 16 is 33 and so the sine is root 33/7."},{"Start":"03:36.050 ","End":"03:40.095","Text":"Now we have the sine and the cosine."},{"Start":"03:40.095 ","End":"03:42.455","Text":"We want the cotangent."},{"Start":"03:42.455 ","End":"03:47.110","Text":"The cotangent is the reciprocal of tangent."},{"Start":"03:47.110 ","End":"03:49.180","Text":"Tangent is sine over cosine."},{"Start":"03:49.180 ","End":"03:53.525","Text":"This is cosine over sine,"},{"Start":"03:53.525 ","End":"03:56.030","Text":"the opposite of the tangent,"},{"Start":"03:56.030 ","End":"04:04.470","Text":"which is the cosine 4/7."},{"Start":"04:04.470 ","End":"04:09.750","Text":"The sine from here, root 33/7."},{"Start":"04:09.750 ","End":"04:15.495","Text":"The answer is 4 over root 33."},{"Start":"04:15.495 ","End":"04:17.875","Text":"That\u0027s part b."},{"Start":"04:17.875 ","End":"04:20.505","Text":"Now part c,"},{"Start":"04:20.505 ","End":"04:22.180","Text":"we have the tangent."},{"Start":"04:22.180 ","End":"04:24.360","Text":"We want the cosine and secant."},{"Start":"04:24.360 ","End":"04:29.570","Text":"I\u0027ll go for the secant because there is a formula that 1"},{"Start":"04:29.570 ","End":"04:37.230","Text":"plus tangent^2 Theta is equal to secant^2 Theta."},{"Start":"04:38.440 ","End":"04:44.480","Text":"Secant Theta is equal to plus or"},{"Start":"04:44.480 ","End":"04:49.890","Text":"minus the square root of 1 plus tangent squared Theta."},{"Start":"04:49.890 ","End":"04:51.260","Text":"But in this case,"},{"Start":"04:51.260 ","End":"04:52.910","Text":"the secant is negative."},{"Start":"04:52.910 ","End":"04:55.820","Text":"This will be what we want."},{"Start":"04:55.820 ","End":"05:02.420","Text":"The tangent is minus 5 over 13."},{"Start":"05:02.420 ","End":"05:06.409","Text":"But when we square it, it doesn\u0027t matter."},{"Start":"05:06.409 ","End":"05:15.310","Text":"It\u0027ll be 1 plus 5^2/13^2,"},{"Start":"05:17.300 ","End":"05:19.390","Text":"25/169."},{"Start":"05:19.390 ","End":"05:23.680","Text":"That is equal to minus the square root,"},{"Start":"05:23.680 ","End":"05:33.845","Text":"169 plus 25 is 194/169."},{"Start":"05:33.845 ","End":"05:39.120","Text":"So the answer is minus root 194/13,"},{"Start":"05:39.120 ","End":"05:42.610","Text":"that\u0027s square root of 169."},{"Start":"05:42.610 ","End":"05:44.960","Text":"Now we have the secant."},{"Start":"05:44.960 ","End":"05:47.315","Text":"Now we just need the cosine."},{"Start":"05:47.315 ","End":"05:52.480","Text":"But the cosine is the reciprocal of the secant."},{"Start":"05:52.480 ","End":"06:01.300","Text":"This is equal to 1 over minus root 194/13."},{"Start":"06:01.300 ","End":"06:11.570","Text":"It\u0027s minus 13 over the root of 194."},{"Start":"06:12.270 ","End":"06:20.270","Text":"That was part c and now part d,"},{"Start":"06:20.270 ","End":"06:26.975","Text":"where we\u0027re given the secant and we have to find the cosine and the tangent,"},{"Start":"06:26.975 ","End":"06:31.790","Text":"the easiest is to find the cosine of Theta,"},{"Start":"06:31.790 ","End":"06:38.285","Text":"which is the reciprocal of the secant and vice versa."},{"Start":"06:38.285 ","End":"06:41.480","Text":"This is equal to just the reciprocal of this."},{"Start":"06:41.480 ","End":"06:47.770","Text":"We just flip it minus 15/7."},{"Start":"06:47.770 ","End":"06:53.270","Text":"Now something is wrong here because the cosine Theta has to"},{"Start":"06:53.270 ","End":"06:58.720","Text":"be between 1 and minus 1 and this is less than minus 1."},{"Start":"06:58.720 ","End":"07:04.115","Text":"I must have missed copied the exercise and there\u0027s an error."},{"Start":"07:04.115 ","End":"07:07.350","Text":"I\u0027ll just stop here."}],"ID":14332},{"Watched":false,"Name":"Exercise 4","Duration":"3m 58s","ChapterTopicVideoID":13614,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13614.jpeg","UploadDate":"2018-10-21T13:41:06.4770000","DurationForVideoObject":"PT3M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have to find the exact value of the expressions."},{"Start":"00:04.320 ","End":"00:06.540","Text":"There\u0027s 4 of them, a through d,"},{"Start":"00:06.540 ","End":"00:08.895","Text":"do not use a calculator."},{"Start":"00:08.895 ","End":"00:10.530","Text":"What can we use?"},{"Start":"00:10.530 ","End":"00:13.410","Text":"Is a table of special angles on"},{"Start":"00:13.410 ","End":"00:17.505","Text":"their trigonometric functions and I happen to have 1 handy."},{"Start":"00:17.505 ","End":"00:19.290","Text":"Sometimes you\u0027re just given the sine,"},{"Start":"00:19.290 ","End":"00:20.954","Text":"cosine, and tangent."},{"Start":"00:20.954 ","End":"00:24.450","Text":"Then you have to use reciprocals to get to the other 3."},{"Start":"00:24.450 ","End":"00:27.090","Text":"You of course means undefined."},{"Start":"00:27.090 ","End":"00:34.750","Text":"Okay. Part a, sine 45 root 2 over 2."},{"Start":"00:34.970 ","End":"00:40.665","Text":"Cosine 45 root 2 over 2."},{"Start":"00:40.665 ","End":"00:43.380","Text":"The answer is root 2."},{"Start":"00:43.380 ","End":"00:46.740","Text":"Part b."},{"Start":"00:46.740 ","End":"00:50.610","Text":"We have 3 times sine of 60,"},{"Start":"00:50.610 ","End":"00:56.175","Text":"60 sine is root 3 over 2"},{"Start":"00:56.175 ","End":"01:04.425","Text":"minus 6 cosine of 30 is also root 3 over 2."},{"Start":"01:04.425 ","End":"01:09.020","Text":"That\u0027s no surprise because there is the co-function rule at"},{"Start":"01:09.020 ","End":"01:13.520","Text":"the sine of 60 is the cosine of its complement, so yeah."},{"Start":"01:13.520 ","End":"01:20.160","Text":"Okay 3 of these minus 6 of these is just minus 3 of them."},{"Start":"01:20.160 ","End":"01:22.845","Text":"Root 3 over 2."},{"Start":"01:22.845 ","End":"01:25.620","Text":"Yeah, you can leave it like that,"},{"Start":"01:25.620 ","End":"01:33.140","Text":"or I suppose we could write minus 3 root 3 over 2."},{"Start":"01:33.140 ","End":"01:42.240","Text":"Part c sine of 30 is 1.5"},{"Start":"01:42.240 ","End":"01:47.600","Text":"plus 1"},{"Start":"01:47.600 ","End":"01:54.035","Text":"squared over cosine of 30."},{"Start":"01:54.035 ","End":"02:04.020","Text":"We had here was root 3 over 2 minus 1 squared."},{"Start":"02:04.020 ","End":"02:06.615","Text":"Let\u0027s see what that equals."},{"Start":"02:06.615 ","End":"02:11.150","Text":"Using a squared plus 2ab plus b squared."},{"Start":"02:11.150 ","End":"02:12.169","Text":"Well, on the numerator,"},{"Start":"02:12.169 ","End":"02:14.690","Text":"we just have to say what 1.5 squared is,"},{"Start":"02:14.690 ","End":"02:16.975","Text":"is 2 and a quarter."},{"Start":"02:16.975 ","End":"02:19.035","Text":"2 and a quarter."},{"Start":"02:19.035 ","End":"02:22.020","Text":"I\u0027ll write it as 9 over 4."},{"Start":"02:22.020 ","End":"02:24.800","Text":"Let\u0027s see. Here we\u0027ll use the rule."},{"Start":"02:24.800 ","End":"02:26.540","Text":"It\u0027s this thing squared,"},{"Start":"02:26.540 ","End":"02:31.180","Text":"which is 3-quarters minus twice this times this."},{"Start":"02:31.180 ","End":"02:36.580","Text":"It\u0027s minus twice root 3 over 2."},{"Start":"02:36.580 ","End":"02:39.635","Text":"You know what? I\u0027ll write it straight away as root 3."},{"Start":"02:39.635 ","End":"02:43.530","Text":"There I save a step, plus 1."},{"Start":"02:43.570 ","End":"02:46.625","Text":"Can this be simplified?"},{"Start":"02:46.625 ","End":"02:51.905","Text":"Well, we can make the 1 together with the 3-quarters."},{"Start":"02:51.905 ","End":"02:55.130","Text":"So we can say and we can put the 4, yeah,"},{"Start":"02:55.130 ","End":"02:57.700","Text":"we can multiply top and bottom by 4,"},{"Start":"02:57.700 ","End":"03:00.225","Text":"so we have 9 over,"},{"Start":"03:00.225 ","End":"03:01.470","Text":"multiply it by 4,"},{"Start":"03:01.470 ","End":"03:02.565","Text":"this becomes 3,"},{"Start":"03:02.565 ","End":"03:04.860","Text":"this becomes 4, altogether,"},{"Start":"03:04.860 ","End":"03:10.560","Text":"7 minus 4 root 3."},{"Start":"03:10.560 ","End":"03:18.330","Text":"Okay. Now, part D. Let\u0027s see,"},{"Start":"03:18.330 ","End":"03:25.180","Text":"sine of 90 is"},{"Start":"03:25.400 ","End":"03:36.345","Text":"1 times cosine of 45 is root 2 over 2,"},{"Start":"03:36.345 ","End":"03:41.435","Text":"plus sine of 30 is 1.5."},{"Start":"03:41.435 ","End":"03:46.530","Text":"Cosine of 90 is 0."},{"Start":"03:46.940 ","End":"03:55.235","Text":"This part is 0 and all we\u0027re left with is root 2 over 2."},{"Start":"03:55.235 ","End":"03:58.080","Text":"Okay, that\u0027s it."}],"ID":14333},{"Watched":false,"Name":"Exercise 5","Duration":"7m 12s","ChapterTopicVideoID":13615,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13615.jpeg","UploadDate":"2018-10-21T13:42:31.5370000","DurationForVideoObject":"PT7M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, we have to evaluate these expressions without a calculator,"},{"Start":"00:04.620 ","End":"00:10.905","Text":"so we\u0027ll need a table for special angles. Here it is."},{"Start":"00:10.905 ","End":"00:13.920","Text":"Let\u0027s start, part a,"},{"Start":"00:13.920 ","End":"00:16.830","Text":"tangent 45 is 1."},{"Start":"00:16.830 ","End":"00:19.860","Text":"Some of them I just know by heart without looking at the table."},{"Start":"00:19.860 ","End":"00:26.385","Text":"Tangent 30 is 1 over the square root of 3."},{"Start":"00:26.385 ","End":"00:28.665","Text":"We can look it up here,"},{"Start":"00:28.665 ","End":"00:34.240","Text":"tangent of 30 degrees."},{"Start":"00:34.240 ","End":"00:37.120","Text":"Yes, some people write it as root 3/3,"},{"Start":"00:37.120 ","End":"00:46.350","Text":"it\u0027s the same thing as 1/ root 3 minus sin 45 root 2/2,"},{"Start":"00:46.350 ","End":"00:49.560","Text":"which is the same as 1/ root 2,"},{"Start":"00:49.560 ","End":"00:54.070","Text":"and sin 30, is 1.5."},{"Start":"00:58.970 ","End":"01:02.925","Text":"I\u0027ll do it like they\u0027re."},{"Start":"01:02.925 ","End":"01:07.820","Text":"Generally mathematicians like square roots only in the numerator,"},{"Start":"01:07.820 ","End":"01:09.245","Text":"not in the denominator."},{"Start":"01:09.245 ","End":"01:13.520","Text":"What we get is root"},{"Start":"01:13.520 ","End":"01:20.775","Text":"3/3 minus root 2/4."},{"Start":"01:20.775 ","End":"01:24.165","Text":"Now b, root 2,"},{"Start":"01:24.165 ","End":"01:31.755","Text":"cosine 45 is root 2/2 minus root 3,"},{"Start":"01:31.755 ","End":"01:35.655","Text":"tan 60 is root 3,"},{"Start":"01:35.655 ","End":"01:43.680","Text":"plus root 18,"},{"Start":"01:43.680 ","End":"01:54.360","Text":"cot 60 is root 3/ 3 or 1/ root 3, same thing."},{"Start":"01:54.360 ","End":"02:02.520","Text":"Let\u0027s see what this gives us root 2 times root 2 is 2, 2/2 is 1."},{"Start":"02:02.520 ","End":"02:05.805","Text":"Root 3 times root 3 is 3."},{"Start":"02:05.805 ","End":"02:12.180","Text":"The last one, we have 18"},{"Start":"02:12.180 ","End":"02:20.540","Text":"times 3 is 54."},{"Start":"02:20.540 ","End":"02:24.420","Text":"But there is another way we can do it."},{"Start":"02:24.760 ","End":"02:28.292","Text":"If I had another root 3 here, that would help me."},{"Start":"02:28.292 ","End":"02:35.450","Text":"So I can write this root 18 as root 3 times root 6."},{"Start":"02:35.450 ","End":"02:38.890","Text":"Then if I do it that way,"},{"Start":"02:38.890 ","End":"02:44.670","Text":"then the root 3 with the root 3 will give us,"},{"Start":"02:44.670 ","End":"02:52.110","Text":"I\u0027ll write it root 6 times root 3 times root 3/ 3,"},{"Start":"02:52.110 ","End":"02:57.120","Text":"and then the root 3 times root 3 gives us 3 which cancels with this."},{"Start":"02:57.120 ","End":"03:01.365","Text":"So altogether, our answer will be,"},{"Start":"03:01.365 ","End":"03:03.360","Text":"1 minus 3 is minus 2,"},{"Start":"03:03.360 ","End":"03:06.520","Text":"so we\u0027ll have root 6 minus 2."},{"Start":"03:06.520 ","End":"03:09.215","Text":"I like to put the positive first."},{"Start":"03:09.215 ","End":"03:15.260","Text":"Now part c, sec 60,"},{"Start":"03:15.260 ","End":"03:23.390","Text":"1/cos 60, it should be equal to 0.5, 1/0.5 is 2."},{"Start":"03:23.390 ","End":"03:28.170","Text":"Let\u0027s see sec of 60 is 2."},{"Start":"03:28.170 ","End":"03:37.000","Text":"We have 2, cosecant 45 should be root 2,"},{"Start":"03:40.340 ","End":"03:47.340","Text":"sin 60 is root 3/ 2,"},{"Start":"03:47.340 ","End":"03:55.055","Text":"and cos 45 is root 2/2."},{"Start":"03:55.055 ","End":"03:58.660","Text":"Let\u0027s see if we can simplify this."},{"Start":"03:58.660 ","End":"04:01.750","Text":"Technically, you could say that this is the answer,"},{"Start":"04:01.750 ","End":"04:03.745","Text":"but I\u0027d like to simplify it."},{"Start":"04:03.745 ","End":"04:06.160","Text":"Multiply top and bottom by 2,"},{"Start":"04:06.160 ","End":"04:07.870","Text":"get rid of some of the fractions."},{"Start":"04:07.870 ","End":"04:15.420","Text":"4 minus 2 root 2/root 3 minus root 2."},{"Start":"04:15.420 ","End":"04:20.760","Text":"Now, I\u0027d like to get rid of the square roots in the denominator,"},{"Start":"04:20.760 ","End":"04:25.240","Text":"so what we can do is the trick of multiplying by the conjugate top and bottom,"},{"Start":"04:25.240 ","End":"04:33.120","Text":"and multiply it by root 3 plus root 2/ root 3 plus root 2, this is equal to 1."},{"Start":"04:33.120 ","End":"04:35.175","Text":"So I haven\u0027t changed anything."},{"Start":"04:35.175 ","End":"04:36.930","Text":"On the numerator,"},{"Start":"04:36.930 ","End":"04:40.790","Text":"I get multiplied by 4,"},{"Start":"04:40.790 ","End":"04:45.160","Text":"4 root 3 plus 4 root 2,"},{"Start":"04:45.160 ","End":"04:53.025","Text":"then minus 2 root 2 root 3 is minus 2 root 6."},{"Start":"04:53.025 ","End":"05:00.135","Text":"Then minus 2 root 2 root 2 is minus 4 over,"},{"Start":"05:00.135 ","End":"05:03.480","Text":"now this is difference of squares, a, minus b,"},{"Start":"05:03.480 ","End":"05:06.450","Text":"a plus b is a^2 minus b^2,"},{"Start":"05:06.450 ","End":"05:09.430","Text":"so it\u0027s 3 minus 2."},{"Start":"05:10.340 ","End":"05:15.375","Text":"The answer is just what is in the numerator."},{"Start":"05:15.375 ","End":"05:18.090","Text":"Won\u0027t write it out again, 3 minus 2 is 1,"},{"Start":"05:18.090 ","End":"05:21.110","Text":"so all we\u0027re left with is this,"},{"Start":"05:21.110 ","End":"05:25.130","Text":"and that\u0027s the answer to part c. If I marked already,"},{"Start":"05:25.130 ","End":"05:28.160","Text":"then you might as well highlight all the results,"},{"Start":"05:28.160 ","End":"05:31.590","Text":"part a, part b,"},{"Start":"05:31.590 ","End":"05:38.639","Text":"we still have part d. Last part d,"},{"Start":"05:38.639 ","End":"05:46.120","Text":"cot 60 is root 3/3,"},{"Start":"05:47.960 ","End":"05:52.740","Text":"cosecant 60 is 1/ sin 60,"},{"Start":"05:52.740 ","End":"05:54.269","Text":"should be 2/ root"},{"Start":"05:54.269 ","End":"06:02.415","Text":"3 minus"},{"Start":"06:02.415 ","End":"06:04.830","Text":"sec 30,"},{"Start":"06:04.830 ","End":"06:07.890","Text":"cos 30 is root 3/2,"},{"Start":"06:07.890 ","End":"06:10.830","Text":"so again, 2/root 3,"},{"Start":"06:10.830 ","End":"06:19.950","Text":"and that sec of 30, sin of 30,"},{"Start":"06:19.950 ","End":"06:22.410","Text":"I know is 1.5."},{"Start":"06:22.410 ","End":"06:24.360","Text":"Let\u0027s see what cancels,"},{"Start":"06:24.360 ","End":"06:26.595","Text":"here root 3 with root 3,"},{"Start":"06:26.595 ","End":"06:28.965","Text":"that gives us 2/3,"},{"Start":"06:28.965 ","End":"06:32.025","Text":"and here 2 with 2,"},{"Start":"06:32.025 ","End":"06:39.120","Text":"which gives us 1/ root 3."},{"Start":"06:39.120 ","End":"06:41.270","Text":"Now, 1/ root 3, I said before,"},{"Start":"06:41.270 ","End":"06:44.165","Text":"is the same as root 3/ 3,"},{"Start":"06:44.165 ","End":"06:45.920","Text":"so you could leave it like this,"},{"Start":"06:45.920 ","End":"06:54.055","Text":"but you could also say it\u0027s 2/3 times root 3/3,"},{"Start":"06:54.055 ","End":"07:00.660","Text":"which is equal to 2 root 3/9,"},{"Start":"07:00.660 ","End":"07:05.810","Text":"but as I said, you could have stopped here and that would have been fine."},{"Start":"07:05.810 ","End":"07:12.240","Text":"Just for consistency, I\u0027ll highlight the result and we\u0027re done."}],"ID":14334},{"Watched":false,"Name":"Exercise 6","Duration":"6m 54s","ChapterTopicVideoID":13616,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13616.jpeg","UploadDate":"2018-10-21T13:43:55.6570000","DurationForVideoObject":"PT6M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this exercise, we have to use"},{"Start":"00:02.565 ","End":"00:09.390","Text":"sum and difference formulas to find the exact value of these expressions."},{"Start":"00:09.390 ","End":"00:13.650","Text":"I\u0027ve also included the table of special angles."},{"Start":"00:13.650 ","End":"00:17.910","Text":"Here I brought in the table of sum and difference formulas."},{"Start":"00:17.910 ","End":"00:22.094","Text":"We won\u0027t be needing the tangent ones."},{"Start":"00:22.094 ","End":"00:26.140","Text":"We just need the sine and cosine you will see."},{"Start":"00:26.270 ","End":"00:31.170","Text":"In part a, 15 is not one of the special angles,"},{"Start":"00:31.170 ","End":"00:40.665","Text":"but we can write 15 as 45 minus 30 degrees."},{"Start":"00:40.665 ","End":"00:43.640","Text":"Then we can use the formula,"},{"Start":"00:43.640 ","End":"00:47.735","Text":"this one for sine of Alpha minus Beta,"},{"Start":"00:47.735 ","End":"00:51.320","Text":"which is equal to,"},{"Start":"00:51.320 ","End":"00:55.055","Text":"let\u0027s see, sine Alpha cosine Beta."},{"Start":"00:55.055 ","End":"00:59.180","Text":"We need sine 45,"},{"Start":"00:59.180 ","End":"01:05.900","Text":"cosine 30 minus cosine"},{"Start":"01:05.900 ","End":"01:11.740","Text":"45, sine of 30."},{"Start":"01:11.740 ","End":"01:14.250","Text":"Now, we use the special angles,"},{"Start":"01:14.250 ","End":"01:16.290","Text":"1 over root 2."},{"Start":"01:16.290 ","End":"01:20.340","Text":"Cosine of 30 is root 3 over 2."},{"Start":"01:20.340 ","End":"01:27.610","Text":"Cosine 45 is root 2 over 2."},{"Start":"01:27.610 ","End":"01:29.750","Text":"Sorry here, this is correct,"},{"Start":"01:29.750 ","End":"01:33.440","Text":"but I\u0027ll use the other form, root 2 over 2,"},{"Start":"01:33.440 ","End":"01:36.988","Text":"there\u0027s 2 versions just for consistency,"},{"Start":"01:36.988 ","End":"01:41.890","Text":"and sine 30 is 1/2."},{"Start":"01:41.890 ","End":"01:46.570","Text":"What we get is everything\u0027s going to be over 4."},{"Start":"01:46.570 ","End":"01:49.420","Text":"We have root 2, root 3."},{"Start":"01:49.420 ","End":"01:52.610","Text":"You can leave it like that or you can write it as root 6,"},{"Start":"01:52.610 ","End":"01:55.835","Text":"root 2 times root 1 is root 2."},{"Start":"01:55.835 ","End":"01:58.355","Text":"This is part a."},{"Start":"01:58.355 ","End":"02:01.345","Text":"Let\u0027s get to part b."},{"Start":"02:01.345 ","End":"02:05.475","Text":"Part b, cosine of 75."},{"Start":"02:05.475 ","End":"02:08.030","Text":"75 is not a special angle,"},{"Start":"02:08.030 ","End":"02:13.685","Text":"but I can write it as 45 plus 30."},{"Start":"02:13.685 ","End":"02:15.905","Text":"Then use the cosine."},{"Start":"02:15.905 ","End":"02:19.070","Text":"This rule here, cosine,"},{"Start":"02:19.070 ","End":"02:21.605","Text":"cosine minus sine, sine."},{"Start":"02:21.605 ","End":"02:32.720","Text":"Cosine 45, cosine 30 minus sine 45, sine 30."},{"Start":"02:32.720 ","End":"02:36.290","Text":"Better put the degree sign in."},{"Start":"02:36.290 ","End":"02:43.080","Text":"This is equal to root 2 over 2."},{"Start":"02:43.080 ","End":"02:45.420","Text":"This is root 2 over 2."},{"Start":"02:45.420 ","End":"02:51.760","Text":"Cosine 30 is root 3 over 2 sine 30 is 1/2."},{"Start":"02:52.010 ","End":"02:57.135","Text":"What we get is going to be over 4."},{"Start":"02:57.135 ","End":"03:06.630","Text":"Root 2, root 3 is root 6 and root 2 times 1 is root 2."},{"Start":"03:06.630 ","End":"03:08.800","Text":"Isn\u0027t that strange?"},{"Start":"03:08.800 ","End":"03:13.405","Text":"We had the same answer as part a,"},{"Start":"03:13.405 ","End":"03:16.855","Text":"but it isn\u0027t really strange because we could have used"},{"Start":"03:16.855 ","End":"03:22.255","Text":"the co-function identity and gotten that"},{"Start":"03:22.255 ","End":"03:32.435","Text":"cosine of 75 degrees is sine of 90 minus 75 degrees."},{"Start":"03:32.435 ","End":"03:35.845","Text":"90 minus 75 is 15."},{"Start":"03:35.845 ","End":"03:38.710","Text":"We could have straightaway done that,"},{"Start":"03:38.710 ","End":"03:42.190","Text":"but they did say use the sum and difference formulas."},{"Start":"03:42.190 ","End":"03:49.855","Text":"Okay. Now, let\u0027s get on to part c and need a bit of space here."},{"Start":"03:49.855 ","End":"03:52.700","Text":"We want tangent of 15,"},{"Start":"03:52.700 ","End":"03:58.520","Text":"and they\u0027ve given us a hint that tangent of 15 is going"},{"Start":"03:58.520 ","End":"04:05.555","Text":"to be sine 15 over cosine of 15."},{"Start":"04:05.555 ","End":"04:11.280","Text":"But we don\u0027t have the cosine of 15 degrees."},{"Start":"04:11.290 ","End":"04:17.540","Text":"We only have the sine of 15 degrees, which is here."},{"Start":"04:17.540 ","End":"04:25.145","Text":"We\u0027ll have to use a separate calculation to find cosine of 15."},{"Start":"04:25.145 ","End":"04:28.085","Text":"We can either find the cosine from the sign."},{"Start":"04:28.085 ","End":"04:32.449","Text":"But since we\u0027re practicing sum and difference of angles,"},{"Start":"04:32.449 ","End":"04:36.260","Text":"I couldn\u0027t decide, say that cosine 15,"},{"Start":"04:36.260 ","End":"04:38.195","Text":"just like I did with the sine,"},{"Start":"04:38.195 ","End":"04:44.815","Text":"is cosine of 45 minus 30,"},{"Start":"04:44.815 ","End":"04:49.790","Text":"which is equal to the cosine of a difference is cost,"},{"Start":"04:49.790 ","End":"04:51.755","Text":"cost plus sine, sine."},{"Start":"04:51.755 ","End":"05:00.665","Text":"That is cosine 45 degrees,"},{"Start":"05:00.665 ","End":"05:10.355","Text":"cosine 30 degrees plus sine 45, sine 30."},{"Start":"05:10.355 ","End":"05:16.665","Text":"That is equal to root 2 over 2."},{"Start":"05:16.665 ","End":"05:23.390","Text":"This is root 3 over 2."},{"Start":"05:23.390 ","End":"05:30.929","Text":"Then plus root 2 over 2 times 1/2."},{"Start":"05:30.929 ","End":"05:40.155","Text":"This comes out that C over 4 root 6 plus root 2 over 4."},{"Start":"05:40.155 ","End":"05:42.760","Text":"Now, we can get back here."},{"Start":"05:42.760 ","End":"05:47.750","Text":"We can say that this equals sine of 15, where is it?"},{"Start":"05:47.750 ","End":"05:56.705","Text":"Yeah, here is root 6 minus root 2 over 4 and a bigger dividing line."},{"Start":"05:56.705 ","End":"06:02.410","Text":"Root 6 plus root 2 over 4."},{"Start":"06:02.410 ","End":"06:05.615","Text":"What do we get altogether, the 4s will cancel."},{"Start":"06:05.615 ","End":"06:09.930","Text":"It\u0027s root 6 minus root"},{"Start":"06:09.930 ","End":"06:17.010","Text":"2 over root 6 plus root 2."},{"Start":"06:17.010 ","End":"06:20.795","Text":"We could simplify this using conjugates,"},{"Start":"06:20.795 ","End":"06:24.450","Text":"but I\u0027ll let it go at that."},{"Start":"06:24.560 ","End":"06:26.873","Text":"Now, part d,"},{"Start":"06:26.873 ","End":"06:28.490","Text":"see I\u0027ll squeeze it in here."},{"Start":"06:28.490 ","End":"06:36.050","Text":"Secant of 75 is 1 over the cosine of 75 degrees."},{"Start":"06:36.050 ","End":"06:40.543","Text":"The cosine of 75 degrees was part b,"},{"Start":"06:40.543 ","End":"06:42.065","Text":"and it was this."},{"Start":"06:42.065 ","End":"06:51.470","Text":"We just have to take the inverse and that will be 4 over root 6 minus root 2 there."},{"Start":"06:51.470 ","End":"06:54.510","Text":"Got everything in. We\u0027re done."}],"ID":14335},{"Watched":false,"Name":"Exercise 7","Duration":"6m 10s","ChapterTopicVideoID":13617,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13617.jpeg","UploadDate":"2018-10-21T13:45:09.1300000","DurationForVideoObject":"PT6M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this exercise, we have 4 expressions and we have to"},{"Start":"00:03.090 ","End":"00:06.970","Text":"find their value without a calculator."},{"Start":"00:07.280 ","End":"00:12.465","Text":"It means that you should use a table of special angles,"},{"Start":"00:12.465 ","End":"00:20.415","Text":"and this one has both degrees and in this case we\u0027re going to use the radian form."},{"Start":"00:20.415 ","End":"00:23.605","Text":"Let\u0027s start with a."},{"Start":"00:23.605 ","End":"00:29.510","Text":"What we have sine Pi/6 is a 1/2,"},{"Start":"00:29.510 ","End":"00:33.545","Text":"cosine Pi/4, root 2/2."},{"Start":"00:33.545 ","End":"00:39.890","Text":"Tangent of Pi/3 is root 3."},{"Start":"00:39.890 ","End":"00:51.390","Text":"Cotangent of Pi/2 is 0."},{"Start":"00:51.390 ","End":"00:54.010","Text":"We just have to add these up."},{"Start":"00:55.190 ","End":"01:06.420","Text":"No real simplification, I could just say it\u0027s 1 plus root 2/2 plus root 3."},{"Start":"01:07.570 ","End":"01:16.470","Text":"Now let\u0027s go for part b. Let\u0027s see."},{"Start":"01:16.470 ","End":"01:21.615","Text":"2 cosine Pi/3 is a 1/2,"},{"Start":"01:21.615 ","End":"01:26.940","Text":"sine of Pi/3 is root 3/2."},{"Start":"01:26.940 ","End":"01:34.695","Text":"To multiply this by cosine Pi/3 is again 1/2,"},{"Start":"01:34.695 ","End":"01:42.850","Text":"and sine Pi/3 is still root 3 over 2."},{"Start":"01:43.640 ","End":"01:47.560","Text":"Let\u0027s just do the multiplication."},{"Start":"01:48.260 ","End":"01:50.925","Text":"2 times a 1/2 is 1,"},{"Start":"01:50.925 ","End":"01:56.290","Text":"now 1 times a 1/2 is 1/2."},{"Start":"01:56.350 ","End":"01:59.965","Text":"Let\u0027s take the last one with the last one."},{"Start":"01:59.965 ","End":"02:02.700","Text":"We have minus 2,"},{"Start":"02:02.700 ","End":"02:05.340","Text":"then root 3/2,"},{"Start":"02:05.340 ","End":"02:14.055","Text":"root 3/2 is minus 2 times 3/4 is root 3/2 ^2 is 3/4."},{"Start":"02:14.055 ","End":"02:21.240","Text":"Now let us see we have this with this gives us minus root 3/4,"},{"Start":"02:21.240 ","End":"02:24.540","Text":"and this with this, well,"},{"Start":"02:24.540 ","End":"02:26.060","Text":"the 2 and the 2 cancel,"},{"Start":"02:26.060 ","End":"02:29.580","Text":"so it just gives us plus root 3."},{"Start":"02:30.100 ","End":"02:38.635","Text":"I\u0027ll just write it minus root 3/4 plus root 3 times 1."},{"Start":"02:38.635 ","End":"02:42.675","Text":"Does this simplify? Yes,"},{"Start":"02:42.675 ","End":"02:47.700","Text":"because this is a 1/2 minus 3/2,"},{"Start":"02:47.700 ","End":"02:50.010","Text":"which is minus 1,"},{"Start":"02:50.010 ","End":"02:54.240","Text":"and minus a 1/4 plus 1 is 3/4 so I can write"},{"Start":"02:54.240 ","End":"03:01.710","Text":"it as 3 root 3/4."},{"Start":"03:01.710 ","End":"03:08.700","Text":"Now part c. Oh,"},{"Start":"03:08.700 ","End":"03:11.290","Text":"this is a complicated one."},{"Start":"03:14.150 ","End":"03:24.210","Text":"Let\u0027s see. We can"},{"Start":"03:24.210 ","End":"03:31.210","Text":"see that now secant of Pi/4 is root 2,"},{"Start":"03:32.120 ","End":"03:40.185","Text":"and cosecant of Pi/4 is also root 2."},{"Start":"03:40.185 ","End":"03:46.335","Text":"This plus this and this is squared plus 1"},{"Start":"03:46.335 ","End":"03:53.110","Text":"over tangent of Pi/ 4 is 1,"},{"Start":"03:53.720 ","End":"03:58.590","Text":"and cotangent of Pi/4 is also 1,"},{"Start":"03:58.590 ","End":"04:01.475","Text":"and this is squared."},{"Start":"04:01.475 ","End":"04:07.100","Text":"What do we have? Root 2 plus root 2 is twice root 2,"},{"Start":"04:07.100 ","End":"04:08.435","Text":"and if you square it,"},{"Start":"04:08.435 ","End":"04:13.190","Text":"we get 4 times 2 which is 8."},{"Start":"04:13.190 ","End":"04:18.930","Text":"This is 1/8 and this is 1 plus 1 is 2,"},{"Start":"04:18.930 ","End":"04:23.385","Text":"2 ^2 is 4 plus 1/4,"},{"Start":"04:23.385 ","End":"04:29.595","Text":"so the answer is 3/8."},{"Start":"04:29.595 ","End":"04:37.370","Text":"Now part d. I don\u0027t want to scroll, I\u0027ll do it over here."},{"Start":"04:37.370 ","End":"04:46.830","Text":"Part d, we have sine Pi/4 is root 2/2."},{"Start":"04:48.220 ","End":"04:50.615","Text":"Cosecant of"},{"Start":"04:50.615 ","End":"04:55.280","Text":"Pi/2"},{"Start":"04:55.880 ","End":"05:00.740","Text":"it\u0027s 1,"},{"Start":"05:00.740 ","End":"05:10.355","Text":"and minus secant of Pi/4 is root 2."},{"Start":"05:10.355 ","End":"05:20.180","Text":"Cosine of Pi/2 is 0."},{"Start":"05:20.180 ","End":"05:24.100","Text":"This also is squared."},{"Start":"05:24.100 ","End":"05:27.930","Text":"What we have let\u0027s see,"},{"Start":"05:27.930 ","End":"05:29.190","Text":"if we square this,"},{"Start":"05:29.190 ","End":"05:31.905","Text":"we get root 2/2^2,"},{"Start":"05:31.905 ","End":"05:39.075","Text":"comes out to be a 1/2 twice this times this comes out root 2."},{"Start":"05:39.075 ","End":"05:42.580","Text":"This one squared is 1."},{"Start":"05:43.070 ","End":"05:46.380","Text":"Root 2 plus 0 is root 2 and if you square it,"},{"Start":"05:46.380 ","End":"05:50.050","Text":"it\u0027s 2, so it\u0027s minus 2."},{"Start":"05:51.020 ","End":"05:53.865","Text":"What we get is,"},{"Start":"05:53.865 ","End":"05:56.430","Text":"let\u0027s see, 1 and a 1/2 minus 2,"},{"Start":"05:56.430 ","End":"06:02.710","Text":"minus a 1/2 plus root 2."},{"Start":"06:03.260 ","End":"06:10.120","Text":"That\u0027s all four of them and we\u0027re done."}],"ID":14336},{"Watched":false,"Name":"Exercise 8","Duration":"8m 25s","ChapterTopicVideoID":13618,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13618.jpeg","UploadDate":"2018-10-21T13:46:51.6330000","DurationForVideoObject":"PT8M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.430","Text":"In this exercise, we have to find the exact value of"},{"Start":"00:03.430 ","End":"00:06.430","Text":"the expressions without a calculator,"},{"Start":"00:06.430 ","End":"00:08.740","Text":"so that\u0027s why I\u0027ve included a table."},{"Start":"00:08.740 ","End":"00:12.610","Text":"But the table doesn\u0027t help us right away here because look"},{"Start":"00:12.610 ","End":"00:17.125","Text":"all these angles are outside the range of 0-90 degrees."},{"Start":"00:17.125 ","End":"00:20.930","Text":"We\u0027re going to have to use some other rules."},{"Start":"00:20.930 ","End":"00:23.800","Text":"Perhaps I\u0027ll just write them as we need them."},{"Start":"00:23.800 ","End":"00:29.170","Text":"In part a, we have the cosine of 135,"},{"Start":"00:29.170 ","End":"00:34.805","Text":"so we\u0027ll need the rule for the cosine of a supplement of an angle,"},{"Start":"00:34.805 ","End":"00:43.275","Text":"180 minus Theta happens to be minus cosine Theta."},{"Start":"00:43.275 ","End":"00:49.055","Text":"In a, this is a 180 minus 45."},{"Start":"00:49.055 ","End":"00:56.675","Text":"This will be minus cosine of 45 degrees."},{"Start":"00:56.675 ","End":"01:00.530","Text":"Now for this, I\u0027ll use another rule."},{"Start":"01:00.530 ","End":"01:10.445","Text":"That is the tangent of Theta plus a 180 degrees is equal to tangent of Theta,"},{"Start":"01:10.445 ","End":"01:14.065","Text":"because the tangent has a period of a 180 degrees."},{"Start":"01:14.065 ","End":"01:24.575","Text":"In this case, it will be plus the tangent of 30 degrees because this is a 180 plus 30."},{"Start":"01:24.575 ","End":"01:32.495","Text":"Now I can use the table and this will be minus root 2 over 2."},{"Start":"01:32.495 ","End":"01:39.420","Text":"Tangent of 30 is root 3 over 3, and that\u0027s it."},{"Start":"01:39.420 ","End":"01:48.855","Text":"In b, sine of 270 is,"},{"Start":"01:48.855 ","End":"01:51.520","Text":"I could subtract 360,"},{"Start":"01:51.590 ","End":"01:56.090","Text":"I won\u0027t even write that the sine and the cosine of a period of 360,"},{"Start":"01:56.090 ","End":"02:01.070","Text":"so this is the same as the sine of minus 90 degrees,"},{"Start":"02:01.070 ","End":"02:05.220","Text":"270 is the same position as minus 90,"},{"Start":"02:05.220 ","End":"02:13.235","Text":"minus cotangent, and I can write this using a supplement."},{"Start":"02:13.235 ","End":"02:17.120","Text":"This is a 180 minus 45,"},{"Start":"02:17.120 ","End":"02:19.530","Text":"just like we had here."},{"Start":"02:20.350 ","End":"02:26.420","Text":"This rule also applies to the cotangent in the sense that the cotangent of"},{"Start":"02:26.420 ","End":"02:35.130","Text":"a 180 minus an angle is minus the cotangent of the angle."},{"Start":"02:38.690 ","End":"02:40.700","Text":"Let\u0027s see, yeah,"},{"Start":"02:40.700 ","End":"02:42.365","Text":"there\u0027s another rule we need."},{"Start":"02:42.365 ","End":"02:45.260","Text":"Sine of minus Theta,"},{"Start":"02:45.260 ","End":"02:46.909","Text":"sine is an odd function,"},{"Start":"02:46.909 ","End":"02:49.565","Text":"is minus sine Theta."},{"Start":"02:49.565 ","End":"02:55.410","Text":"We have minus sine of 90 degrees,"},{"Start":"02:55.410 ","End":"03:00.345","Text":"and minus with a minus is a plus,"},{"Start":"03:00.345 ","End":"03:06.450","Text":"plus cotangent of 45 degrees."},{"Start":"03:06.450 ","End":"03:09.420","Text":"This comes out,"},{"Start":"03:09.420 ","End":"03:16.055","Text":"this is equal to minus 1 cotangent of 45."},{"Start":"03:16.055 ","End":"03:17.780","Text":"Look it up here."},{"Start":"03:17.780 ","End":"03:21.750","Text":"It\u0027s equal to 1,"},{"Start":"03:22.780 ","End":"03:27.695","Text":"so the answer is 0."},{"Start":"03:27.695 ","End":"03:32.810","Text":"Now part C. Again,"},{"Start":"03:32.810 ","End":"03:38.990","Text":"we\u0027ll use the supplement formulas we\u0027ve had the one for cosine."},{"Start":"03:38.990 ","End":"03:41.165","Text":"I\u0027ll continue over here."},{"Start":"03:41.165 ","End":"03:45.275","Text":"The supplement with a sine works a bit differently."},{"Start":"03:45.275 ","End":"03:46.910","Text":"In the case of a sine,"},{"Start":"03:46.910 ","End":"03:52.539","Text":"the sine of the supplement is the same as the sine of the angle."},{"Start":"03:52.539 ","End":"03:56.320","Text":"In part c we\u0027ll have,"},{"Start":"03:56.950 ","End":"03:59.600","Text":"for the cosine, it\u0027s minus,"},{"Start":"03:59.600 ","End":"04:04.310","Text":"so this will be minus cosine of 60 degrees."},{"Start":"04:04.310 ","End":"04:07.790","Text":"Here it\u0027s a 180 minus 60,"},{"Start":"04:07.790 ","End":"04:10.880","Text":"but it\u0027ll be plus sine of 60."},{"Start":"04:10.880 ","End":"04:15.155","Text":"Sine of 60 minus."},{"Start":"04:15.155 ","End":"04:21.470","Text":"Now the tangent, like the cotangent also works with a negative."},{"Start":"04:21.470 ","End":"04:29.760","Text":"A tangent of 180 minus an angle is minus the tangent of the angle,"},{"Start":"04:30.050 ","End":"04:36.065","Text":"so this is equal to minus,"},{"Start":"04:36.065 ","End":"04:38.830","Text":"minus, that\u0027s a plus."},{"Start":"04:39.280 ","End":"04:42.335","Text":"I\u0027ll leave it as minus,"},{"Start":"04:42.335 ","End":"04:45.440","Text":"minus tangent of 60,"},{"Start":"04:45.440 ","End":"04:50.165","Text":"and the cotangent we had that before also works like a minus."},{"Start":"04:50.165 ","End":"04:57.800","Text":"That is minus cotangent of 60."},{"Start":"04:57.800 ","End":"05:01.310","Text":"I could\u0027ve taken a shortcut here because"},{"Start":"05:01.310 ","End":"05:04.340","Text":"tangent of anything times cotangent of anything is 1,"},{"Start":"05:04.340 ","End":"05:06.620","Text":"but we needed the practice."},{"Start":"05:06.620 ","End":"05:15.810","Text":"This is equal to cosine of 60 is 1/2."},{"Start":"05:15.810 ","End":"05:20.335","Text":"Sine of 60 is root 3 over 2."},{"Start":"05:20.335 ","End":"05:21.890","Text":"We have minus, minus,"},{"Start":"05:21.890 ","End":"05:24.470","Text":"minus, so it\u0027s minus."},{"Start":"05:24.470 ","End":"05:34.800","Text":"Tangent of 60 is root 3 and cotangent of 60 is root 3 over 3."},{"Start":"05:34.800 ","End":"05:41.545","Text":"What we get all together is minus root 3 over 4."},{"Start":"05:41.545 ","End":"05:44.735","Text":"This times this is 1 because it\u0027s 3 over 3."},{"Start":"05:44.735 ","End":"05:47.540","Text":"Also, this is the same as 1 over root 3."},{"Start":"05:47.540 ","End":"05:49.340","Text":"Anyway, like I said,"},{"Start":"05:49.340 ","End":"05:55.720","Text":"the cotangent times the tangent, they\u0027re going to give 1."},{"Start":"05:55.880 ","End":"05:59.220","Text":"Let\u0027s see. Now, let\u0027s see part"},{"Start":"05:59.220 ","End":"06:07.170","Text":"d. In this one,"},{"Start":"06:07.170 ","End":"06:14.785","Text":"I want to subtract a 180 degrees from this."},{"Start":"06:14.785 ","End":"06:23.075","Text":"These two I want to subtract 360 and it won\u0027t change them."},{"Start":"06:23.075 ","End":"06:24.755","Text":"What I\u0027m saying is,"},{"Start":"06:24.755 ","End":"06:29.090","Text":"I have here secant"},{"Start":"06:29.090 ","End":"06:37.460","Text":"of 180 degrees plus 60 degrees."},{"Start":"06:37.460 ","End":"06:42.125","Text":"Here I have minus cosine."},{"Start":"06:42.125 ","End":"06:44.540","Text":"If I subtract 360,"},{"Start":"06:44.540 ","End":"06:46.645","Text":"which won\u0027t change it,"},{"Start":"06:46.645 ","End":"06:53.350","Text":"that will be minus 45 degrees."},{"Start":"06:54.830 ","End":"07:05.340","Text":"Here I\u0027ll also subtract 360 and get cotangent of minus 60 degrees."},{"Start":"07:05.750 ","End":"07:09.270","Text":"Now in the case of a secant,"},{"Start":"07:09.270 ","End":"07:11.580","Text":"when you add 180,"},{"Start":"07:11.580 ","End":"07:14.190","Text":"it just makes it negative,"},{"Start":"07:14.190 ","End":"07:18.500","Text":"so I have minus secant of 60 degrees."},{"Start":"07:18.500 ","End":"07:21.965","Text":"I know this because I know it works for cosine,"},{"Start":"07:21.965 ","End":"07:25.115","Text":"so it also is going to work for secant."},{"Start":"07:25.115 ","End":"07:29.045","Text":"Now here, cosine is an even function."},{"Start":"07:29.045 ","End":"07:34.575","Text":"This is the same as minus cosine of 45 degrees,"},{"Start":"07:34.575 ","End":"07:38.475","Text":"and cotangent like tangent is an odd function,"},{"Start":"07:38.475 ","End":"07:44.045","Text":"so that is minus cotangent of 60 degrees."},{"Start":"07:44.045 ","End":"07:46.320","Text":"Now everything\u0027s minus,"},{"Start":"07:46.320 ","End":"07:49.530","Text":"so it\u0027s going to be minus,"},{"Start":"07:49.530 ","End":"07:56.250","Text":"and let\u0027s see, secant of 60 from the table is 2."},{"Start":"07:56.250 ","End":"08:03.045","Text":"Then cosine of 45 is root 2 over 2,"},{"Start":"08:03.045 ","End":"08:13.540","Text":"and cotangent of 60 is root 3 over 3,"},{"Start":"08:16.250 ","End":"08:20.195","Text":"and possibly we could simplify this,"},{"Start":"08:20.195 ","End":"08:23.300","Text":"but let\u0027s just leave it at that."},{"Start":"08:23.300 ","End":"08:26.100","Text":"Okay, we\u0027re done."}],"ID":14337},{"Watched":false,"Name":"Exercise 9 - Part a","Duration":"6m 26s","ChapterTopicVideoID":13619,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13619.jpeg","UploadDate":"2018-10-21T13:48:04.8800000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we have to compute sine of Alpha plus"},{"Start":"00:03.660 ","End":"00:07.905","Text":"Beta and also cosine of Alpha plus Beta."},{"Start":"00:07.905 ","End":"00:12.415","Text":"We\u0027re given some conditions on Alpha and Beta."},{"Start":"00:12.415 ","End":"00:14.880","Text":"We\u0027re given the sine of Alpha, the sine of Beta,"},{"Start":"00:14.880 ","End":"00:19.585","Text":"and possible ranges for Alpha and Beta."},{"Start":"00:19.585 ","End":"00:25.939","Text":"Here I have the formulas for sum and difference of sine and cosine."},{"Start":"00:25.939 ","End":"00:30.290","Text":"What we\u0027ll need is this formula first,"},{"Start":"00:30.290 ","End":"00:33.155","Text":"and then this formula."},{"Start":"00:33.155 ","End":"00:36.125","Text":"Now, in both of them,"},{"Start":"00:36.125 ","End":"00:46.205","Text":"we will need cosine of Alpha and cosine of Beta."},{"Start":"00:46.205 ","End":"00:50.980","Text":"We have sine Alpha and sine Beta, these are given."},{"Start":"00:50.980 ","End":"00:54.705","Text":"In either case, we\u0027ll need these two."},{"Start":"00:54.705 ","End":"00:57.480","Text":"Let\u0027s compute those first."},{"Start":"00:57.480 ","End":"01:03.890","Text":"Cosine Alpha from sine Alpha is plus or"},{"Start":"01:03.890 ","End":"01:14.430","Text":"minus the square root of 1 minus sin^2 Alpha in general because sin^2 plus cosin^2 is 1."},{"Start":"01:14.960 ","End":"01:22.080","Text":"We know that Alpha is between 0 and 90 degrees,"},{"Start":"01:22.080 ","End":"01:25.220","Text":"or Pi over 2, so it\u0027s in the first quadrant."},{"Start":"01:25.220 ","End":"01:28.175","Text":"It\u0027s in quadrant 1."},{"Start":"01:28.175 ","End":"01:38.040","Text":"Beta is in quadrant 2 because it\u0027s between 90 and 180 degrees, quadrant 2."},{"Start":"01:39.680 ","End":"01:43.860","Text":"Now, in quadrant 1,"},{"Start":"01:45.230 ","End":"01:48.469","Text":"all the trigonometric functions are positive,"},{"Start":"01:48.469 ","End":"01:50.390","Text":"so the cosine is positive,"},{"Start":"01:50.390 ","End":"01:52.765","Text":"actually this is plus."},{"Start":"01:52.765 ","End":"01:56.730","Text":"I\u0027m just taking the plus 1 minus,"},{"Start":"01:56.730 ","End":"02:00.223","Text":"sine Alpha is 4/9,"},{"Start":"02:00.223 ","End":"02:06.520","Text":"so this would be 16/81."},{"Start":"02:07.910 ","End":"02:12.760","Text":"Let\u0027s see, this equals"},{"Start":"02:16.700 ","End":"02:21.080","Text":"65 over 81 square root."},{"Start":"02:21.080 ","End":"02:23.240","Text":"I can take just the square root of this."},{"Start":"02:23.240 ","End":"02:26.000","Text":"The square root of 81 is 9."},{"Start":"02:26.000 ","End":"02:30.180","Text":"That\u0027s the cosine of Alpha."},{"Start":"02:32.350 ","End":"02:40.815","Text":"The cosine of Beta will be similarly,"},{"Start":"02:40.815 ","End":"02:47.710","Text":"but it\u0027s going to be negative because we said that Beta is in quadrant 2,"},{"Start":"02:47.710 ","End":"02:56.140","Text":"so it\u0027s minus the square root of 1 minus sine squared Beta."},{"Start":"02:57.890 ","End":"03:04.450","Text":"When it\u0027s squared, the minus disappears, so it\u0027s 9/25."},{"Start":"03:05.450 ","End":"03:08.445","Text":"This will equal,"},{"Start":"03:08.445 ","End":"03:11.268","Text":"25 minus 9 is 16,"},{"Start":"03:11.268 ","End":"03:15.435","Text":"16 has a square root."},{"Start":"03:15.435 ","End":"03:18.550","Text":"It\u0027s going to be minus 4/5."},{"Start":"03:19.160 ","End":"03:23.980","Text":"Now, we just have to substitute."},{"Start":"03:24.350 ","End":"03:34.320","Text":"Sine of Alpha plus Beta is equal to, from here,"},{"Start":"03:34.320 ","End":"03:40.215","Text":"sine Alpha, 4/9, cosine Beta,"},{"Start":"03:40.215 ","End":"03:47.085","Text":"minus 4/5, plus cosine Alpha,"},{"Start":"03:47.085 ","End":"03:51.150","Text":"root 65 over 9."},{"Start":"03:51.150 ","End":"03:58.210","Text":"Sine Beta is minus 3/5."},{"Start":"03:58.460 ","End":"04:00.860","Text":"Altogether, let\u0027s see."},{"Start":"04:00.860 ","End":"04:02.240","Text":"We have a minus and a minus,"},{"Start":"04:02.240 ","End":"04:03.940","Text":"so we can make it minus."},{"Start":"04:03.940 ","End":"04:10.795","Text":"It\u0027s all going to be over 9 times 5, which is 45."},{"Start":"04:10.795 ","End":"04:12.740","Text":"What are we going to have here?"},{"Start":"04:12.740 ","End":"04:15.820","Text":"4 times 4 is 16,"},{"Start":"04:15.820 ","End":"04:23.500","Text":"minus 3 root 65."},{"Start":"04:26.150 ","End":"04:28.430","Text":"That will do. Of course,"},{"Start":"04:28.430 ","End":"04:32.540","Text":"we could get rid of the minus if we switch the order of these two,"},{"Start":"04:32.540 ","End":"04:38.115","Text":"but I\u0027ll leave it like this."},{"Start":"04:38.115 ","End":"04:42.875","Text":"I guess we have room. I actually prefer to have it as,"},{"Start":"04:42.875 ","End":"04:50.545","Text":"this looks a bit simpler, 3 root 65 minus 16 over 45."},{"Start":"04:50.545 ","End":"04:54.830","Text":"Anyway, this is bigger than this because root 65 is just over 8,"},{"Start":"04:54.830 ","End":"04:58.235","Text":"so this is just over 24, 25."},{"Start":"04:58.235 ","End":"05:00.920","Text":"Anyway, comes up positive."},{"Start":"05:00.920 ","End":"05:04.640","Text":"That\u0027s one. Now, the other one,"},{"Start":"05:04.640 ","End":"05:08.660","Text":"cosine of Alpha plus Beta."},{"Start":"05:08.660 ","End":"05:10.010","Text":"Again, just formula."},{"Start":"05:10.010 ","End":"05:13.970","Text":"Cosine Alpha cosine Beta, this times this."},{"Start":"05:13.970 ","End":"05:19.590","Text":"It\u0027s minus root 65 over 9 times 4/5,"},{"Start":"05:19.590 ","End":"05:22.005","Text":"you put the minus to the front."},{"Start":"05:22.005 ","End":"05:27.271","Text":"Then minus sine Alpha sine Beta,"},{"Start":"05:27.271 ","End":"05:33.210","Text":"again, the minus I\u0027m bringing to the front and there already is a minus, so it\u0027s plus."},{"Start":"05:33.370 ","End":"05:37.000","Text":"Sine Alpha is 4/9,"},{"Start":"05:37.000 ","End":"05:42.070","Text":"sine Beta, the minus I took care of, so 3/5."},{"Start":"05:42.080 ","End":"05:49.220","Text":"Here also, we will get minus this plus this,"},{"Start":"05:49.220 ","End":"05:51.080","Text":"I\u0027ll write the plus first,"},{"Start":"05:51.080 ","End":"05:53.675","Text":"4 times 3 is 12,"},{"Start":"05:53.675 ","End":"06:05.310","Text":"minus 4 root 65 over 45."},{"Start":"06:06.590 ","End":"06:10.315","Text":"I just compacted this."},{"Start":"06:10.315 ","End":"06:13.820","Text":"I also want to highlight the results."},{"Start":"06:13.820 ","End":"06:23.010","Text":"This is the result of the sine Alpha plus Beta and the cosine of Alpha plus Beta is this."},{"Start":"06:23.010 ","End":"06:27.520","Text":"Done."}],"ID":14338},{"Watched":false,"Name":"Exercise 9 - Part b","Duration":"6m 31s","ChapterTopicVideoID":13620,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13620.jpeg","UploadDate":"2018-10-21T13:49:17.3830000","DurationForVideoObject":"PT6M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this exercise, we have to compute sine of Alpha"},{"Start":"00:03.810 ","End":"00:07.545","Text":"minus Beta and also cosine of Alpha minus Beta."},{"Start":"00:07.545 ","End":"00:10.290","Text":"We\u0027re given conditions on Alpha and Beta,"},{"Start":"00:10.290 ","End":"00:11.790","Text":"we\u0027re given the sine of Alpha,"},{"Start":"00:11.790 ","End":"00:13.860","Text":"we\u0027re given the cosine of Beta,"},{"Start":"00:13.860 ","End":"00:16.890","Text":"and we\u0027re given the quadrants for Alpha and Beta."},{"Start":"00:16.890 ","End":"00:20.445","Text":"This is 180 and 270,"},{"Start":"00:20.445 ","End":"00:29.070","Text":"so that is quadrant number III for Alpha and Beta between 0 and Pi/2,"},{"Start":"00:29.070 ","End":"00:33.405","Text":"0 and 90, that\u0027s the quadrant number I."},{"Start":"00:33.405 ","End":"00:37.770","Text":"This will tell us what sign,"},{"Start":"00:37.770 ","End":"00:39.315","Text":"plus or minus,"},{"Start":"00:39.315 ","End":"00:43.140","Text":"we\u0027ll have for our computations."},{"Start":"00:43.140 ","End":"00:48.730","Text":"We\u0027ll need cos Alpha and will need sin Beta, well, you\u0027ll see."},{"Start":"00:48.890 ","End":"00:51.410","Text":"So which formulas do we need?"},{"Start":"00:51.410 ","End":"00:52.910","Text":"We need this one,"},{"Start":"00:52.910 ","End":"00:55.025","Text":"and we need this one."},{"Start":"00:55.025 ","End":"01:02.410","Text":"Now, we have sine of Alpha and we have cosine of Beta."},{"Start":"01:02.410 ","End":"01:05.205","Text":"We have sin Alpha,"},{"Start":"01:05.205 ","End":"01:07.020","Text":"we have cos Beta."},{"Start":"01:07.020 ","End":"01:08.610","Text":"What we need to find,"},{"Start":"01:08.610 ","End":"01:15.660","Text":"our cos Alpha and we need to find sine of Beta."},{"Start":"01:15.660 ","End":"01:17.955","Text":"Then we can substitute."},{"Start":"01:17.955 ","End":"01:21.510","Text":"So which should we take first?"},{"Start":"01:21.510 ","End":"01:25.965","Text":"Let\u0027s go for cos Alpha."},{"Start":"01:25.965 ","End":"01:31.550","Text":"Cos Alpha is plus or"},{"Start":"01:31.550 ","End":"01:39.780","Text":"minus the square root of 1 minus sin^2 Alpha."},{"Start":"01:40.880 ","End":"01:45.480","Text":"Now, Alpha is in the third quadrant,"},{"Start":"01:45.480 ","End":"01:48.965","Text":"and in the third quadrant,"},{"Start":"01:48.965 ","End":"01:53.670","Text":"the cosine is negative."},{"Start":"01:53.740 ","End":"01:59.300","Text":"If you remember, they\u0027re all positive in"},{"Start":"01:59.300 ","End":"02:02.300","Text":"the first quadrant and in the third quadrant"},{"Start":"02:02.300 ","End":"02:06.750","Text":"only the tangent is positive, there\u0027s that mnemonic."},{"Start":"02:07.300 ","End":"02:12.305","Text":"Anyway, what do we get?"},{"Start":"02:12.305 ","End":"02:19.520","Text":"We get minus the square root of 1 minus"},{"Start":"02:19.520 ","End":"02:21.289","Text":"sin^2 Alpha"},{"Start":"02:21.289 ","End":"02:30.860","Text":"is 16/81."},{"Start":"02:30.860 ","End":"02:36.015","Text":"What am I saying? This is equal to,"},{"Start":"02:36.015 ","End":"02:40.730","Text":"let\u0027s see, 81 minus 16 is 65."},{"Start":"02:40.730 ","End":"02:46.580","Text":"So we have minus root 65 over root 81,"},{"Start":"02:46.580 ","End":"02:50.525","Text":"which is 9, that\u0027s cos Alpha."},{"Start":"02:50.525 ","End":"02:55.075","Text":"Now, we need sine of Beta."},{"Start":"02:55.075 ","End":"02:58.620","Text":"So sine of Beta,"},{"Start":"02:58.620 ","End":"03:03.335","Text":"this is going to be positive because Beta is in the first quadrant."},{"Start":"03:03.335 ","End":"03:07.645","Text":"So it\u0027s going to be the square root"},{"Start":"03:07.645 ","End":"03:16.650","Text":"of 1 minus cos^2 Beta,"},{"Start":"03:18.670 ","End":"03:24.570","Text":"which is equal to the square root of"},{"Start":"03:24.570 ","End":"03:28.655","Text":"1 minus cos^2 Beta"},{"Start":"03:28.655 ","End":"03:30.150","Text":"is"},{"Start":"03:37.690 ","End":"03:38.945","Text":"25/144."},{"Start":"03:38.945 ","End":"03:41.790","Text":"What does that give us?"},{"Start":"03:42.310 ","End":"03:49.140","Text":"It gives us the square root of"},{"Start":"03:52.220 ","End":"04:03.930","Text":"119/12."},{"Start":"04:03.930 ","End":"04:06.970","Text":"We have cos Alpha, we have sin Beta."},{"Start":"04:06.970 ","End":"04:17.330","Text":"Now we can substitute sine of Alpha minus Beta is sin Alpha minus 4/9."},{"Start":"04:17.330 ","End":"04:28.725","Text":"Cos Beta 5/12 minus cos Alpha."},{"Start":"04:28.725 ","End":"04:31.920","Text":"Cos Alpha is this,"},{"Start":"04:31.920 ","End":"04:33.150","Text":"so I\u0027ll make it a plus,"},{"Start":"04:33.150 ","End":"04:38.730","Text":"minus minus is plus, root 65/9,"},{"Start":"04:38.730 ","End":"04:42.165","Text":"and sin Beta"},{"Start":"04:42.165 ","End":"04:49.530","Text":"root 119/12."},{"Start":"04:49.530 ","End":"04:52.755","Text":"Let\u0027s just collect it under 1 common denominator,"},{"Start":"04:52.755 ","End":"04:57.290","Text":"9 times 12 is 108."},{"Start":"04:57.290 ","End":"05:00.440","Text":"So we have minus 20,"},{"Start":"05:00.440 ","End":"05:02.165","Text":"I\u0027ll write this here,"},{"Start":"05:02.165 ","End":"05:08.280","Text":"plus square root of 65,"},{"Start":"05:08.280 ","End":"05:10.785","Text":"square root of 119,"},{"Start":"05:10.785 ","End":"05:13.506","Text":"or you could multiply 65 by 119,"},{"Start":"05:13.506 ","End":"05:14.520","Text":"I don\u0027t want to do that,"},{"Start":"05:14.520 ","End":"05:16.495","Text":"I leave it like this."},{"Start":"05:16.495 ","End":"05:20.370","Text":"That\u0027s one part."},{"Start":"05:21.470 ","End":"05:23.810","Text":"Just highlight it."},{"Start":"05:23.810 ","End":"05:26.465","Text":"Now, the next part,"},{"Start":"05:26.465 ","End":"05:33.995","Text":"we get that cosine of Alpha minus Beta is cos Alpha cos Beta."},{"Start":"05:33.995 ","End":"05:41.325","Text":"Cos Alpha is here minus root 65/9."},{"Start":"05:41.325 ","End":"05:43.650","Text":"Cos Beta was given,"},{"Start":"05:43.650 ","End":"05:50.820","Text":"is 5/12 plus sin Alpha,"},{"Start":"05:50.820 ","End":"05:54.370","Text":"which is minus 4/9."},{"Start":"05:54.530 ","End":"05:57.855","Text":"So I just made this into a minus."},{"Start":"05:57.855 ","End":"06:02.910","Text":"That\u0027s 4/9 and sin Beta,"},{"Start":"06:02.910 ","End":"06:10.030","Text":"which is root 119/12."},{"Start":"06:10.580 ","End":"06:12.780","Text":"They\u0027re both minus,"},{"Start":"06:12.780 ","End":"06:16.920","Text":"so we\u0027ll put the minus in front and then I have 5 root"},{"Start":"06:16.920 ","End":"06:25.390","Text":"65 minus 4 root of 119 over 108,"},{"Start":"06:25.430 ","End":"06:31.480","Text":"and highlight the answer, and we\u0027re done."}],"ID":14339},{"Watched":false,"Name":"Exercise 9 - Part c","Duration":"9m 20s","ChapterTopicVideoID":13621,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13621.jpeg","UploadDate":"2018-10-21T13:51:08.4030000","DurationForVideoObject":"PT9M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.430","Text":"In this exercise, we have to compute tangent of Alpha minus"},{"Start":"00:05.430 ","End":"00:10.470","Text":"Beta and we\u0027re given conditions on Alpha and Beta that,"},{"Start":"00:10.470 ","End":"00:12.270","Text":"we\u0027re given the cosine of Alpha,"},{"Start":"00:12.270 ","End":"00:17.625","Text":"we\u0027re given the sine of Beta and we\u0027re given quadrant information."},{"Start":"00:17.625 ","End":"00:22.350","Text":"Alpha is in the first quadrant based on this inequality,"},{"Start":"00:22.350 ","End":"00:28.530","Text":"and Beta is in the fourth quadrant based on this inequality,"},{"Start":"00:28.530 ","End":"00:31.575","Text":"270 and 360 degrees."},{"Start":"00:31.575 ","End":"00:40.175","Text":"The quadrants will help us to decide whether sine or cosine are plus or minus."},{"Start":"00:40.175 ","End":"00:42.290","Text":"Now we\u0027re given a hint tool."},{"Start":"00:42.290 ","End":"00:43.985","Text":"Because we have a tangent here,"},{"Start":"00:43.985 ","End":"00:46.860","Text":"the tangent is sine over cosine."},{"Start":"00:46.910 ","End":"00:52.410","Text":"What I\u0027m saying is that tangent of Alpha minus Beta,"},{"Start":"00:52.410 ","End":"00:58.320","Text":"where we let Theta equals Alpha minus Beta will be sine of"},{"Start":"00:58.320 ","End":"01:04.590","Text":"Alpha minus Beta over cosine of Alpha minus Beta."},{"Start":"01:04.590 ","End":"01:10.535","Text":"These two will compute with this formula and this formula."},{"Start":"01:10.535 ","End":"01:13.460","Text":"Now note we\u0027re given cosine Alpha,"},{"Start":"01:13.460 ","End":"01:15.459","Text":"so we have this."},{"Start":"01:15.459 ","End":"01:17.955","Text":"We\u0027re given sine of Beta,"},{"Start":"01:17.955 ","End":"01:20.005","Text":"so we have this."},{"Start":"01:20.005 ","End":"01:22.940","Text":"But the other two we need to compute,"},{"Start":"01:22.940 ","End":"01:26.180","Text":"we need to compute sine Alpha and we need to"},{"Start":"01:26.180 ","End":"01:30.755","Text":"compute cosine Beta before we can use the formulas."},{"Start":"01:30.755 ","End":"01:33.950","Text":"Let\u0027s start with sine Alpha."},{"Start":"01:33.950 ","End":"01:38.575","Text":"Sine Alpha is plus or minus."},{"Start":"01:38.575 ","End":"01:43.805","Text":"But I know it\u0027s going to be plus or minus because Alpha is in the first quadrant."},{"Start":"01:43.805 ","End":"01:46.730","Text":"In the first quadrant the sine is positive."},{"Start":"01:46.730 ","End":"01:48.500","Text":"So it\u0027s a plus."},{"Start":"01:48.500 ","End":"01:50.465","Text":"I\u0027ll just write plus to show I didn\u0027t forget,"},{"Start":"01:50.465 ","End":"01:59.640","Text":"plus the square root of 1 minus cosine squared Alpha,"},{"Start":"01:59.640 ","End":"02:05.325","Text":"which is 1 over 100."},{"Start":"02:05.325 ","End":"02:07.245","Text":"That\u0027s 1/10^2."},{"Start":"02:07.245 ","End":"02:10.605","Text":"It a root of 99 over 100."},{"Start":"02:10.605 ","End":"02:14.390","Text":"It\u0027s root 99 over root 100."},{"Start":"02:14.390 ","End":"02:17.630","Text":"This is what sine Alpha is."},{"Start":"02:17.630 ","End":"02:22.055","Text":"Then cosine Beta."},{"Start":"02:22.055 ","End":"02:29.540","Text":"Cosine Beta is, Beta is in the fourth quadrant,"},{"Start":"02:29.540 ","End":"02:33.320","Text":"and cosine is positive in the fourth quadrant,"},{"Start":"02:33.320 ","End":"02:38.750","Text":"it\u0027s positive in the first and fourth quadrants and so we get also"},{"Start":"02:38.750 ","End":"02:45.855","Text":"plus the square root of 1 minus sine squared Beta."},{"Start":"02:45.855 ","End":"02:50.355","Text":"When I square it, it\u0027s plus 1 over 25,"},{"Start":"02:50.355 ","End":"02:53.805","Text":"which is the root of 24 over 25."},{"Start":"02:53.805 ","End":"02:59.050","Text":"It\u0027s root 24 over 5."},{"Start":"02:59.120 ","End":"03:07.445","Text":"Now, we can compute sine of"},{"Start":"03:07.445 ","End":"03:17.415","Text":"Alpha minus Beta equals sine Alpha root 99 over 10,"},{"Start":"03:17.415 ","End":"03:23.670","Text":"cosine Beta root 24 over"},{"Start":"03:23.670 ","End":"03:33.495","Text":"5 minus cosine Alpha, which is 1/10."},{"Start":"03:33.495 ","End":"03:37.860","Text":"Sine Beta is minus 1/5."},{"Start":"03:37.860 ","End":"03:39.595","Text":"Now instead of minus 1/5,"},{"Start":"03:39.595 ","End":"03:41.285","Text":"I\u0027ll make that a plus,"},{"Start":"03:41.285 ","End":"03:43.790","Text":"and then it will be 1/5."},{"Start":"03:43.790 ","End":"03:53.705","Text":"Altogether, if I put it over 50 it\u0027s root 99,"},{"Start":"03:53.705 ","End":"04:03.810","Text":"root 24 plus 1."},{"Start":"04:03.810 ","End":"04:07.035","Text":"Cosine of Alpha minus Beta,"},{"Start":"04:07.035 ","End":"04:09.465","Text":"cos Alpha cos Beta."},{"Start":"04:09.465 ","End":"04:15.060","Text":"It\u0027s 1/10 times root"},{"Start":"04:15.060 ","End":"04:22.810","Text":"24 over 5 plus sine Alpha."},{"Start":"04:26.150 ","End":"04:29.810","Text":"Well, I see that sine Beta is negative,"},{"Start":"04:29.810 ","End":"04:32.240","Text":"so already I\u0027m going to write the minus here."},{"Start":"04:32.240 ","End":"04:39.740","Text":"So sine Alpha is route 99 over 10,"},{"Start":"04:39.740 ","End":"04:45.695","Text":"and sine Beta without the minus is 1/5."},{"Start":"04:45.695 ","End":"04:49.175","Text":"That\u0027s also over 50."},{"Start":"04:49.175 ","End":"04:55.860","Text":"That would be root 24 minus root 99."},{"Start":"04:55.860 ","End":"04:58.310","Text":"That\u0027s going to be negative never mind."},{"Start":"04:58.310 ","End":"05:01.890","Text":"Anyway, that gives us these two."},{"Start":"05:02.630 ","End":"05:08.530","Text":"Now we can say what tangent of Alpha minus Beta is."},{"Start":"05:09.550 ","End":"05:13.160","Text":"I\u0027ll continue over here."},{"Start":"05:13.160 ","End":"05:23.604","Text":"We get that tangent of Alpha minus Beta is equal to this over this,"},{"Start":"05:23.604 ","End":"05:31.715","Text":"so the 50 in the denominator cancels and we get root 99,"},{"Start":"05:31.715 ","End":"05:36.955","Text":"root 24 plus 1"},{"Start":"05:36.955 ","End":"05:46.600","Text":"over root 24 minus root 99."},{"Start":"05:47.030 ","End":"05:51.185","Text":"We\u0027re done. But don\u0027t go."},{"Start":"05:51.185 ","End":"05:52.924","Text":"I have an alternative solution,"},{"Start":"05:52.924 ","End":"05:54.845","Text":"if you would like."},{"Start":"05:54.845 ","End":"06:01.430","Text":"That is, there is another formula for the tangent of the difference."},{"Start":"06:01.430 ","End":"06:03.560","Text":"I\u0027ll just like to show it to you."},{"Start":"06:03.560 ","End":"06:13.280","Text":"There is a formula that tangent of Alpha minus Beta equals tangent Alpha minus"},{"Start":"06:13.280 ","End":"06:23.680","Text":"tangent Beta over 1 plus tangent Alpha tangent Beta."},{"Start":"06:23.680 ","End":"06:35.920","Text":"I\u0027ll"},{"Start":"06:35.920 ","End":"06:36.760","Text":"just write a bit more."},{"Start":"06:36.760 ","End":"06:40.810","Text":"This is sine Alpha over cosine Alpha."},{"Start":"06:40.810 ","End":"06:42.730","Text":"That\u0027s the tangent of Alpha."},{"Start":"06:42.730 ","End":"06:48.150","Text":"We can compute that or maybe I\u0027ll go on with this."},{"Start":"06:48.150 ","End":"06:53.370","Text":"Let\u0027s just see, sine Alpha"},{"Start":"06:53.370 ","End":"06:59.130","Text":"where was it now?"},{"Start":"06:59.130 ","End":"07:04.330","Text":"Yeah, here it is. Is root 99 over 10."},{"Start":"07:06.050 ","End":"07:08.280","Text":"I didn\u0027t organize it right."},{"Start":"07:08.280 ","End":"07:11.185","Text":"I meant to say tangent Alpha is this."},{"Start":"07:11.185 ","End":"07:15.760","Text":"Now sine Alpha is root 99 over 10"},{"Start":"07:15.760 ","End":"07:22.285","Text":"and cosine of Alpha is 1/10."},{"Start":"07:22.285 ","End":"07:31.140","Text":"We just get root 99 over 10 divided by 1/10 is this and"},{"Start":"07:31.140 ","End":"07:39.975","Text":"tangent of Beta is sine Beta over cosine Beta,"},{"Start":"07:39.975 ","End":"07:44.943","Text":"which is, sine of Beta was minus 1/5,"},{"Start":"07:44.943 ","End":"07:48.060","Text":"cosine Beta is root 24 over 5."},{"Start":"07:48.060 ","End":"07:56.350","Text":"The 5 cancels and we get minus 1 over root 24."},{"Start":"07:57.400 ","End":"08:02.030","Text":"If we substitute in the formula,"},{"Start":"08:02.030 ","End":"08:09.920","Text":"we get that this is equal to tangent of Alpha root 99."},{"Start":"08:09.920 ","End":"08:16.303","Text":"Tangent Beta minus 1 over root 24,"},{"Start":"08:16.303 ","End":"08:23.415","Text":"it will be plus divided by 1 plus tangent Alpha"},{"Start":"08:23.415 ","End":"08:33.625","Text":"is root 99 times tangent Beta is minus 1 over root 24."},{"Start":"08:33.625 ","End":"08:37.790","Text":"I\u0027ll change this to a minus."},{"Start":"08:37.790 ","End":"08:43.040","Text":"Now multiply top and bottom by"},{"Start":"08:43.040 ","End":"08:49.475","Text":"root 24 and we get root 99,"},{"Start":"08:49.475 ","End":"09:00.490","Text":"root 24 plus 1 over root 24 minus root 99."},{"Start":"09:00.490 ","End":"09:05.360","Text":"I would say that this is the same answer that we got before,"},{"Start":"09:05.360 ","End":"09:08.300","Text":"which is comforting to know that if you do something two ways,"},{"Start":"09:08.300 ","End":"09:10.130","Text":"you get the same answer."},{"Start":"09:10.130 ","End":"09:12.650","Text":"But I thought I\u0027d give you the alternative."},{"Start":"09:12.650 ","End":"09:16.200","Text":"You could have just stopped here of course."},{"Start":"09:16.770 ","End":"09:19.880","Text":"We\u0027re more than done."}],"ID":14340},{"Watched":false,"Name":"Exercise 9 - Part d","Duration":"4m 34s","ChapterTopicVideoID":13622,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13622.jpeg","UploadDate":"2018-10-21T13:51:58.8730000","DurationForVideoObject":"PT4M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we have to compute sine of Alpha"},{"Start":"00:04.890 ","End":"00:09.570","Text":"plus Beta and also cosine of Alpha minus Beta."},{"Start":"00:09.570 ","End":"00:14.040","Text":"We\u0027re given certain conditions on Alpha and Beta."},{"Start":"00:14.040 ","End":"00:16.785","Text":"We are given the value of cosine Alpha,"},{"Start":"00:16.785 ","End":"00:23.865","Text":"we\u0027re given the value of cosine Beta and we\u0027re given the quadrant."},{"Start":"00:23.865 ","End":"00:30.045","Text":"This translates to saying that Alpha is in the first quadrant."},{"Start":"00:30.045 ","End":"00:36.355","Text":"This says that Beta is in the second quadrant,"},{"Start":"00:36.355 ","End":"00:40.130","Text":"I mean, it\u0027s between 90 and 180 degrees."},{"Start":"00:40.130 ","End":"00:41.705","Text":"First of all, does it make sense?"},{"Start":"00:41.705 ","End":"00:46.320","Text":"Yes. In the second quadrant, cosine is negative."},{"Start":"00:47.420 ","End":"00:51.395","Text":"We\u0027re going to use to sine Alpha plus Beta,"},{"Start":"00:51.395 ","End":"00:55.745","Text":"this formula, for cosine of Alpha minus Beta, this formula."},{"Start":"00:55.745 ","End":"00:58.025","Text":"We have cosine Alpha,"},{"Start":"00:58.025 ","End":"01:00.290","Text":"which is this and this."},{"Start":"01:00.290 ","End":"01:03.545","Text":"We have cosine Beta,"},{"Start":"01:03.545 ","End":"01:06.755","Text":"which is this and this."},{"Start":"01:06.755 ","End":"01:14.680","Text":"What we need are sine Alpha and sine Beta."},{"Start":"01:15.010 ","End":"01:18.245","Text":"Of course, we\u0027re going to use the formula,"},{"Start":"01:18.245 ","End":"01:22.460","Text":"I don\u0027t want to use Alpha or Beta,"},{"Start":"01:22.460 ","End":"01:31.765","Text":"I\u0027ll say sine of Theta is plus or minus the square root of 1 minus cosine squared Theta."},{"Start":"01:31.765 ","End":"01:34.160","Text":"I\u0027ll use it once with Theta being Alpha,"},{"Start":"01:34.160 ","End":"01:36.155","Text":"once with Theta being Beta."},{"Start":"01:36.155 ","End":"01:41.944","Text":"Here we are, sine of Alpha."},{"Start":"01:41.944 ","End":"01:45.650","Text":"Now, I don\u0027t want to leave it as plus or minus."},{"Start":"01:45.650 ","End":"01:50.165","Text":"I want to decide which Alpha is in the first quadrant."},{"Start":"01:50.165 ","End":"01:53.164","Text":"Everything\u0027s positive in the first quadrant."},{"Start":"01:53.164 ","End":"01:58.040","Text":"It\u0027s going to be the square root of 1 minus cosine squared,"},{"Start":"01:58.040 ","End":"02:03.280","Text":"which is 1 minus cosine squared."},{"Start":"02:03.860 ","End":"02:08.920","Text":"Cosine squared Alpha is 1\\9."},{"Start":"02:09.220 ","End":"02:13.220","Text":"That is equal to the square root of 8\\9,"},{"Start":"02:13.220 ","End":"02:16.895","Text":"which is the square root of 8 over the square root of 9."},{"Start":"02:16.895 ","End":"02:19.140","Text":"It\u0027s like this."},{"Start":"02:20.350 ","End":"02:28.460","Text":"Sine of Beta will be the square root."},{"Start":"02:28.460 ","End":"02:37.950","Text":"Now this time it\u0027s going to be negative because we are in the second quadrant."},{"Start":"02:42.080 ","End":"02:46.429","Text":"Sine is also positive in the second quadrant."},{"Start":"02:46.429 ","End":"02:52.430","Text":"We have 1 minus cosine squared Beta."},{"Start":"02:52.430 ","End":"02:57.175","Text":"When I square this, it still comes out to be 1/9."},{"Start":"02:57.175 ","End":"02:59.610","Text":"I guess that\u0027s the same thing,"},{"Start":"02:59.610 ","End":"03:05.260","Text":"root 8 over 3."},{"Start":"03:05.260 ","End":"03:08.450","Text":"Now we can apply the formulas that we need."},{"Start":"03:08.450 ","End":"03:16.815","Text":"Sine of Alpha plus Beta is sine Alpha root 8 over 3."},{"Start":"03:16.815 ","End":"03:21.810","Text":"Cosine Beta minus 1\\3"},{"Start":"03:21.810 ","End":"03:27.420","Text":"plus cosine Alpha 1\\3,"},{"Start":"03:27.420 ","End":"03:35.190","Text":"sine Beta root 8\\3."},{"Start":"03:35.190 ","End":"03:39.740","Text":"It looks like this is equal to this opposite signs."},{"Start":"03:39.740 ","End":"03:42.745","Text":"It seems to be 0."},{"Start":"03:42.745 ","End":"03:45.645","Text":"That\u0027s the first one."},{"Start":"03:45.645 ","End":"03:47.935","Text":"Now the other one,"},{"Start":"03:47.935 ","End":"03:52.880","Text":"cosine of Alpha minus Beta."},{"Start":"03:52.880 ","End":"03:56.200","Text":"Cosine, cosine plus sine, sine,"},{"Start":"03:56.200 ","End":"04:04.620","Text":"and the 2 cosines 1\\3 and minus 1\\3 plus sine,"},{"Start":"04:04.620 ","End":"04:07.320","Text":"sine root 8,"},{"Start":"04:07.320 ","End":"04:11.680","Text":"it over 3 root 8 over 3."},{"Start":"04:11.680 ","End":"04:14.000","Text":"That is equal to,"},{"Start":"04:14.000 ","End":"04:19.610","Text":"it\u0027s all going to be over 9 minus 1 plus 8."},{"Start":"04:19.610 ","End":"04:22.430","Text":"That\u0027s 7\\9, the 8 is root 8,"},{"Start":"04:22.430 ","End":"04:25.740","Text":"and root 8 is 8 minus 1 is 7."},{"Start":"04:25.750 ","End":"04:29.120","Text":"That\u0027s the second part, that\u0027s the cosine."},{"Start":"04:29.120 ","End":"04:31.285","Text":"The first one was sine of this."},{"Start":"04:31.285 ","End":"04:34.430","Text":"That\u0027s it, we\u0027re done."}],"ID":14341},{"Watched":false,"Name":"Exercise 10 - Part a","Duration":"2m 4s","ChapterTopicVideoID":13623,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13623.jpeg","UploadDate":"2018-10-21T13:52:20.9170000","DurationForVideoObject":"PT2M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:06.180","Text":"In this exercise, we have to find the value of sine 2 Theta,"},{"Start":"00:06.180 ","End":"00:11.130","Text":"but we\u0027re given that sine of Theta is 3/5,"},{"Start":"00:11.130 ","End":"00:13.830","Text":"and that Theta is in this range,"},{"Start":"00:13.830 ","End":"00:15.360","Text":"which if you think about it,"},{"Start":"00:15.360 ","End":"00:22.260","Text":"just means that Theta is in quadrant 1 between 0 and 90 degrees."},{"Start":"00:22.260 ","End":"00:25.785","Text":"You\u0027ll see why we need this in a moment."},{"Start":"00:25.785 ","End":"00:32.880","Text":"Now, here I have the formulas for the double angle where working with sine,"},{"Start":"00:32.880 ","End":"00:35.680","Text":"we need this formula."},{"Start":"00:37.190 ","End":"00:40.605","Text":"We\u0027re going to replace Alpha with Theta,"},{"Start":"00:40.605 ","End":"00:45.255","Text":"we have the sine, we don\u0027t have the cosine."},{"Start":"00:45.255 ","End":"00:51.205","Text":"The first thing we\u0027ll do is find the cosine Theta."},{"Start":"00:51.205 ","End":"01:00.485","Text":"Now, normally, it\u0027s plus or minus the square root of 1 minus sine squared Theta."},{"Start":"01:00.485 ","End":"01:04.640","Text":"But we don\u0027t need the plus or minus because we\u0027re in the first quadrant,"},{"Start":"01:04.640 ","End":"01:08.195","Text":"so I\u0027ll just write a plus to emphasize."},{"Start":"01:08.195 ","End":"01:11.909","Text":"I didn\u0027t forget about the sine."},{"Start":"01:12.130 ","End":"01:14.360","Text":"Now we substitute."},{"Start":"01:14.360 ","End":"01:17.645","Text":"Sine of Theta is 3/5,"},{"Start":"01:17.645 ","End":"01:27.490","Text":"so that is equal to the square root of 1 minus 9/25,"},{"Start":"01:27.490 ","End":"01:30.190","Text":"which is root 16 over 25,"},{"Start":"01:30.190 ","End":"01:33.520","Text":"which comes out to be 4/5."},{"Start":"01:33.520 ","End":"01:36.580","Text":"Now I have the sine and I have the cosine,"},{"Start":"01:36.580 ","End":"01:40.090","Text":"so I know that sine of 2 Theta,"},{"Start":"01:40.090 ","End":"01:46.260","Text":"which is 2 sine Theta cosine Theta is equal to"},{"Start":"01:46.260 ","End":"01:52.410","Text":"2 times 3/5 times 4/5,"},{"Start":"01:52.410 ","End":"02:00.570","Text":"which is equal to 2 times 3 times 4 is 24/25."},{"Start":"02:00.570 ","End":"02:04.420","Text":"That\u0027s the answer. We\u0027re done."}],"ID":14342},{"Watched":false,"Name":"Exercise 10 - Part b","Duration":"2m 13s","ChapterTopicVideoID":13624,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13624.jpeg","UploadDate":"2018-10-21T13:52:44.7930000","DurationForVideoObject":"PT2M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.025","Text":"In this exercise, we have to find the value of cosine 2 theta."},{"Start":"00:05.025 ","End":"00:09.375","Text":"What we\u0027re given is that cosine theta is 3/5."},{"Start":"00:09.375 ","End":"00:13.860","Text":"We\u0027re also given a range of possible values for theta."},{"Start":"00:13.860 ","End":"00:18.030","Text":"If you think about it just means that theta is in"},{"Start":"00:18.030 ","End":"00:25.305","Text":"quadrant number 1, first quadrant, 0-90."},{"Start":"00:25.305 ","End":"00:30.120","Text":"Now, we have the formulas for the double angle."},{"Start":"00:30.120 ","End":"00:31.710","Text":"We need the cosine of 2 theta."},{"Start":"00:31.710 ","End":"00:38.025","Text":"We\u0027ll use this formula only with a theta instead of alpha."},{"Start":"00:38.025 ","End":"00:43.460","Text":"Now, there\u0027s 3 possible formulas depending on what you\u0027re given."},{"Start":"00:43.460 ","End":"00:48.110","Text":"Usually, we use this or this because were either given the cosine or sine."},{"Start":"00:48.110 ","End":"00:50.870","Text":"In this case, we\u0027re given the cosine."},{"Start":"00:50.870 ","End":"00:54.830","Text":"We shall use this formula,"},{"Start":"00:54.830 ","End":"01:00.570","Text":"gives you the cosine of 2 theta in terms of the cosine of theta alpha."},{"Start":"01:01.750 ","End":"01:06.660","Text":"We have that cosine of"},{"Start":"01:08.210 ","End":"01:18.250","Text":"2 theta is equal to 2 cosine squared theta minus 1."},{"Start":"01:18.380 ","End":"01:26.850","Text":"This is equal to 2X (3/5)^2-1."},{"Start":"01:26.850 ","End":"01:30.300","Text":"That is equal to, let\u0027s see,"},{"Start":"01:30.300 ","End":"01:36.820","Text":"3^2 is 9 times 2 is 18 over 25-1."},{"Start":"01:40.160 ","End":"01:47.870","Text":"That is -7/25. As it happens,"},{"Start":"01:47.870 ","End":"01:51.210","Text":"we didn\u0027t need to use this."},{"Start":"01:52.000 ","End":"01:54.320","Text":"Very well. It was given to us."},{"Start":"01:54.320 ","End":"01:55.745","Text":"We just didn\u0027t use it."},{"Start":"01:55.745 ","End":"02:02.495","Text":"If we wanted to find sine theta and use 1 of these 2 formulas,"},{"Start":"02:02.495 ","End":"02:04.550","Text":"maybe would\u0027ve used it,"},{"Start":"02:04.550 ","End":"02:06.605","Text":"I guess even then not because it\u0027s squared."},{"Start":"02:06.605 ","End":"02:09.620","Text":"It was just redundant information."},{"Start":"02:09.620 ","End":"02:13.470","Text":"This is the answer. We\u0027re done."}],"ID":14343},{"Watched":false,"Name":"Exercise 10 - Part c","Duration":"3m 23s","ChapterTopicVideoID":13625,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13625.jpeg","UploadDate":"2018-10-21T13:53:22.1430000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.045","Text":"In this exercise, we have to compute the value of cosine 2 Theta,"},{"Start":"00:06.045 ","End":"00:13.830","Text":"and we\u0027re given tangent of Theta and we\u0027re also given that Theta is in this range,"},{"Start":"00:13.830 ","End":"00:21.075","Text":"which means that it is in quadrant number 3,"},{"Start":"00:21.075 ","End":"00:26.010","Text":"between 180 and 270."},{"Start":"00:26.010 ","End":"00:29.220","Text":"Now I have double angle formulas."},{"Start":"00:29.220 ","End":"00:33.840","Text":"We need the cosine of a double angle."},{"Start":"00:33.840 ","End":"00:36.885","Text":"The thing is that we have the tangent,"},{"Start":"00:36.885 ","End":"00:39.975","Text":"we don\u0027t have the cosine or the sine."},{"Start":"00:39.975 ","End":"00:42.015","Text":"I\u0027ll tell you what I\u0027m going to do."},{"Start":"00:42.015 ","End":"00:45.125","Text":"I\u0027m going to use this formula,"},{"Start":"00:45.125 ","End":"00:51.725","Text":"but I\u0027m going to find the cosine using other trigonometric identities."},{"Start":"00:51.725 ","End":"01:01.200","Text":"Another trigonometric identity is that tangent^2,"},{"Start":"01:01.200 ","End":"01:08.240","Text":"let\u0027s say Alpha plus 1 is equal to secant^2 Alpha."},{"Start":"01:08.240 ","End":"01:10.160","Text":"Now, how\u0027s this going to help us?"},{"Start":"01:10.160 ","End":"01:11.825","Text":"If I have tangent Theta,"},{"Start":"01:11.825 ","End":"01:13.835","Text":"I\u0027ll find secant Theta."},{"Start":"01:13.835 ","End":"01:16.970","Text":"If I have secant Theta for writing,"},{"Start":"01:16.970 ","End":"01:18.770","Text":"even though, of course you know this,"},{"Start":"01:18.770 ","End":"01:26.980","Text":"the cosine Theta is 1 over secant Theta or Alpha and vice versa,"},{"Start":"01:26.980 ","End":"01:28.390","Text":"the reciprocals of each other."},{"Start":"01:28.390 ","End":"01:34.420","Text":"I\u0027m going to go a circuitous route to get from tangent to secant,"},{"Start":"01:34.420 ","End":"01:37.929","Text":"to cosine and finally to this formula."},{"Start":"01:37.929 ","End":"01:40.340","Text":"That\u0027s the plan."},{"Start":"01:40.350 ","End":"01:43.235","Text":"We have that,"},{"Start":"01:43.235 ","End":"01:51.535","Text":"secant^2 Theta is equal to tangent^2."},{"Start":"01:51.535 ","End":"01:57.440","Text":"That\u0027s 4/3^2 plus 1."},{"Start":"01:57.440 ","End":"02:00.530","Text":"I\u0027m talking about this formula now."},{"Start":"02:00.660 ","End":"02:05.920","Text":"That is equal to 16 over 9 plus"},{"Start":"02:05.920 ","End":"02:12.780","Text":"1 is 25 over 9."},{"Start":"02:12.780 ","End":"02:18.315","Text":"Now that means that secant Theta"},{"Start":"02:18.315 ","End":"02:25.395","Text":"is plus or minus the square root of this,"},{"Start":"02:25.395 ","End":"02:29.145","Text":"which is 5 over 3."},{"Start":"02:29.145 ","End":"02:31.140","Text":"Now, which is it, plus or minus?"},{"Start":"02:31.140 ","End":"02:37.460","Text":"We are in the third quadrant and in the third quadrant, cosine is negative."},{"Start":"02:37.460 ","End":"02:40.490","Text":"Cosine is negative and secant is negative,"},{"Start":"02:40.490 ","End":"02:45.405","Text":"so it is minus 5/3,"},{"Start":"02:45.405 ","End":"02:47.230","Text":"and if that\u0027s the secant,"},{"Start":"02:47.230 ","End":"02:49.970","Text":"then the cosine is the reciprocal,"},{"Start":"02:49.970 ","End":"02:53.465","Text":"is going to be minus 3/5."},{"Start":"02:53.465 ","End":"02:55.850","Text":"Now I have the cosine."},{"Start":"02:55.850 ","End":"03:02.735","Text":"Now I can use this formula and say that cosine 2 Theta is 2 cosine^2."},{"Start":"03:02.735 ","End":"03:09.705","Text":"Cosine^2 is 9 over 25 for the plus, minus 1,"},{"Start":"03:09.705 ","End":"03:14.265","Text":"which is 18 over 25 minus 1,"},{"Start":"03:14.265 ","End":"03:19.395","Text":"which is minus 7 over 25,"},{"Start":"03:19.395 ","End":"03:23.440","Text":"and that\u0027s the answer. We\u0027re done."}],"ID":14344},{"Watched":false,"Name":"Exercise 10 - Part d","Duration":"2m 36s","ChapterTopicVideoID":13626,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13626.jpeg","UploadDate":"2018-10-21T13:53:51.5270000","DurationForVideoObject":"PT2M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, we have to find sine(2)Theta."},{"Start":"00:04.470 ","End":"00:09.375","Text":"What we\u0027re given is secant of Theta minus 3,"},{"Start":"00:09.375 ","End":"00:12.330","Text":"and we\u0027re also given that Theta is in this range now,"},{"Start":"00:12.330 ","End":"00:16.095","Text":"which quadrant is that between 90 and 180 degrees?"},{"Start":"00:16.095 ","End":"00:25.020","Text":"This is quadrant number 2 and let\u0027s see,"},{"Start":"00:25.020 ","End":"00:27.120","Text":"we\u0027re given the double angle formulas."},{"Start":"00:27.120 ","End":"00:32.670","Text":"Sine (2) Theta is this 1,"},{"Start":"00:32.670 ","End":"00:39.790","Text":"so we need sine of Theta and cosine of Theta."},{"Start":"00:40.220 ","End":"00:43.910","Text":"What I suggest is we have the secant,"},{"Start":"00:43.910 ","End":"00:45.575","Text":"the reciprocal is the cosine,"},{"Start":"00:45.575 ","End":"00:47.300","Text":"and then from the cosine, we\u0027ll find the sine,"},{"Start":"00:47.300 ","End":"00:49.560","Text":"and then we\u0027ll use this formula."},{"Start":"00:50.210 ","End":"00:57.480","Text":"Cosine of Theta is 1 over secant Theta,"},{"Start":"00:57.480 ","End":"01:00.405","Text":"so that will be minus 1/3,"},{"Start":"01:00.405 ","End":"01:06.529","Text":"and then sine of Theta is generally"},{"Start":"01:06.529 ","End":"01:12.575","Text":"plus or minus the square root of 1 minus cosine squared Theta."},{"Start":"01:12.575 ","End":"01:15.185","Text":"But we are in quadrant 2."},{"Start":"01:15.185 ","End":"01:16.370","Text":"In quadrant 2,"},{"Start":"01:16.370 ","End":"01:18.350","Text":"the sign is positive,"},{"Start":"01:18.350 ","End":"01:23.790","Text":"so I\u0027m going to take the plus here, so this is equal."},{"Start":"01:23.790 ","End":"01:27.205","Text":"Now, cosine Theta is minus 1/3."},{"Start":"01:27.205 ","End":"01:34.555","Text":"Here we get the square root of 1 minus 1/9,"},{"Start":"01:34.555 ","End":"01:41.070","Text":"and that is equal to root 8 over root 9,"},{"Start":"01:41.070 ","End":"01:43.500","Text":"root 8 over 3."},{"Start":"01:43.500 ","End":"01:46.155","Text":"Now we have cosine Theta,"},{"Start":"01:46.155 ","End":"01:48.760","Text":"and we have sine Theta."},{"Start":"01:48.760 ","End":"01:54.480","Text":"We can use the formula sine(2)Theta is"},{"Start":"01:54.480 ","End":"02:00.600","Text":"2 sine Theta cosine Theta which is equal to 2 sine"},{"Start":"02:00.600 ","End":"02:07.605","Text":"Theta root 8 over 3 cosine Theta minus 1/3,"},{"Start":"02:07.605 ","End":"02:12.255","Text":"and so this is equal to, let\u0027s see,"},{"Start":"02:12.255 ","End":"02:19.555","Text":"minus 2 root 8 over 9."},{"Start":"02:19.555 ","End":"02:22.625","Text":"That\u0027s the answer for those who like to simplify,"},{"Start":"02:22.625 ","End":"02:25.550","Text":"root 8 is root 2 root 4 anyway,"},{"Start":"02:25.550 ","End":"02:29.640","Text":"it comes out to minus 4 root 2 over 9."},{"Start":"02:29.640 ","End":"02:33.285","Text":"But this is good enough,"},{"Start":"02:33.285 ","End":"02:35.980","Text":"that\u0027s it we\u0027re done."}],"ID":14345},{"Watched":false,"Name":"Exercise 11","Duration":"4m 28s","ChapterTopicVideoID":13627,"CourseChapterTopicPlaylistID":257204,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13627.jpeg","UploadDate":"2018-10-21T13:54:38.2830000","DurationForVideoObject":"PT4M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this exercise, we\u0027re given some information about an angle Theta,"},{"Start":"00:05.220 ","End":"00:09.150","Text":"we\u0027re given that sine of 2 Theta is 0.8,"},{"Start":"00:09.150 ","End":"00:13.695","Text":"and that 2 Theta is in this range,"},{"Start":"00:13.695 ","End":"00:20.910","Text":"which means that it\u0027s in quadrant number 2,"},{"Start":"00:20.910 ","End":"00:24.960","Text":"because it\u0027s between 90 and 180."},{"Start":"00:24.960 ","End":"00:28.320","Text":"Our task is to find the cosecant of Theta,"},{"Start":"00:28.320 ","End":"00:31.650","Text":"the question is which formulas are we going to use."},{"Start":"00:31.650 ","End":"00:35.495","Text":"Well, if I have the sine Theta,"},{"Start":"00:35.495 ","End":"00:40.410","Text":"then I can take the reciprocal and find cosecant,"},{"Start":"00:40.410 ","End":"00:43.780","Text":"so how do I find sine Theta?"},{"Start":"00:44.870 ","End":"00:48.450","Text":"I\u0027ve brought in some formulas,"},{"Start":"00:48.450 ","End":"00:58.485","Text":"they\u0027re also called half-angle identities because we find the cosine or sine of x,"},{"Start":"00:58.485 ","End":"01:01.660","Text":"if we have the cosine of 2x."},{"Start":"01:01.670 ","End":"01:04.040","Text":"Theta is going to replace x,"},{"Start":"01:04.040 ","End":"01:06.050","Text":"but we don\u0027t have cosine 2x,"},{"Start":"01:06.050 ","End":"01:07.675","Text":"we have sine 2x,"},{"Start":"01:07.675 ","End":"01:09.120","Text":"that\u0027s no problem,"},{"Start":"01:09.120 ","End":"01:11.450","Text":"when we have the cosine, we can find the sine,"},{"Start":"01:11.450 ","End":"01:15.080","Text":"but the strategy is first to find cosine 2 Theta,"},{"Start":"01:15.080 ","End":"01:21.590","Text":"so cosine 2 Theta is plus or minus the square root"},{"Start":"01:21.590 ","End":"01:29.230","Text":"of 1 minus sine^2 of the same thing, 2 Theta."},{"Start":"01:29.450 ","End":"01:31.790","Text":"We have sine 2 Theta,"},{"Start":"01:31.790 ","End":"01:34.010","Text":"but I\u0027ll be going to take plus or minus,"},{"Start":"01:34.010 ","End":"01:36.290","Text":"well, we\u0027re in the second quadrant."},{"Start":"01:36.290 ","End":"01:38.660","Text":"In the second quadrant,"},{"Start":"01:38.660 ","End":"01:41.290","Text":"the cosine is negative,"},{"Start":"01:41.290 ","End":"01:45.435","Text":"so we\u0027re going to take the minus,"},{"Start":"01:45.435 ","End":"01:55.345","Text":"and this is equal to minus the square root of 1 minus 0.8^2,"},{"Start":"01:55.345 ","End":"02:02.205","Text":"which is 0.8^2 is 0.64 from 1 is 0.36,"},{"Start":"02:02.205 ","End":"02:07.950","Text":"square root is 0.6, so minus 0.6."},{"Start":"02:07.950 ","End":"02:12.300","Text":"That is cosine of 2 Theta, I\u0027ll write it again."},{"Start":"02:12.300 ","End":"02:15.990","Text":"Cosine 2 Theta is this,"},{"Start":"02:15.990 ","End":"02:22.520","Text":"and now all we have to do is use this formula and say that sine^2"},{"Start":"02:22.520 ","End":"02:32.145","Text":"Theta is 1/2 of 1 minus,"},{"Start":"02:32.145 ","End":"02:39.390","Text":"minus 0.6 is 1 plus 0.6,"},{"Start":"02:39.390 ","End":"02:48.435","Text":"and that is equal to 1.6 over 2 is 0.8,"},{"Start":"02:48.435 ","End":"02:56.505","Text":"so sine Theta will be plus or minus the square root of 0.8."},{"Start":"02:56.505 ","End":"02:58.250","Text":"Again, we have to ask,"},{"Start":"02:58.250 ","End":"03:00.650","Text":"do we take the plus or the minus?"},{"Start":"03:00.650 ","End":"03:03.560","Text":"Well, we also have a range for Theta."},{"Start":"03:03.560 ","End":"03:05.485","Text":"We divide this by 2,"},{"Start":"03:05.485 ","End":"03:09.840","Text":"then Theta is between Pi over 2."},{"Start":"03:09.840 ","End":"03:11.610","Text":"Well, other way round."},{"Start":"03:11.610 ","End":"03:13.935","Text":"Between Pi over 4 and Pi over 2,"},{"Start":"03:13.935 ","End":"03:18.920","Text":"and that is between 45 and 90 degrees."},{"Start":"03:18.920 ","End":"03:25.070","Text":"In any event that this gives us that this is quarter a quadrant, quadrant number 1."},{"Start":"03:25.070 ","End":"03:26.450","Text":"In quadrant number 1,"},{"Start":"03:26.450 ","End":"03:27.910","Text":"everything is positive,"},{"Start":"03:27.910 ","End":"03:31.440","Text":"so we can get rid of this minus,"},{"Start":"03:31.440 ","End":"03:35.280","Text":"and minus will get rid of the plus 2."},{"Start":"03:35.280 ","End":"03:38.645","Text":"That\u0027s the answer, I\u0027m not going to evaluate it numerically with,"},{"Start":"03:38.645 ","End":"03:42.185","Text":"just leave it as a square root of 0.8."},{"Start":"03:42.185 ","End":"03:44.835","Text":"Oh sorry, I haven\u0027t finished,"},{"Start":"03:44.835 ","End":"03:48.450","Text":"oops, we want the cosecant."},{"Start":"03:48.450 ","End":"03:58.200","Text":"The cosecant of Theta will just be 1 over the square root of 0.8."},{"Start":"03:58.200 ","End":"04:01.425","Text":"We can simplify this,"},{"Start":"04:01.425 ","End":"04:02.475","Text":"you could leave it like that."},{"Start":"04:02.475 ","End":"04:05.070","Text":"0.8 is 4/5,"},{"Start":"04:05.070 ","End":"04:10.800","Text":"so I can say that this is if it\u0027s 1 over the square root of 4/5."},{"Start":"04:10.800 ","End":"04:14.325","Text":"Square root of 4/5 is root 2 over 5,"},{"Start":"04:14.325 ","End":"04:20.260","Text":"so it\u0027s root 5 over 2,"},{"Start":"04:20.390 ","End":"04:24.405","Text":"but this is fine anyway,"},{"Start":"04:24.405 ","End":"04:28.900","Text":"so we are done."}],"ID":14346}],"Thumbnail":null,"ID":257204},{"Name":"Inverse Trigonometric Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Inverse Trigonometric Functions - Part 1","Duration":"8m 15s","ChapterTopicVideoID":10475,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10475.jpeg","UploadDate":"2021-06-29T12:52:20.6800000","DurationForVideoObject":"PT8M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.654","Text":"Starting a new topic,"},{"Start":"00:02.654 ","End":"00:08.020","Text":"the Inverse Trigonometric Functions and Their Graphs."},{"Start":"00:08.510 ","End":"00:14.476","Text":"The main thing here is something called the"},{"Start":"00:14.476 ","End":"00:20.910","Text":"restricted domain or principal branch for each of the trigonometric functions."},{"Start":"00:20.910 ","End":"00:23.370","Text":"Rather than explain it in general,"},{"Start":"00:23.370 ","End":"00:25.620","Text":"I\u0027ll just go over each of the functions."},{"Start":"00:25.620 ","End":"00:28.545","Text":"Mostly we\u0027ll be concerned with the sine,"},{"Start":"00:28.545 ","End":"00:31.290","Text":"the cosine, and the tangent."},{"Start":"00:31.290 ","End":"00:34.245","Text":"Less important will be the other three."},{"Start":"00:34.245 ","End":"00:37.065","Text":"We\u0027ll just mention those at the end briefly."},{"Start":"00:37.065 ","End":"00:39.225","Text":"Let\u0027s start with the sine."},{"Start":"00:39.225 ","End":"00:44.795","Text":"Here\u0027s part of the graph of y = sin(x)."},{"Start":"00:44.795 ","End":"00:49.790","Text":"Now, the reason we can\u0027t talk about an inverse sine function,"},{"Start":"00:49.790 ","End":"00:52.415","Text":"inverse in the sense of inverse function,"},{"Start":"00:52.415 ","End":"00:55.760","Text":"is because this function is not 1 to 1."},{"Start":"00:55.760 ","End":"00:58.040","Text":"I mean, for example,"},{"Start":"00:58.040 ","End":"01:01.820","Text":"the value 0 is attained when x is 0,"},{"Start":"01:01.820 ","End":"01:03.620","Text":"when x is Pi, when x is 2 Pi,"},{"Start":"01:03.620 ","End":"01:05.420","Text":"when x is minus Pi."},{"Start":"01:05.420 ","End":"01:09.860","Text":"Different values of x could give the same value of y."},{"Start":"01:09.860 ","End":"01:13.790","Text":"In order to have a chance of getting an inverse function,"},{"Start":"01:13.790 ","End":"01:19.010","Text":"we want to restrict the domain in such a way that it is 1 to 1."},{"Start":"01:19.010 ","End":"01:24.114","Text":"We\u0027d also like not to restrict the range."},{"Start":"01:24.114 ","End":"01:30.260","Text":"Here, the domain is all x and the range is from minus 1 to 1."},{"Start":"01:30.260 ","End":"01:32.750","Text":"I want to keep the minus 1 to 1."},{"Start":"01:32.750 ","End":"01:38.663","Text":"One way of doing it would be to restrict the domain,"},{"Start":"01:38.663 ","End":"01:44.100","Text":"so it\u0027s just from minus Pi over 2 to Pi over 2."},{"Start":"01:44.100 ","End":"01:54.330","Text":"Then this part of the graph is from minus 1 to 1."},{"Start":"01:54.330 ","End":"01:56.660","Text":"The range is now the same,"},{"Start":"01:56.660 ","End":"02:01.610","Text":"but the domain is being clipped to just minus Pi over 2 to Pi over 2."},{"Start":"02:01.610 ","End":"02:05.280","Text":"In this part, we can take an inverse."},{"Start":"02:05.350 ","End":"02:12.230","Text":"This is the graph of the inverse just gotten by flipping x and y."},{"Start":"02:12.230 ","End":"02:15.140","Text":"This function, which is the inverse,"},{"Start":"02:15.140 ","End":"02:19.370","Text":"is sometimes written as sin(x)^-1."},{"Start":"02:20.730 ","End":"02:28.380","Text":"Sometimes we write it as y = arcsine(x)."},{"Start":"02:28.380 ","End":"02:32.930","Text":"This will be so with all the other trigonometric functions, the notation,"},{"Start":"02:32.930 ","End":"02:39.270","Text":"they\u0027ll either be a minus 1 here to indicate inverse or the word or prefix arc."},{"Start":"02:39.860 ","End":"02:45.520","Text":"This is called a principal branch or,"},{"Start":"02:45.520 ","End":"02:50.910","Text":"if you like, the restricted domain sine, either way."},{"Start":"02:51.020 ","End":"02:54.080","Text":"This is the inverse of this,"},{"Start":"02:54.080 ","End":"02:56.870","Text":"which means that the domain and range are reversed."},{"Start":"02:56.870 ","End":"02:58.535","Text":"Let\u0027s just write those down."},{"Start":"02:58.535 ","End":"03:05.015","Text":"Here the domain for the sine is that"},{"Start":"03:05.015 ","End":"03:11.990","Text":"x is between minus Pi over 2 and Pi over 2 inclusive,"},{"Start":"03:11.990 ","End":"03:21.210","Text":"and the range is from minus 1 to 1."},{"Start":"03:21.210 ","End":"03:24.620","Text":"In the inverse function, it\u0027s just reversed."},{"Start":"03:24.620 ","End":"03:29.950","Text":"I\u0027ll just write D for domain would be from minus 1 to 1."},{"Start":"03:29.950 ","End":"03:33.255","Text":"But, of course, now it\u0027s x not y."},{"Start":"03:33.255 ","End":"03:38.130","Text":"The range would be from"},{"Start":"03:38.130 ","End":"03:44.475","Text":"minus Pi over 2 to Pi over 2."},{"Start":"03:44.475 ","End":"03:46.275","Text":"We just flip them."},{"Start":"03:46.275 ","End":"03:48.830","Text":"At the end, when we\u0027ve done all three of them,"},{"Start":"03:48.830 ","End":"03:52.730","Text":"I\u0027ll put all this information in a table."},{"Start":"03:52.730 ","End":"03:56.285","Text":"Now, let\u0027s move on to the cosine."},{"Start":"03:56.285 ","End":"03:59.810","Text":"This is the cosine."},{"Start":"03:59.810 ","End":"04:06.520","Text":"Once again, we have to restrict it in order to make it 1 to 1."},{"Start":"04:06.520 ","End":"04:10.340","Text":"Again, the range is going to be from minus 1 to 1."},{"Start":"04:10.340 ","End":"04:12.544","Text":"In fact, you can even write that already,"},{"Start":"04:12.544 ","End":"04:19.655","Text":"that the range of the cosine will be from minus 1 to 1 inclusive."},{"Start":"04:19.655 ","End":"04:22.775","Text":"But we have to restrict the domain."},{"Start":"04:22.775 ","End":"04:24.920","Text":"There\u0027s more than one way to do this,"},{"Start":"04:24.920 ","End":"04:27.890","Text":"but the usual is to go from here to here."},{"Start":"04:27.890 ","End":"04:33.300","Text":"I\u0027ll draw a rectangle again from here to here."},{"Start":"04:35.230 ","End":"04:42.855","Text":"That makes it from 0 to Pi."},{"Start":"04:42.855 ","End":"04:46.425","Text":"For extra emphasis, this is how it goes."},{"Start":"04:46.425 ","End":"04:48.355","Text":"It\u0027s decreasing."},{"Start":"04:48.355 ","End":"04:52.850","Text":"This is the principle branch, this bit here."},{"Start":"04:52.850 ","End":"04:55.445","Text":"Now, we can invert it."},{"Start":"04:55.445 ","End":"04:57.770","Text":"This is what we get,"},{"Start":"04:57.770 ","End":"05:07.860","Text":"y= cosine(x)^-1, also written as arccosine(x)."},{"Start":"05:08.090 ","End":"05:13.760","Text":"The domain and range are just reverse this time,"},{"Start":"05:13.760 ","End":"05:19.920","Text":"the domain would be like x between minus 1 and"},{"Start":"05:19.920 ","End":"05:28.415","Text":"1 and the range from 0 to Pi."},{"Start":"05:28.415 ","End":"05:30.850","Text":"Now that\u0027s two out of the three."},{"Start":"05:30.850 ","End":"05:33.760","Text":"We\u0027ve got the sine and we\u0027ve got the cosine."},{"Start":"05:33.760 ","End":"05:39.740","Text":"Next we want to go on to the tangent."},{"Start":"05:39.740 ","End":"05:46.470","Text":"Here we are with y = tan(x) or part of it,"},{"Start":"05:46.470 ","End":"05:49.930","Text":"it goes on to Infinity both ways."},{"Start":"05:50.660 ","End":"05:56.455","Text":"This time, the range is from minus Infinity to Infinity."},{"Start":"05:56.455 ","End":"05:59.530","Text":"But in order to make it 1 to 1,"},{"Start":"05:59.530 ","End":"06:05.680","Text":"we pick the obvious choice which is this bit."},{"Start":"06:05.680 ","End":"06:12.920","Text":"Pick this time as the principal branch."},{"Start":"06:12.920 ","End":"06:16.520","Text":"As before, for extra emphasis,"},{"Start":"06:16.520 ","End":"06:19.115","Text":"I\u0027ll also highlight it."},{"Start":"06:19.115 ","End":"06:22.145","Text":"You really can\u0027t miss it."},{"Start":"06:22.145 ","End":"06:27.860","Text":"Then we can take the inverse of just this branch and we get this,"},{"Start":"06:27.860 ","End":"06:35.660","Text":"which is tan(x)^-1 or arctan(x)."},{"Start":"06:35.660 ","End":"06:38.120","Text":"You should know both these notations,"},{"Start":"06:38.120 ","End":"06:42.575","Text":"they\u0027re both very common and I\u0027ll sometimes use one and sometimes the other."},{"Start":"06:42.575 ","End":"06:48.500","Text":"As for the domain and range of this,"},{"Start":"06:48.500 ","End":"06:51.500","Text":"this is a good opportunity to,"},{"Start":"06:51.500 ","End":"06:54.000","Text":"bring in that table."},{"Start":"06:54.000 ","End":"06:59.245","Text":"Here, the restricted or principal branch of the tangent"},{"Start":"06:59.245 ","End":"07:04.835","Text":"has a domain from minus Pi over 2 to Pi over 2,"},{"Start":"07:04.835 ","End":"07:07.990","Text":"but not inclusive because that\u0027s where the asymptote is."},{"Start":"07:07.990 ","End":"07:11.845","Text":"This is the interval notation here."},{"Start":"07:11.845 ","End":"07:14.620","Text":"The square brackets mean inclusive of"},{"Start":"07:14.620 ","End":"07:19.595","Text":"the endpoint and the round bracket means it does not include the endpoint."},{"Start":"07:19.595 ","End":"07:23.410","Text":"The range here is all the numbers,"},{"Start":"07:23.410 ","End":"07:25.555","Text":"but sometimes right off the rails,"},{"Start":"07:25.555 ","End":"07:30.215","Text":"sometimes you would write minus Infinity to Infinity."},{"Start":"07:30.215 ","End":"07:33.115","Text":"Similarly, here when you invert it,"},{"Start":"07:33.115 ","End":"07:35.810","Text":"you just swap these two around."},{"Start":"07:35.810 ","End":"07:38.530","Text":"I could have said here,"},{"Start":"07:38.530 ","End":"07:39.980","Text":"x between this and this,"},{"Start":"07:39.980 ","End":"07:41.210","Text":"y between this and this,"},{"Start":"07:41.210 ","End":"07:47.615","Text":"or use the interval notation R or if you prefer minus infinity to infinity."},{"Start":"07:47.615 ","End":"07:49.130","Text":"Similarly, like before,"},{"Start":"07:49.130 ","End":"07:53.820","Text":"I wrote it with the inequality notation."},{"Start":"07:53.820 ","End":"07:57.120","Text":"This one I wrote it as minus Pi"},{"Start":"07:57.120 ","End":"08:00.470","Text":"over 2 less than or equal to x less than or equal to Pi over 2."},{"Start":"08:00.470 ","End":"08:03.695","Text":"This table just brings it in interval notation."},{"Start":"08:03.695 ","End":"08:06.664","Text":"This is a summary of the sine and its inverse,"},{"Start":"08:06.664 ","End":"08:08.555","Text":"the cosine and its inverse,"},{"Start":"08:08.555 ","End":"08:11.825","Text":"the tangent and its inverse."},{"Start":"08:11.825 ","End":"08:14.910","Text":"We\u0027ll take a break now."}],"ID":10895},{"Watched":false,"Name":"Inverse Trigonometric Functions - Part 2","Duration":"7m 31s","ChapterTopicVideoID":10476,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10476.jpeg","UploadDate":"2021-06-29T12:52:50.1770000","DurationForVideoObject":"PT7M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:04.830","Text":"We\u0027re continuing with the inverse trig functions"},{"Start":"00:04.830 ","End":"00:09.195","Text":"and the principal branch related to the restricted domain."},{"Start":"00:09.195 ","End":"00:10.640","Text":"We\u0027ve covered the cosine,"},{"Start":"00:10.640 ","End":"00:12.030","Text":"sine, and tangent,"},{"Start":"00:12.030 ","End":"00:15.930","Text":"what we\u0027re missing are the cosecant,"},{"Start":"00:15.930 ","End":"00:20.055","Text":"the secant, and the cotangent."},{"Start":"00:20.055 ","End":"00:21.680","Text":"There are 2 differences."},{"Start":"00:21.680 ","End":"00:24.260","Text":"First of all, these are less important,"},{"Start":"00:24.260 ","End":"00:27.085","Text":"and secondly, there is some controversy."},{"Start":"00:27.085 ","End":"00:30.750","Text":"Not everyone universally agrees on what is the principal branch,"},{"Start":"00:30.750 ","End":"00:32.250","Text":"well, you\u0027ll see that."},{"Start":"00:32.250 ","End":"00:35.010","Text":"Let\u0027s start with the cosecant,"},{"Start":"00:35.010 ","End":"00:39.885","Text":"is part of the graph of the cosecant,"},{"Start":"00:39.885 ","End":"00:44.190","Text":"and the range,"},{"Start":"00:44.190 ","End":"00:48.030","Text":"you can see is not all the numbers,"},{"Start":"00:48.030 ","End":"00:49.730","Text":"this we talked about it,"},{"Start":"00:49.730 ","End":"00:51.770","Text":"the minimum point is 1."},{"Start":"00:51.770 ","End":"00:57.125","Text":"It could have values from 1 upwards or from minus 1 downwards."},{"Start":"00:57.125 ","End":"00:59.405","Text":"If I do that in interval notation,"},{"Start":"00:59.405 ","End":"01:05.010","Text":"the range is from minus infinity to minus 1 inclusive,"},{"Start":"01:05.010 ","End":"01:12.830","Text":"and like or union from 1 to infinity."},{"Start":"01:12.830 ","End":"01:16.579","Text":"Now we want to restrict the domain but keep the range."},{"Start":"01:16.579 ","End":"01:19.025","Text":"There\u0027s more than 1 way to do this,"},{"Start":"01:19.025 ","End":"01:22.850","Text":"I\u0027m going to choose the principle branch from here,"},{"Start":"01:22.850 ","End":"01:27.920","Text":"and then going all the way down to minus infinity,"},{"Start":"01:27.920 ","End":"01:36.720","Text":"and coming up from infinity all the way to this point here."},{"Start":"01:37.570 ","End":"01:43.190","Text":"That makes the domain as"},{"Start":"01:43.190 ","End":"01:50.690","Text":"follows that we have from here to here with a missing 0.0,"},{"Start":"01:50.690 ","End":"01:52.295","Text":"or we can just say in 2 bits,"},{"Start":"01:52.295 ","End":"01:55.830","Text":"it\u0027s from minus Pi over 2,"},{"Start":"01:55.830 ","End":"01:57.930","Text":"including minus Pi over 2,"},{"Start":"01:57.930 ","End":"02:02.235","Text":"up to 0, not including union,"},{"Start":"02:02.235 ","End":"02:08.450","Text":"and then from 0 excluding 0 up to Pi over 2."},{"Start":"02:08.450 ","End":"02:11.435","Text":"As I said this is not standard."},{"Start":"02:11.435 ","End":"02:15.530","Text":"The main other convention is to keep this part,"},{"Start":"02:15.530 ","End":"02:17.465","Text":"but instead of this,"},{"Start":"02:17.465 ","End":"02:21.215","Text":"to take this instead,"},{"Start":"02:21.215 ","End":"02:25.640","Text":"the left-hand part, the right-hand path."},{"Start":"02:25.640 ","End":"02:27.530","Text":"Anyway, we\u0027ll stick with this."},{"Start":"02:27.530 ","End":"02:31.710","Text":"But in case you see it otherwise, don\u0027t be alarmed."},{"Start":"02:31.870 ","End":"02:34.040","Text":"I forgot to label it."},{"Start":"02:34.040 ","End":"02:37.130","Text":"This is the cosX,"},{"Start":"02:37.130 ","End":"02:43.565","Text":"and if we take the secant inverse,"},{"Start":"02:43.565 ","End":"02:48.200","Text":"then its range and domain will be reversed."},{"Start":"02:48.200 ","End":"02:49.880","Text":"I\u0027ll give a table at the end,"},{"Start":"02:49.880 ","End":"02:57.979","Text":"so I don\u0027t want to rewrite it with just the opposite secant inverse or the arc secant."},{"Start":"02:57.979 ","End":"03:01.375","Text":"Get the idea you either put a minus 1 here or you put"},{"Start":"03:01.375 ","End":"03:05.465","Text":"the prefix arc in front to make it inverse."},{"Start":"03:05.465 ","End":"03:08.240","Text":"But let\u0027s put the graph in."},{"Start":"03:08.240 ","End":"03:18.284","Text":"Here it is, That\u0027s the graph Y=sec-1X,"},{"Start":"03:18.284 ","End":"03:21.410","Text":"and that\u0027s the inverse of the part I highlighted in yellow,"},{"Start":"03:21.410 ","End":"03:23.800","Text":"not this, that was just an alternative."},{"Start":"03:23.800 ","End":"03:25.980","Text":"Let\u0027s go on to the next,"},{"Start":"03:25.980 ","End":"03:29.760","Text":"the next will be the secant."},{"Start":"03:29.760 ","End":"03:35.460","Text":"Here\u0027s the graph Y=secX."},{"Start":"03:35.460 ","End":"03:44.880","Text":"In this case, the principal branch that we\u0027ll choose will be this bit,"},{"Start":"03:47.090 ","End":"03:52.310","Text":"and this bit up to here,"},{"Start":"03:52.310 ","End":"03:57.320","Text":"and the alternative, which we won\u0027t use,"},{"Start":"03:57.320 ","End":"04:01.775","Text":"but is mentioned in other places would be this."},{"Start":"04:01.775 ","End":"04:05.930","Text":"You can see the ranges from 1 to infinity,"},{"Start":"04:05.930 ","End":"04:09.545","Text":"and from minus infinity to 1, just like before."},{"Start":"04:09.545 ","End":"04:11.330","Text":"The domain is a bit different,"},{"Start":"04:11.330 ","End":"04:14.330","Text":"it\u0027s from 0 to Pi over 2,"},{"Start":"04:14.330 ","End":"04:16.655","Text":"and from Pi over 2 to Pi."},{"Start":"04:16.655 ","End":"04:20.480","Text":"I\u0027ll leave that for the summary table."},{"Start":"04:20.480 ","End":"04:25.415","Text":"Anyway, I\u0027ll give you the picture of what happens when we invert the yellow bit,"},{"Start":"04:25.415 ","End":"04:27.160","Text":"the highlighted bit,"},{"Start":"04:27.160 ","End":"04:33.710","Text":"so here\u0027s the graph Y=sec-1X,"},{"Start":"04:33.710 ","End":"04:39.020","Text":"or the arc secant and I\u0027m just rushing through these as I said,"},{"Start":"04:39.020 ","End":"04:47.195","Text":"the less important and as I also noted, they\u0027re not universal."},{"Start":"04:47.195 ","End":"04:54.920","Text":"The one that we\u0027re missing that now would be the inverse cotangent,"},{"Start":"04:54.920 ","End":"05:00.390","Text":"and here we are with the cotangent."},{"Start":"05:00.390 ","End":"05:04.515","Text":"First of all, Y=cotX,"},{"Start":"05:04.515 ","End":"05:08.705","Text":"and we need to choose a principal branch."},{"Start":"05:08.705 ","End":"05:11.825","Text":"Again, there\u0027s controversy."},{"Start":"05:11.825 ","End":"05:17.360","Text":"We need to keep the range from infinity to minus infinity,"},{"Start":"05:17.360 ","End":"05:18.535","Text":"I mean all the real numbers,"},{"Start":"05:18.535 ","End":"05:21.470","Text":"we don\u0027t want any repetition."},{"Start":"05:21.470 ","End":"05:29.340","Text":"To me, the simplest thing seems to be to take just this as the principal branch."},{"Start":"05:29.340 ","End":"05:33.705","Text":"But there are books,"},{"Start":"05:33.705 ","End":"05:40.860","Text":"professors who use this bit up to here,"},{"Start":"05:40.860 ","End":"05:43.335","Text":"and then not this bit,"},{"Start":"05:43.335 ","End":"05:52.700","Text":"but here, and excluding this because we\u0027ve already included it here."},{"Start":"05:52.700 ","End":"05:55.805","Text":"Either take this and this or all of this,"},{"Start":"05:55.805 ","End":"06:00.270","Text":"as I said, I\u0027m going with the one highlighted in yellow."},{"Start":"06:00.580 ","End":"06:06.050","Text":"That gives us the inverse of the cotangent function,"},{"Start":"06:06.050 ","End":"06:08.210","Text":"also the arc cotangent,"},{"Start":"06:08.210 ","End":"06:10.430","Text":"and as I promised,"},{"Start":"06:10.430 ","End":"06:16.195","Text":"I\u0027ll put all the results for domain and range for these 3 in a table."},{"Start":"06:16.195 ","End":"06:19.610","Text":"Now here\u0027s the table."},{"Start":"06:19.610 ","End":"06:22.180","Text":"But before I even go into it,"},{"Start":"06:22.180 ","End":"06:24.295","Text":"let me fix the discrepancy."},{"Start":"06:24.295 ","End":"06:29.800","Text":"This is one of those cases where they chose differently in the case of the cotangent,"},{"Start":"06:29.800 ","End":"06:32.120","Text":"so I\u0027m going to fix that."},{"Start":"06:32.120 ","End":"06:37.110","Text":"This should be from 0 to Pi,"},{"Start":"06:37.110 ","End":"06:40.800","Text":"non-inclusive and the same thing here."},{"Start":"06:40.800 ","End":"06:46.575","Text":"All these domains and ranges,"},{"Start":"06:46.575 ","End":"06:49.200","Text":"when we have the inverse,"},{"Start":"06:49.200 ","End":"06:54.730","Text":"we just flip the domain and range like the cosecant domain and range is this."},{"Start":"06:54.730 ","End":"06:58.680","Text":"This is the same as this and this is the same as this for the inverse function."},{"Start":"06:58.680 ","End":"07:00.530","Text":"Similarly, this and this are the same,"},{"Start":"07:00.530 ","End":"07:03.700","Text":"this and this are the same, and here also."},{"Start":"07:03.700 ","End":"07:09.980","Text":"it\u0027s one of the properties of inverse functions that it flips domain and range."},{"Start":"07:09.980 ","End":"07:11.690","Text":"Go back and check,"},{"Start":"07:11.690 ","End":"07:15.620","Text":"I\u0027ll leave it to you that what we have written in this table"},{"Start":"07:15.620 ","End":"07:20.180","Text":"corresponds to what we saw above with the pictures of the graphs,"},{"Start":"07:20.180 ","End":"07:22.040","Text":"and other than that,"},{"Start":"07:22.040 ","End":"07:24.790","Text":"we\u0027re done with inverse functions."},{"Start":"07:24.790 ","End":"07:31.180","Text":"Of course, there is the matter of the solved exercises which you should take a look at."}],"ID":10896},{"Watched":false,"Name":"Exercise 1","Duration":"1m 52s","ChapterTopicVideoID":10399,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10399.jpeg","UploadDate":"2017-11-02T16:03:25.4170000","DurationForVideoObject":"PT1M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this exercise, we have 3 similar computations to make."},{"Start":"00:04.380 ","End":"00:09.240","Text":"In each case, we have to take an inverse cosine and then apply the cosine."},{"Start":"00:09.240 ","End":"00:12.150","Text":"The inverse cosine,"},{"Start":"00:12.150 ","End":"00:14.895","Text":"written like this with a minus 1 here,"},{"Start":"00:14.895 ","End":"00:23.710","Text":"is defined as long as x is between minus 1 and 1 inclusive."},{"Start":"00:24.290 ","End":"00:26.744","Text":"Whenever this is so,"},{"Start":"00:26.744 ","End":"00:31.049","Text":"the cosine of the inverse cosine will take us back to the start."},{"Start":"00:31.049 ","End":"00:34.740","Text":"For the first 1, I take a look and see 1/3,"},{"Start":"00:34.740 ","End":"00:37.260","Text":"it is between minus 1 and 1."},{"Start":"00:37.260 ","End":"00:42.660","Text":"Cosine of inverse cosine just takes me back to 1/3."},{"Start":"00:42.660 ","End":"00:45.075","Text":"Here also, 1/2."},{"Start":"00:45.075 ","End":"00:47.660","Text":"No problem, it\u0027s between minus 1 and 1,"},{"Start":"00:47.660 ","End":"00:50.890","Text":"so cosine of inverse cosine is 1/2."},{"Start":"00:50.890 ","End":"00:53.935","Text":"But in this case, I\u0027d like to show you an alternative method,"},{"Start":"00:53.935 ","End":"00:56.935","Text":"because 1/2 is one of those special numbers,"},{"Start":"00:56.935 ","End":"01:04.845","Text":"1/2 is cosine of Pi over 360 degrees."},{"Start":"01:04.845 ","End":"01:10.185","Text":"I could say that this here is"},{"Start":"01:10.185 ","End":"01:15.741","Text":"equal to cosine of Pi over 3,"},{"Start":"01:15.741 ","End":"01:25.590","Text":"because the inverse cosine of 1/2 is Pi over 3 and cosine of Pi over 3 is equal to 1/2."},{"Start":"01:25.590 ","End":"01:29.535","Text":"It\u0027s just an alternative."},{"Start":"01:29.535 ","End":"01:31.230","Text":"Usually, we will just do it this way."},{"Start":"01:31.230 ","End":"01:33.680","Text":"The cosine of inverse cosine is the thing itself."},{"Start":"01:33.680 ","End":"01:36.009","Text":"The third one you have to watch out,"},{"Start":"01:36.009 ","End":"01:42.410","Text":"because 3/2 is not between minus 1 and 1."},{"Start":"01:42.410 ","End":"01:45.020","Text":"This is 1.5, it\u0027s outside the range."},{"Start":"01:45.020 ","End":"01:48.050","Text":"This is undefined."},{"Start":"01:48.050 ","End":"01:52.710","Text":"There is no answer. That\u0027s it."}],"ID":10759},{"Watched":false,"Name":"Exercise 2","Duration":"3m 21s","ChapterTopicVideoID":10400,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10400.jpeg","UploadDate":"2017-11-02T16:03:38.2970000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"In this exercise, we have"},{"Start":"00:01.950 ","End":"00:07.230","Text":"3 similar problems involving the tangent and the inverse tangent."},{"Start":"00:07.230 ","End":"00:10.320","Text":"I use the notation arc tangent here,"},{"Start":"00:10.320 ","End":"00:17.925","Text":"because remember that arc tangent is just another way of saying inverse tangent."},{"Start":"00:17.925 ","End":"00:20.370","Text":"Some books use this,"},{"Start":"00:20.370 ","End":"00:23.200","Text":"some use that, I use both."},{"Start":"00:23.360 ","End":"00:31.380","Text":"Also, I\u0027d like you to remember that the restricted tangent function, tangent x,"},{"Start":"00:31.380 ","End":"00:39.600","Text":"and the restricted tangent function restricts the domain to minus Pi over 2,"},{"Start":"00:39.600 ","End":"00:41.505","Text":"to Pi over 2,"},{"Start":"00:41.505 ","End":"00:44.695","Text":"not including the endpoints."},{"Start":"00:44.695 ","End":"00:48.770","Text":"If we happen to be in the restricted domain,"},{"Start":"00:48.770 ","End":"00:52.190","Text":"then if you take the tangent and the inverse tangent,"},{"Start":"00:52.190 ","End":"00:55.230","Text":"you get back to where you started from."},{"Start":"00:55.820 ","End":"00:59.035","Text":"Let\u0027s do that."},{"Start":"00:59.035 ","End":"01:03.800","Text":"The first case minus a third is certainly within"},{"Start":"01:03.800 ","End":"01:09.480","Text":"this range because minus Pi over 2 is minus 1.6 something."},{"Start":"01:09.480 ","End":"01:12.180","Text":"Certainly minus a 1/3 is in here."},{"Start":"01:12.180 ","End":"01:17.869","Text":"Arc tangent of tangent is just this itself."},{"Start":"01:17.869 ","End":"01:22.835","Text":"Similarly, Pi over 3 is certainly less than Pi over 2."},{"Start":"01:22.835 ","End":"01:27.815","Text":"I can say that this is equal to Pi over 3."},{"Start":"01:27.815 ","End":"01:30.800","Text":"In this particular case, we can also do it another way,"},{"Start":"01:30.800 ","End":"01:32.315","Text":"because Pi over 3,"},{"Start":"01:32.315 ","End":"01:33.710","Text":"or if you want it in degrees,"},{"Start":"01:33.710 ","End":"01:36.560","Text":"60 degrees, we know the tangent of it."},{"Start":"01:36.560 ","End":"01:44.205","Text":"We could say that it\u0027s arc tangent of 60 degrees gives us square root of 3,"},{"Start":"01:44.205 ","End":"01:50.985","Text":"and the arc tangent of square root of 3 is 60 degrees or Pi over 3."},{"Start":"01:50.985 ","End":"01:53.435","Text":"Yeah, we could do it this way also."},{"Start":"01:53.435 ","End":"01:56.030","Text":"The third is slightly different,"},{"Start":"01:56.030 ","End":"02:04.725","Text":"because 5 Pi over 6 is not between these 2,"},{"Start":"02:04.725 ","End":"02:11.884","Text":"but 5 pi over 6 is in the domain of the general tangent function."},{"Start":"02:11.884 ","End":"02:13.460","Text":"What we can do,"},{"Start":"02:13.460 ","End":"02:17.285","Text":"is first compute tangent of 5 Pi over 6."},{"Start":"02:17.285 ","End":"02:20.240","Text":"Now, if you\u0027d like to do it in degrees,"},{"Start":"02:20.240 ","End":"02:22.925","Text":"this is 150 degrees."},{"Start":"02:22.925 ","End":"02:26.495","Text":"You could use trigonometric identities."},{"Start":"02:26.495 ","End":"02:31.295","Text":"I could take the supplement of the angle and get 30 degrees or Pi over 6."},{"Start":"02:31.295 ","End":"02:35.130","Text":"The tangent of that is 1 over the square root of 3."},{"Start":"02:35.130 ","End":"02:39.220","Text":"This will be minus 1 over square root of 3."},{"Start":"02:39.220 ","End":"02:42.905","Text":"Sorry, I didn\u0027t take the arc tangent of that."},{"Start":"02:42.905 ","End":"02:48.535","Text":"Yeah, I need arc tangent of this."},{"Start":"02:48.535 ","End":"02:53.515","Text":"The arc tangent of this is equal to,"},{"Start":"02:53.515 ","End":"02:56.480","Text":"it\u0027s going to be between this and this."},{"Start":"02:56.480 ","End":"03:02.465","Text":"Actually it\u0027s minus 30 degrees or minus Pi over 6."},{"Start":"03:02.465 ","End":"03:04.745","Text":"This is the answer."},{"Start":"03:04.745 ","End":"03:06.785","Text":"There is another way of doing it,"},{"Start":"03:06.785 ","End":"03:10.460","Text":"you could also say that the period of tangent is"},{"Start":"03:10.460 ","End":"03:14.900","Text":"Pi and you could have subtracted Pi from this and got minus Pi over 6."},{"Start":"03:14.900 ","End":"03:17.945","Text":"Anyway, this is the obvious way to do it,"},{"Start":"03:17.945 ","End":"03:20.850","Text":"and so yes, we\u0027ve done all 3."}],"ID":10760},{"Watched":false,"Name":"Exercise 3","Duration":"7m 22s","ChapterTopicVideoID":10401,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10401.jpeg","UploadDate":"2017-11-02T16:04:05.8300000","DurationForVideoObject":"PT7M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"This exercise has 4 parts."},{"Start":"00:03.225 ","End":"00:05.130","Text":"In each case, we take"},{"Start":"00:05.130 ","End":"00:09.765","Text":"an inverse trigonometric function then apply a trigonometric function."},{"Start":"00:09.765 ","End":"00:12.329","Text":"But note that these are mixed."},{"Start":"00:12.329 ","End":"00:13.740","Text":"Previously we would have say,"},{"Start":"00:13.740 ","End":"00:16.470","Text":"inverse cosine and then take the cosine,"},{"Start":"00:16.470 ","End":"00:18.510","Text":"inverse sine then take the sine."},{"Start":"00:18.510 ","End":"00:20.580","Text":"Here we\u0027re mixing, we take the inverse cosine,"},{"Start":"00:20.580 ","End":"00:21.990","Text":"but then we take the sine."},{"Start":"00:21.990 ","End":"00:29.670","Text":"Let\u0027s just remember what the domains are for each of these inverse functions."},{"Start":"00:29.670 ","End":"00:31.770","Text":"For inverse cosine,"},{"Start":"00:31.770 ","End":"00:36.795","Text":"call it cosine to the minus 1 of x."},{"Start":"00:36.795 ","End":"00:43.020","Text":"This is good for when x is between minus 1 and 1."},{"Start":"00:43.020 ","End":"00:45.180","Text":"As for the inverse sine,"},{"Start":"00:45.180 ","End":"00:46.970","Text":"it\u0027s shorter to write this way,"},{"Start":"00:46.970 ","End":"00:53.595","Text":"it\u0027s also for minus 1 less than or equal to x less than or equal to 1."},{"Start":"00:53.595 ","End":"00:57.075","Text":"As for the arc secant,"},{"Start":"00:57.075 ","End":"00:59.610","Text":"secant to the minus 1."},{"Start":"00:59.610 ","End":"01:04.350","Text":"That\u0027s good for x,"},{"Start":"01:04.350 ","End":"01:13.015","Text":"either bigger or equal to 1 or less than or equal to minus 1."},{"Start":"01:13.015 ","End":"01:14.910","Text":"Sine again."},{"Start":"01:14.910 ","End":"01:16.975","Text":"Inverse sine,"},{"Start":"01:16.975 ","End":"01:19.559","Text":"well, we already have that here."},{"Start":"01:19.559 ","End":"01:22.440","Text":"You know what, I\u0027ll just copy it again."},{"Start":"01:22.480 ","End":"01:26.120","Text":"Now let\u0027s see how that applies here."},{"Start":"01:26.120 ","End":"01:28.715","Text":"Let\u0027s start with the first one."},{"Start":"01:28.715 ","End":"01:36.620","Text":"Now, 3/5 is in the domain of the inverse cosine or arc cosine,"},{"Start":"01:36.620 ","End":"01:40.255","Text":"and so if I call this thing u,"},{"Start":"01:40.255 ","End":"01:46.990","Text":"I know that cosine u is equal to 3/5,"},{"Start":"01:46.990 ","End":"01:48.755","Text":"but that\u0027s not what we want."},{"Start":"01:48.755 ","End":"01:53.315","Text":"We want to know what sine u is equal to."},{"Start":"01:53.315 ","End":"01:57.110","Text":"Here we\u0027ll use trigonometric identities."},{"Start":"01:57.110 ","End":"02:01.760","Text":"This will be equal to the square root of"},{"Start":"02:01.760 ","End":"02:09.220","Text":"1 minus cosine squared u and actually could be plus or minus."},{"Start":"02:09.220 ","End":"02:14.315","Text":"But I claim that it has to be a plus here."},{"Start":"02:14.315 ","End":"02:19.025","Text":"The inverse cosine, it has to be in the first or second quadrants,"},{"Start":"02:19.025 ","End":"02:20.960","Text":"and if the argument here is positive,"},{"Start":"02:20.960 ","End":"02:23.840","Text":"it\u0027s the first quadrant, and if it\u0027s negative is a second quadrant."},{"Start":"02:23.840 ","End":"02:28.625","Text":"We know that u is in the first quadrant."},{"Start":"02:28.625 ","End":"02:31.160","Text":"Since u is in the first quadrant,"},{"Start":"02:31.160 ","End":"02:33.770","Text":"the sine u has to be positive."},{"Start":"02:33.770 ","End":"02:43.435","Text":"This is equal to the square root of 1 minus cosine squared u is 3/5 squared."},{"Start":"02:43.435 ","End":"02:46.260","Text":"It\u0027s 1 minus 9/25,"},{"Start":"02:46.260 ","End":"02:51.850","Text":"16/25, and the square root is 4/5."},{"Start":"02:54.590 ","End":"03:02.190","Text":"In this case with the arcsine minus 3/4 is between minus 1 and 1,"},{"Start":"03:02.190 ","End":"03:06.870","Text":"so we are in the domain"},{"Start":"03:06.870 ","End":"03:13.660","Text":"of the inverse sine."},{"Start":"03:13.660 ","End":"03:20.150","Text":"Let\u0027s call this u the arcsine of this,"},{"Start":"03:20.150 ","End":"03:28.990","Text":"we know that sine of u is equal to minus 3/4."},{"Start":"03:29.450 ","End":"03:33.620","Text":"It\u0027s always that the trigonometric function"},{"Start":"03:33.620 ","End":"03:36.905","Text":"of the inverse trigonometric function always takes us back."},{"Start":"03:36.905 ","End":"03:39.170","Text":"The other way round, not always."},{"Start":"03:39.170 ","End":"03:42.380","Text":"Now, we don\u0027t want sine u,"},{"Start":"03:42.380 ","End":"03:45.110","Text":"we want cosine of u."},{"Start":"03:45.110 ","End":"03:49.950","Text":"Again, we\u0027ll use Pythagorean identity and get it."},{"Start":"03:49.950 ","End":"03:56.720","Text":"So square root of 1 minus sine squared u and plus or minus."},{"Start":"03:56.720 ","End":"04:02.635","Text":"The arcsine gives us something in the first or fourth quadrants,"},{"Start":"04:02.635 ","End":"04:04.070","Text":"and if this is positive,"},{"Start":"04:04.070 ","End":"04:05.195","Text":"it\u0027s the first quadrant,"},{"Start":"04:05.195 ","End":"04:06.470","Text":"and if it\u0027s negative,"},{"Start":"04:06.470 ","End":"04:08.375","Text":"it\u0027s the fourth quadrant."},{"Start":"04:08.375 ","End":"04:10.445","Text":"Since this is negative,"},{"Start":"04:10.445 ","End":"04:14.270","Text":"then u is in the fourth quadrant."},{"Start":"04:14.270 ","End":"04:19.625","Text":"But cosine of something in the fourth quadrant is positive."},{"Start":"04:19.625 ","End":"04:26.660","Text":"Let\u0027s just compute the square root of 1 minus sine squared,"},{"Start":"04:26.660 ","End":"04:33.550","Text":"I\u0027ll write straightaway minus 9/16 because it\u0027s is 3^2 over 4^2."},{"Start":"04:33.550 ","End":"04:36.480","Text":"We\u0027re left with square root of 7/16,"},{"Start":"04:36.480 ","End":"04:40.920","Text":"which is the square root of 7 over the square root of 16."},{"Start":"04:40.920 ","End":"04:43.255","Text":"That\u0027s the answer."},{"Start":"04:43.255 ","End":"04:45.950","Text":"In the third part,"},{"Start":"04:45.950 ","End":"04:52.825","Text":"the arc secant is defined because this happens to be less than minus 1."},{"Start":"04:52.825 ","End":"04:55.399","Text":"Let the arc secant be u."},{"Start":"04:55.399 ","End":"05:02.595","Text":"Then we know that the secant of u is equal to minus 7/3."},{"Start":"05:02.595 ","End":"05:05.300","Text":"But we don\u0027t want the secant of u,"},{"Start":"05:05.300 ","End":"05:07.945","Text":"we want the tangent of u."},{"Start":"05:07.945 ","End":"05:11.680","Text":"We\u0027re going to use trigonometric identities."},{"Start":"05:11.680 ","End":"05:20.845","Text":"The tangent squared is equal to secant squared minus 1."},{"Start":"05:20.845 ","End":"05:22.540","Text":"When I just take the tangent,"},{"Start":"05:22.540 ","End":"05:26.460","Text":"I need to take the square root of that and a plus or minus."},{"Start":"05:26.460 ","End":"05:29.995","Text":"Let\u0027s see which of the signs it is."},{"Start":"05:29.995 ","End":"05:37.130","Text":"The arc secant is in the first or third quadrants."},{"Start":"05:37.130 ","End":"05:40.400","Text":"If you look at where arc secant takes us,"},{"Start":"05:40.400 ","End":"05:43.190","Text":"it\u0027s first or third quadrant,"},{"Start":"05:43.190 ","End":"05:45.230","Text":"depending on the sign of this."},{"Start":"05:45.230 ","End":"05:46.550","Text":"If this were positive,"},{"Start":"05:46.550 ","End":"05:48.470","Text":"it would be in the first quadrant,"},{"Start":"05:48.470 ","End":"05:50.000","Text":"but it\u0027s negative,"},{"Start":"05:50.000 ","End":"05:55.340","Text":"so we are in the third quadrant."},{"Start":"05:55.340 ","End":"05:58.925","Text":"So we need tangent of something in the third quadrant."},{"Start":"05:58.925 ","End":"06:03.170","Text":"But tangent is positive in the third quadrant."},{"Start":"06:03.170 ","End":"06:06.636","Text":"Look it up, it\u0027s positive in the first third,"},{"Start":"06:06.636 ","End":"06:11.715","Text":"and so we\u0027re going to go for the plus part as before."},{"Start":"06:11.715 ","End":"06:16.760","Text":"Now we just do the computation plus the square root of secant"},{"Start":"06:16.760 ","End":"06:22.325","Text":"squared would be 49 over 9 minus 1."},{"Start":"06:22.325 ","End":"06:26.445","Text":"That would be 40/9 if I put it over 9."},{"Start":"06:26.445 ","End":"06:29.560","Text":"Square root of 40 over 3,"},{"Start":"06:30.140 ","End":"06:33.925","Text":"which is the square root of 9."},{"Start":"06:33.925 ","End":"06:36.305","Text":"That\u0027s the answer."},{"Start":"06:36.305 ","End":"06:38.390","Text":"For those who like simplifying it further,"},{"Start":"06:38.390 ","End":"06:40.280","Text":"I could take a 4 out of"},{"Start":"06:40.280 ","End":"06:45.210","Text":"the square root sign and make it equal to 2, is totally optional."},{"Start":"06:45.400 ","End":"06:48.515","Text":"It\u0027s another way of writing it."},{"Start":"06:48.515 ","End":"06:53.930","Text":"The last one, kind of a trick question, not really though."},{"Start":"06:53.930 ","End":"07:00.440","Text":"You have to always check if the arcsine of x that you\u0027re taking,"},{"Start":"07:00.440 ","End":"07:06.245","Text":"that x is in between minus 1 and 1,"},{"Start":"07:06.245 ","End":"07:10.535","Text":"and minus 2 is most certainly not in here."},{"Start":"07:10.535 ","End":"07:17.160","Text":"This thing is going to be undefined because the arcsine already is undefined."},{"Start":"07:17.860 ","End":"07:21.870","Text":"That finishes all 4 of them. We\u0027re done."}],"ID":10761},{"Watched":false,"Name":"Exercise 4","Duration":"5m 23s","ChapterTopicVideoID":10402,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10402.jpeg","UploadDate":"2017-11-02T16:04:24.0700000","DurationForVideoObject":"PT5M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we have four calculations or"},{"Start":"00:04.890 ","End":"00:09.975","Text":"evaluations to perform they\u0027re all inverse trigonometric functions."},{"Start":"00:09.975 ","End":"00:16.065","Text":"They all have a negative argument and if you look at the numbers that appear here,"},{"Start":"00:16.065 ","End":"00:19.200","Text":"they\u0027re all quotients involving root 3,"},{"Start":"00:19.200 ","End":"00:20.580","Text":"2, and 1."},{"Start":"00:20.580 ","End":"00:24.750","Text":"That should remind you of something and I"},{"Start":"00:24.750 ","End":"00:29.800","Text":"brought a little sketch that will show you what I mean."},{"Start":"00:29.840 ","End":"00:36.105","Text":"I borrowed this sketch from the chapter on Angles in degrees."},{"Start":"00:36.105 ","End":"00:42.380","Text":"Really should write the radian equivalent here"},{"Start":"00:42.380 ","End":"00:49.525","Text":"but it\u0027s not going to make any difference for the final answer if it is sorry."},{"Start":"00:49.525 ","End":"00:54.530","Text":"Let\u0027s just write here pi over"},{"Start":"00:54.530 ","End":"01:01.640","Text":"6 and here pi over 3 and of course there\u0027s the 90 degrees here."},{"Start":"01:01.640 ","End":"01:07.775","Text":"I\u0027ll use the right to the triangle that\u0027s in the right half here."},{"Start":"01:07.775 ","End":"01:15.360","Text":"Let\u0027s see, just scroll a bit to get some space for these."},{"Start":"01:15.360 ","End":"01:24.380","Text":"Now, sine if you remember is opposite over hypotenuse."},{"Start":"01:24.380 ","End":"01:27.480","Text":"If we\u0027re in the first quadrant that is."},{"Start":"01:27.530 ","End":"01:32.700","Text":"If we have root 3 over 2 and we want to make root 3 the opposite"},{"Start":"01:32.700 ","End":"01:37.250","Text":"side then we are talking about this angle here of pi over 3."},{"Start":"01:37.250 ","End":"01:41.765","Text":"Then opposite over hypotenuse is root 3 over 2."},{"Start":"01:41.765 ","End":"01:49.745","Text":"However, I can\u0027t write 60 degrees or pi over 3 here because there\u0027s a minus."},{"Start":"01:49.745 ","End":"01:53.720","Text":"Now the inverse sine is also besides the first quadrant,"},{"Start":"01:53.720 ","End":"01:59.615","Text":"it\u0027s also in the fourth quadrant and that\u0027s the one we have to take for the minus."},{"Start":"01:59.615 ","End":"02:05.929","Text":"It\u0027s actually going to be minus 60 degrees although I should talk in radians."},{"Start":"02:05.929 ","End":"02:13.010","Text":"It\u0027s minus pi over 3 in order to get this to be minus,"},{"Start":"02:13.010 ","End":"02:20.230","Text":"using the identity, sine of minus an angle is minus the sine of the angle."},{"Start":"02:20.230 ","End":"02:22.955","Text":"Okay, what about the cotangent?"},{"Start":"02:22.955 ","End":"02:27.690","Text":"Cotangent, in the first quadrant it\u0027s"},{"Start":"02:27.690 ","End":"02:34.830","Text":"the adjacent over opposite so again,"},{"Start":"02:34.830 ","End":"02:38.130","Text":"it\u0027s going to be the 60 degrees because the 1 is adjacent to"},{"Start":"02:38.130 ","End":"02:43.060","Text":"60 degrees and the opposite is root 3."},{"Start":"02:43.340 ","End":"02:49.140","Text":"This would be pi over 3 except"},{"Start":"02:49.140 ","End":"02:57.215","Text":"that the minus here makes it in the second quadrant."},{"Start":"02:57.215 ","End":"02:59.150","Text":"The way it works with cotangent,"},{"Start":"02:59.150 ","End":"03:04.220","Text":"the way we get this into the second quadrant is by taking the supplement."},{"Start":"03:04.220 ","End":"03:12.000","Text":"We take pi minus pi over 3 which is 2 pi over 3,"},{"Start":"03:12.000 ","End":"03:17.900","Text":"that\u0027s like 120 degrees and that\u0027s in the second quadrant."},{"Start":"03:17.900 ","End":"03:22.700","Text":"The cotangent of a supplement is minus the cotangent of the angle"},{"Start":"03:22.700 ","End":"03:28.270","Text":"so the cotangent of this will be minus 1 over root 3."},{"Start":"03:28.270 ","End":"03:32.685","Text":"The next one is the inverse tangent,"},{"Start":"03:32.685 ","End":"03:36.750","Text":"once again we start as if it was in the first quadrant and if it was 1 over"},{"Start":"03:36.750 ","End":"03:41.000","Text":"root 3 and we need an opposite over adjacent."},{"Start":"03:41.000 ","End":"03:42.530","Text":"We\u0027d have to take this angle,"},{"Start":"03:42.530 ","End":"03:44.940","Text":"the Pi over 6."},{"Start":"03:45.320 ","End":"03:50.440","Text":"If it wasn\u0027t a minus it would be pi over 6."},{"Start":"03:50.440 ","End":"03:53.420","Text":"But it is a minus and the tangent is defined in"},{"Start":"03:53.420 ","End":"03:58.250","Text":"the first fourth quadrants and we need the fourth quadrant part to"},{"Start":"03:58.250 ","End":"04:01.970","Text":"make it negative and the way we do it is just by putting a minus in"},{"Start":"04:01.970 ","End":"04:06.455","Text":"front and that makes the tangent also negative."},{"Start":"04:06.455 ","End":"04:13.160","Text":"That\u0027s this one and finally for"},{"Start":"04:13.160 ","End":"04:20.240","Text":"cosine we want adjacent over hypotenuse."},{"Start":"04:20.240 ","End":"04:29.330","Text":"If there wasn\u0027t the minus there we could take root 3 over 2 using this angle."},{"Start":"04:29.330 ","End":"04:32.420","Text":"For this angle, the root 3 is the adjacent side,"},{"Start":"04:32.420 ","End":"04:35.700","Text":"and this is the hypotenuse in either case."},{"Start":"04:36.620 ","End":"04:47.450","Text":"If it wasn\u0027t the minus the inverse cosine would be pi over 6 but there is a minus here."},{"Start":"04:47.450 ","End":"04:49.520","Text":"The inverse cosine takes us to"},{"Start":"04:49.520 ","End":"04:55.880","Text":"the first or second quadrants and so we\u0027ll have to take the second quadrant for minus."},{"Start":"04:55.880 ","End":"04:58.955","Text":"What we do is just like what we did with the cotangent."},{"Start":"04:58.955 ","End":"05:04.710","Text":"We take the supplement of the angle so instead of pi"},{"Start":"05:04.710 ","End":"05:10.395","Text":"over 6 it\u0027s pi minus pi over 6 which is 5 pi over 6."},{"Start":"05:10.395 ","End":"05:14.340","Text":"Or if you\u0027re thinking in degrees 180 minus 30 is"},{"Start":"05:14.340 ","End":"05:22.990","Text":"150 degrees and okay that\u0027s the fourth one and so we are done."}],"ID":10762},{"Watched":false,"Name":"Exercise 5","Duration":"1m 43s","ChapterTopicVideoID":10403,"CourseChapterTopicPlaylistID":257205,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10403.jpeg","UploadDate":"2017-11-02T16:04:32.5870000","DurationForVideoObject":"PT1M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.245","Text":"This exercise is similar to some previous ones we\u0027ve had."},{"Start":"00:04.245 ","End":"00:09.510","Text":"In each case, we take first the inverse trigonometric function,"},{"Start":"00:09.510 ","End":"00:16.920","Text":"and then the regular trigonometric function applies to that, like cos(cos^-1(2/3))."},{"Start":"00:16.920 ","End":"00:19.545","Text":"Now, in all these cases,"},{"Start":"00:19.545 ","End":"00:22.620","Text":"it will take us to the original argument,"},{"Start":"00:22.620 ","End":"00:24.745","Text":"like here the answer would be 2/3,"},{"Start":"00:24.745 ","End":"00:29.270","Text":"but this is provided that the argument is"},{"Start":"00:29.270 ","End":"00:34.519","Text":"in the domain of definition for the inverse function."},{"Start":"00:34.519 ","End":"00:41.225","Text":"Now, this will be true in most cases, it will be an exception."},{"Start":"00:41.225 ","End":"00:43.535","Text":"Let\u0027s just do one at a time and you\u0027ll see."},{"Start":"00:43.535 ","End":"00:46.430","Text":"The first one, I look at the 2/3,"},{"Start":"00:46.430 ","End":"00:51.750","Text":"the argument for inverse cosine is from minus 1 to 1."},{"Start":"00:51.750 ","End":"00:54.630","Text":"This is between minus 1 and 1,"},{"Start":"00:54.630 ","End":"00:58.940","Text":"so the answer is 2/3."},{"Start":"00:58.940 ","End":"01:00.410","Text":"For the inverse sine also,"},{"Start":"01:00.410 ","End":"01:02.180","Text":"it has to be between minus 1 and 1,"},{"Start":"01:02.180 ","End":"01:06.310","Text":"which it is, and so the answer is minus 1/4."},{"Start":"01:06.310 ","End":"01:08.460","Text":"For the tangent,"},{"Start":"01:08.460 ","End":"01:12.155","Text":"the argument is anything from minus infinity to infinity,"},{"Start":"01:12.155 ","End":"01:13.910","Text":"this will always work."},{"Start":"01:13.910 ","End":"01:17.465","Text":"This will just equal the argument itself, which is 1."},{"Start":"01:17.465 ","End":"01:20.180","Text":"The only trouble is in the last one because"},{"Start":"01:20.180 ","End":"01:28.960","Text":"the cosecant inverse is defined from 1 upwards or from minus 1 downwards."},{"Start":"01:28.960 ","End":"01:32.660","Text":"It\u0027s not defined when we\u0027re between minus 1 and 1,"},{"Start":"01:32.660 ","End":"01:38.340","Text":"so this is undefined."},{"Start":"01:38.480 ","End":"01:43.420","Text":"That finishes the four of them and we\u0027re done."}],"ID":10763}],"Thumbnail":null,"ID":257205},{"Name":"Triangles","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Triangles - Part 1","Duration":"13m 19s","ChapterTopicVideoID":10477,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10477.jpeg","UploadDate":"2021-06-29T13:15:29.7370000","DurationForVideoObject":"PT13M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"We\u0027re starting a new topic involving triangles in general,"},{"Start":"00:05.280 ","End":"00:08.310","Text":"but mostly a process called solving triangles."},{"Start":"00:08.310 ","End":"00:09.765","Text":"To start off with,"},{"Start":"00:09.765 ","End":"00:13.005","Text":"we\u0027ll have some definitions and notation."},{"Start":"00:13.005 ","End":"00:16.020","Text":"There are different kinds of triangles,"},{"Start":"00:16.020 ","End":"00:19.125","Text":"we\u0027re going to distinguish 3 kinds."},{"Start":"00:19.125 ","End":"00:23.910","Text":"A right triangle is a triangle which contains a right angle."},{"Start":"00:23.910 ","End":"00:28.304","Text":"A right angle means it\u0027s 90 degrees or Pi over 2 radians."},{"Start":"00:28.304 ","End":"00:32.747","Text":"An obtuse triangle is one which contains an obtuse angle."},{"Start":"00:32.747 ","End":"00:35.580","Text":"Obtuse, meaning bigger than 90 degrees."},{"Start":"00:35.580 ","End":"00:41.095","Text":"An acute triangle is a triangle where all the angles are acute,"},{"Start":"00:41.095 ","End":"00:44.000","Text":"meaning less than 90 degrees."},{"Start":"00:44.000 ","End":"00:47.090","Text":"Everything that\u0027s not a right triangle is"},{"Start":"00:47.090 ","End":"00:52.175","Text":"an oblique triangle of which there are 2 kinds, obtuse and acute."},{"Start":"00:52.175 ","End":"00:54.440","Text":"That\u0027s some definitions about triangles."},{"Start":"00:54.440 ","End":"00:58.610","Text":"Now how do we label things?"},{"Start":"00:58.610 ","End":"01:02.524","Text":"A triangle is made up of 6 parts."},{"Start":"01:02.524 ","End":"01:08.570","Text":"There are 3 sides and 3 angles, sides and angles."},{"Start":"01:08.570 ","End":"01:11.390","Text":"There\u0027s 3 of each and they make up the triangle."},{"Start":"01:11.390 ","End":"01:12.919","Text":"Now, the points,"},{"Start":"01:12.919 ","End":"01:14.810","Text":"they\u0027re not part of the triangle,"},{"Start":"01:14.810 ","End":"01:18.350","Text":"but let\u0027s say a triangle is labeled A,"},{"Start":"01:18.350 ","End":"01:21.020","Text":"B, C. For points,"},{"Start":"01:21.020 ","End":"01:23.150","Text":"we use capital letters."},{"Start":"01:23.150 ","End":"01:31.245","Text":"Then typically, the sides are labeled according to the vertex that\u0027s opposite."},{"Start":"01:31.245 ","End":"01:34.975","Text":"If here I have B, this will be b,"},{"Start":"01:34.975 ","End":"01:37.160","Text":"this one will be a,"},{"Start":"01:37.160 ","End":"01:40.610","Text":"this one will be c. Here we use lowercase letters."},{"Start":"01:40.610 ","End":"01:42.650","Text":"Those are the 3 sides, a, b,"},{"Start":"01:42.650 ","End":"01:45.260","Text":"and c. The angles,"},{"Start":"01:45.260 ","End":"01:47.360","Text":"we typically use Greek letters."},{"Start":"01:47.360 ","End":"01:50.060","Text":"Next to the point A or opposite,"},{"Start":"01:50.060 ","End":"01:52.250","Text":"the side a,"},{"Start":"01:52.250 ","End":"01:54.985","Text":"we call the angle Alpha."},{"Start":"01:54.985 ","End":"01:57.650","Text":"Then B goes with Beta,"},{"Start":"01:57.650 ","End":"01:59.210","Text":"which is opposite b."},{"Start":"01:59.210 ","End":"02:06.210","Text":"C is the third letter of the Greek alphabet that\u0027s Gamma."},{"Start":"02:08.410 ","End":"02:11.514","Text":"Maybe I\u0027ll do it with the others as well,"},{"Start":"02:11.514 ","End":"02:20.730","Text":"Alpha, Gamma, Beta."},{"Start":"02:20.730 ","End":"02:25.120","Text":"That\u0027s how we\u0027re going to label our triangles."},{"Start":"02:25.250 ","End":"02:29.540","Text":"Back here, 3 angles,"},{"Start":"02:29.540 ","End":"02:32.580","Text":"3 sides, 6 parts altogether."},{"Start":"02:35.800 ","End":"02:40.415","Text":"Now, I\u0027m going to explain what solving a triangle means."},{"Start":"02:40.415 ","End":"02:45.420","Text":"Actually, this is the reason trigonometry was invented mainly."},{"Start":"02:45.420 ","End":"02:48.965","Text":"Solving a triangle means I give you part of the information."},{"Start":"02:48.965 ","End":"02:50.870","Text":"I don\u0027t give you all the sides and the angles,"},{"Start":"02:50.870 ","End":"02:52.865","Text":"and you\u0027re going to find the others."},{"Start":"02:52.865 ","End":"03:00.740","Text":"In general, you\u0027ll be given 3 parts of the 6,"},{"Start":"03:00.740 ","End":"03:08.250","Text":"and the idea is that you have to find the rest."},{"Start":"03:10.030 ","End":"03:14.329","Text":"The main condition is that,"},{"Start":"03:14.329 ","End":"03:16.760","Text":"of these 3 parts,"},{"Start":"03:16.760 ","End":"03:22.685","Text":"I have to say that we have to have at least one which is a side."},{"Start":"03:22.685 ","End":"03:28.550","Text":"You can\u0027t do it from 3 angles because you"},{"Start":"03:28.550 ","End":"03:30.200","Text":"could enlarge or shrink and get"},{"Start":"03:30.200 ","End":"03:34.130","Text":"a similar triangle with the same angles but with different sides."},{"Start":"03:34.130 ","End":"03:36.213","Text":"We\u0027re given 3 parts,"},{"Start":"03:36.213 ","End":"03:37.595","Text":"at least 1 of them is a side,"},{"Start":"03:37.595 ","End":"03:39.440","Text":"we have to find the rest."},{"Start":"03:39.440 ","End":"03:45.530","Text":"Before we proceed, I want to make sure that everyone remembers their basic trigonometry,"},{"Start":"03:45.530 ","End":"03:48.140","Text":"sine, cosine, and tangent."},{"Start":"03:48.140 ","End":"03:54.060","Text":"I\u0027m going to show you a strange word which is a mnemonic, SOHCAHTOA."},{"Start":"03:54.130 ","End":"03:58.504","Text":"I\u0027ll explain what this means."},{"Start":"03:58.504 ","End":"04:05.540","Text":"We\u0027re going to deal exclusively in this section with just the sine,"},{"Start":"04:05.540 ","End":"04:06.913","Text":"the cosine, and the tangent."},{"Start":"04:06.913 ","End":"04:10.220","Text":"I know there are 3 other trigonometric functions,"},{"Start":"04:10.220 ","End":"04:12.335","Text":"cosecant, secant, and cotangent."},{"Start":"04:12.335 ","End":"04:13.910","Text":"We\u0027ll just be using these and,"},{"Start":"04:13.910 ","End":"04:17.010","Text":"of these, mostly the sine and cosine."},{"Start":"04:17.810 ","End":"04:20.420","Text":"We had various definitions,"},{"Start":"04:20.420 ","End":"04:23.810","Text":"but one of them involved adjacent,"},{"Start":"04:23.810 ","End":"04:25.655","Text":"opposite, and hypotenuse."},{"Start":"04:25.655 ","End":"04:31.190","Text":"If we have a right triangle and let\u0027s say this angle is Theta."},{"Start":"04:31.190 ","End":"04:32.630","Text":"In practice, in our case,"},{"Start":"04:32.630 ","End":"04:35.570","Text":"it will be Alpha or Beta if you take the diagrams above,"},{"Start":"04:35.570 ","End":"04:37.475","Text":"but we\u0027ll just call this one Theta."},{"Start":"04:37.475 ","End":"04:45.840","Text":"Then the sign of Theta is the side opposite over the hypotenuse."},{"Start":"04:46.040 ","End":"04:49.550","Text":"This arrow indicates this divided by this."},{"Start":"04:49.550 ","End":"04:51.500","Text":"Sometimes you remember it visually."},{"Start":"04:51.500 ","End":"04:55.324","Text":"The sine is this over this or you just say opposite over hypotenuse."},{"Start":"04:55.324 ","End":"04:58.610","Text":"Cosine is adjacent over hypotenuse,"},{"Start":"04:58.610 ","End":"05:03.390","Text":"and the tangent is opposite over adjacent. They\u0027re written here."},{"Start":"05:03.410 ","End":"05:07.145","Text":"The mnemonic is as follows."},{"Start":"05:07.145 ","End":"05:11.171","Text":"This means sine is opposite over hypotenuse,"},{"Start":"05:11.171 ","End":"05:14.570","Text":"cosine is adjacent over hypotenuse,"},{"Start":"05:14.570 ","End":"05:17.810","Text":"and tangent is opposite over adjacent."},{"Start":"05:17.810 ","End":"05:19.430","Text":"If you remember these 3,"},{"Start":"05:19.430 ","End":"05:22.220","Text":"which really means just remembering the mnemonic,"},{"Start":"05:22.220 ","End":"05:25.670","Text":"SOHCAHTOA, then you\u0027ve got your definitions of sine,"},{"Start":"05:25.670 ","End":"05:27.260","Text":"cosine, and tangent."},{"Start":"05:27.260 ","End":"05:30.250","Text":"That might come in useful."},{"Start":"05:30.250 ","End":"05:33.905","Text":"Now in solving triangles,"},{"Start":"05:33.905 ","End":"05:36.110","Text":"we\u0027re going to divide it into 2 cases."},{"Start":"05:36.110 ","End":"05:40.198","Text":"There\u0027s simple case, which is the case of a right triangle,"},{"Start":"05:40.198 ","End":"05:44.990","Text":"and we\u0027ll do that first and then move on to oblique triangles."},{"Start":"05:44.990 ","End":"05:46.796","Text":"For solving right triangles,"},{"Start":"05:46.796 ","End":"05:48.995","Text":"well, I need a little picture."},{"Start":"05:48.995 ","End":"05:51.455","Text":"Here\u0027s a right triangle,"},{"Start":"05:51.455 ","End":"05:54.990","Text":"and this is the one that\u0027s 90 degrees."},{"Start":"05:55.360 ","End":"05:59.606","Text":"We\u0027re given 2 pieces of information here,"},{"Start":"05:59.606 ","End":"06:00.695","Text":"or we should be given."},{"Start":"06:00.695 ","End":"06:01.970","Text":"Because we already have,"},{"Start":"06:01.970 ","End":"06:04.340","Text":"the third one would be the right angle."},{"Start":"06:04.340 ","End":"06:11.510","Text":"The main possibilities are that we\u0027re given 2 sides or we\u0027re given a side and an angle."},{"Start":"06:11.510 ","End":"06:14.180","Text":"When I say an angle, I mean 1 of these 2 acute angles."},{"Start":"06:14.180 ","End":"06:15.965","Text":"This one, we have already."},{"Start":"06:15.965 ","End":"06:18.800","Text":"If we have 2 sides,"},{"Start":"06:18.800 ","End":"06:23.635","Text":"we can find the third side using Pythagoras\u0027s theorem."},{"Start":"06:23.635 ","End":"06:26.865","Text":"Do I need to remind you? Well, just to be safe,"},{"Start":"06:26.865 ","End":"06:31.174","Text":"a^2 plus b^2=c^2 in this case."},{"Start":"06:31.174 ","End":"06:33.695","Text":"Now, there are 3 tools."},{"Start":"06:33.695 ","End":"06:40.355","Text":"I could call this one the first tool, that\u0027s the SOHCAHTOA."},{"Start":"06:40.355 ","End":"06:44.330","Text":"The second tool is Pythagoras\u0027s theorem,"},{"Start":"06:44.330 ","End":"06:46.460","Text":"and there\u0027s also a third tool,"},{"Start":"06:46.460 ","End":"06:52.040","Text":"is that the sum of the angles in a triangle is 180 degrees."},{"Start":"06:52.040 ","End":"06:53.450","Text":"But in this case,"},{"Start":"06:53.450 ","End":"06:55.310","Text":"because this is 90,"},{"Start":"06:55.310 ","End":"07:01.055","Text":"I know that Alpha plus Beta is 90 degrees, or in other words,"},{"Start":"07:01.055 ","End":"07:03.720","Text":"Alpha and Beta are complementary angles,"},{"Start":"07:03.720 ","End":"07:05.145","Text":"we use that term."},{"Start":"07:05.145 ","End":"07:07.640","Text":"When we go on to oblique triangles,"},{"Start":"07:07.640 ","End":"07:13.910","Text":"we\u0027ll write it that Alpha plus Beta plus Gamma is 180 degrees."},{"Start":"07:13.910 ","End":"07:17.555","Text":"But here, because Gamma is 90, we\u0027ll do it this way."},{"Start":"07:17.555 ","End":"07:19.430","Text":"These are the 3 tools."},{"Start":"07:19.430 ","End":"07:22.365","Text":"Now I\u0027ll do an example."},{"Start":"07:22.365 ","End":"07:25.240","Text":"Let\u0027s work in degrees."},{"Start":"07:25.240 ","End":"07:27.050","Text":"For the first example,"},{"Start":"07:27.050 ","End":"07:28.940","Text":"I will give you 2 sides."},{"Start":"07:28.940 ","End":"07:34.760","Text":"I will give you a=1 and"},{"Start":"07:34.760 ","End":"07:41.600","Text":"c=2."},{"Start":"07:41.600 ","End":"07:43.670","Text":"We have to find remaining parts."},{"Start":"07:43.670 ","End":"07:47.510","Text":"We have to find b and Alpha and Beta."},{"Start":"07:47.510 ","End":"07:50.075","Text":"There\u0027s more than 1 way to do this."},{"Start":"07:50.075 ","End":"07:53.015","Text":"I\u0027d like to start with Pythagoras\u0027s theorem,"},{"Start":"07:53.015 ","End":"07:59.900","Text":"and find the side b. I can use Pythagoras to say that"},{"Start":"07:59.900 ","End":"08:05.420","Text":"b^2=c^2 minus a^2 just"},{"Start":"08:05.420 ","End":"08:09.500","Text":"like Pythagoras where I brought the a squared over to the other side."},{"Start":"08:09.500 ","End":"08:13.505","Text":"This would equal c^2 is 4,"},{"Start":"08:13.505 ","End":"08:16.694","Text":"a^2 is 1 which is 3,"},{"Start":"08:16.694 ","End":"08:21.695","Text":"which means that b equals the square root of 3."},{"Start":"08:21.695 ","End":"08:25.490","Text":"It has to be the positive because sides are positive."},{"Start":"08:25.490 ","End":"08:28.620","Text":"Now, a was given,"},{"Start":"08:28.620 ","End":"08:29.880","Text":"c was given, we have b,"},{"Start":"08:29.880 ","End":"08:32.415","Text":"now we need the angles."},{"Start":"08:32.415 ","End":"08:35.855","Text":"Let\u0027s say, we\u0027ll go for Alpha first."},{"Start":"08:35.855 ","End":"08:42.875","Text":"We know that sine Alpha is a over c. I also know that tangent Alpha is a over b,"},{"Start":"08:42.875 ","End":"08:48.100","Text":"but I prefer to work with original data because of rounding errors."},{"Start":"08:48.100 ","End":"08:50.480","Text":"Here I made the numbers easy, but in general,"},{"Start":"08:50.480 ","End":"08:55.415","Text":"it\u0027s better to use original data than computed data for more accuracy."},{"Start":"08:55.415 ","End":"09:01.220","Text":"I\u0027ll go with the sine of Alpha using SOHCAHTOA."},{"Start":"09:01.220 ","End":"09:05.600","Text":"The SOH part is sine is opposite over hypotenuse."},{"Start":"09:05.600 ","End":"09:11.990","Text":"It\u0027s a over c. If we figure a over c,"},{"Start":"09:11.990 ","End":"09:15.485","Text":"that\u0027s equal to 1/2."},{"Start":"09:15.485 ","End":"09:22.105","Text":"Then we need a calculator to look up the inverse of the sine."},{"Start":"09:22.105 ","End":"09:26.150","Text":"Sometimes you press the inverse key or the shift,"},{"Start":"09:26.150 ","End":"09:28.685","Text":"or some key, it depends on calculator."},{"Start":"09:28.685 ","End":"09:35.475","Text":"Then we get from this that Alpha is equal to 30 degrees."},{"Start":"09:35.475 ","End":"09:39.050","Text":"Once we have one angle than the other one,"},{"Start":"09:39.050 ","End":"09:45.290","Text":"we just get from the rule that the angles in the triangle are 180,"},{"Start":"09:45.290 ","End":"09:47.030","Text":"or the 2 acute angles are 90,"},{"Start":"09:47.030 ","End":"09:53.105","Text":"so I just do 90 minus 30 which is 60 degrees."},{"Start":"09:53.105 ","End":"09:55.115","Text":"We found everything."},{"Start":"09:55.115 ","End":"09:57.470","Text":"We had 3 things to find."},{"Start":"09:57.470 ","End":"10:00.500","Text":"We had the right angle and 2 other pieces of information,"},{"Start":"10:00.500 ","End":"10:02.990","Text":"a and c. We found b here,"},{"Start":"10:02.990 ","End":"10:06.350","Text":"and we found Alpha and Beta."},{"Start":"10:06.350 ","End":"10:08.660","Text":"I\u0027ll do another example,"},{"Start":"10:08.660 ","End":"10:11.419","Text":"although there are solved examples after the tutorial,"},{"Start":"10:11.419 ","End":"10:16.405","Text":"but I want to do one where we are given a side and an angle."},{"Start":"10:16.405 ","End":"10:20.960","Text":"I\u0027m going to give you again Alpha equals 30 degrees."},{"Start":"10:20.960 ","End":"10:25.580","Text":"I like this angle because the sine comes out in nice 1/2."},{"Start":"10:25.580 ","End":"10:29.450","Text":"This time we\u0027ll give one of the sides."},{"Start":"10:29.450 ","End":"10:36.950","Text":"Let\u0027s say we take b=5."},{"Start":"10:36.950 ","End":"10:39.965","Text":"There\u0027s more than one way to do this,"},{"Start":"10:39.965 ","End":"10:42.440","Text":"I\u0027ll show you one way."},{"Start":"10:42.440 ","End":"10:46.560","Text":"I have Alpha and I have b,"},{"Start":"10:46.560 ","End":"10:48.485","Text":"which is the adjacent."},{"Start":"10:48.485 ","End":"10:56.480","Text":"Why don\u0027t I go and find c using SOHCAHTOA, the cosine part."},{"Start":"10:56.480 ","End":"11:02.610","Text":"Cosine is the CAH."},{"Start":"11:02.710 ","End":"11:06.950","Text":"Cosine is adjacent over hypotenuse."},{"Start":"11:06.950 ","End":"11:16.240","Text":"I can say that cosine of 30 degrees is b,"},{"Start":"11:16.240 ","End":"11:17.795","Text":"which is 5,"},{"Start":"11:17.795 ","End":"11:24.335","Text":"over the hypotenuse is c. All I don\u0027t know is c,"},{"Start":"11:24.335 ","End":"11:32.100","Text":"so I can say that c is 5 over cosine 30."},{"Start":"11:32.100 ","End":"11:35.015","Text":"Now, cosine 30 is one of those special angles."},{"Start":"11:35.015 ","End":"11:37.025","Text":"You could do it with the calculator,"},{"Start":"11:37.025 ","End":"11:44.370","Text":"or you could just remember that cosine 30 is root 3 over 2."},{"Start":"11:44.600 ","End":"11:51.218","Text":"We could bring the 2 to the top and make this 10 over root 3,"},{"Start":"11:51.218 ","End":"11:54.515","Text":"and you could compute it numerically."},{"Start":"11:54.515 ","End":"11:57.920","Text":"I\u0027ll just leave it as that."},{"Start":"11:57.920 ","End":"12:02.389","Text":"Now, we have c. What else are we missing?"},{"Start":"12:02.389 ","End":"12:04.730","Text":"We\u0027re missing a."},{"Start":"12:04.730 ","End":"12:09.770","Text":"For a, since I have b,"},{"Start":"12:09.770 ","End":"12:13.190","Text":"I could use the tangent formula,"},{"Start":"12:13.190 ","End":"12:16.025","Text":"and just using the original data because"},{"Start":"12:16.025 ","End":"12:20.510","Text":"a over b is the tangent of a, opposite over adjacent."},{"Start":"12:20.510 ","End":"12:25.760","Text":"I can say that tangent 30 degrees is"},{"Start":"12:25.760 ","End":"12:31.710","Text":"the opposite a over b, which is 5."},{"Start":"12:31.710 ","End":"12:41.325","Text":"This gives us that a= 5 times tangent 30."},{"Start":"12:41.325 ","End":"12:45.050","Text":"Tangent 30, you either do it in the calculator or"},{"Start":"12:45.050 ","End":"12:50.400","Text":"I let you know that it\u0027s 1 over square root of 3."},{"Start":"12:50.400 ","End":"12:56.880","Text":"Here we get 5 over the square root of 3."},{"Start":"12:56.880 ","End":"13:01.640","Text":"That\u0027s c, that\u0027s a, so what else are we missing?"},{"Start":"13:01.640 ","End":"13:03.115","Text":"We\u0027re missing Beta."},{"Start":"13:03.115 ","End":"13:06.590","Text":"Beta, we just get from the complementary angle,"},{"Start":"13:06.590 ","End":"13:12.515","Text":"90 minus 30 is 60 degrees."},{"Start":"13:12.515 ","End":"13:16.050","Text":"That solves the triangle."},{"Start":"13:16.810 ","End":"13:20.460","Text":"We\u0027ll take a break now."}],"ID":10897},{"Watched":false,"Name":"Triangles - Part 2","Duration":"12m 37s","ChapterTopicVideoID":10478,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10478.jpeg","UploadDate":"2021-06-29T13:16:22.7030000","DurationForVideoObject":"PT12M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.740","Text":"In the previous clip,"},{"Start":"00:01.740 ","End":"00:05.490","Text":"we solved right triangles."},{"Start":"00:05.490 ","End":"00:08.160","Text":"Now, we\u0027ll learn to solve oblique triangles,"},{"Start":"00:08.160 ","End":"00:10.480","Text":"which means anything Beta right triangle."},{"Start":"00:10.480 ","End":"00:12.750","Text":"Right triangles are relatively easy,"},{"Start":"00:12.750 ","End":"00:15.975","Text":"little bit more tricky with oblique triangles."},{"Start":"00:15.975 ","End":"00:19.020","Text":"Now, we\u0027re going to distinguish different cases."},{"Start":"00:19.020 ","End":"00:22.710","Text":"Remember that there are 6 parts to a triangle,"},{"Start":"00:22.710 ","End":"00:24.405","Text":"there\u0027s three sides,"},{"Start":"00:24.405 ","End":"00:32.769","Text":"and three angles and we have to be given 3 out of the 6 or more,"},{"Start":"00:32.769 ","End":"00:35.610","Text":"but there has to be at least one side."},{"Start":"00:35.610 ","End":"00:38.610","Text":"Now, we\u0027ll classify them as follows,"},{"Start":"00:38.610 ","End":"00:40.365","Text":"if we have one side,"},{"Start":"00:40.365 ","End":"00:47.330","Text":"then there\u0027s two possibilities: It could be that the side is between the two angles,"},{"Start":"00:47.330 ","End":"00:49.205","Text":"let\u0027s say Alpha C,"},{"Start":"00:49.205 ","End":"00:56.465","Text":"and Beta, or it could be that there are two angles and the side not between them,"},{"Start":"00:56.465 ","End":"00:58.639","Text":"like maybe Gamma, Beta,"},{"Start":"00:58.639 ","End":"01:01.430","Text":"and C, that\u0027s one side."},{"Start":"01:01.430 ","End":"01:03.470","Text":"Now, suppose we have two sides,"},{"Start":"01:03.470 ","End":"01:06.511","Text":"then there\u0027s really two possibilities,"},{"Start":"01:06.511 ","End":"01:10.670","Text":"we could have side, angle, side,"},{"Start":"01:10.670 ","End":"01:12.920","Text":"meaning two sides and an included angle,"},{"Start":"01:12.920 ","End":"01:16.700","Text":"like A, and C, and Beta,"},{"Start":"01:16.700 ","End":"01:22.775","Text":"or we could have SSA,"},{"Start":"01:22.775 ","End":"01:24.285","Text":"which would be like, I don\u0027t know,"},{"Start":"01:24.285 ","End":"01:26.730","Text":"B and C and Beta,"},{"Start":"01:26.730 ","End":"01:28.790","Text":"two sides and the non minus inclusive angle."},{"Start":"01:28.790 ","End":"01:32.790","Text":"Then it could also have three sides."},{"Start":"01:32.840 ","End":"01:38.130","Text":"The tools we\u0027re going to have to solve these,"},{"Start":"01:38.330 ","End":"01:40.850","Text":"the first tool you know already,"},{"Start":"01:40.850 ","End":"01:44.285","Text":"it\u0027s just that the sum of the angles in a triangle is 180."},{"Start":"01:44.285 ","End":"01:48.560","Text":"I\u0027ll just write that Alpha plus Beta plus Gamma is 180,"},{"Start":"01:48.560 ","End":"01:50.825","Text":"and we work in degrees here."},{"Start":"01:50.825 ","End":"01:53.970","Text":"If it was radians, you\u0027d put Pi here."},{"Start":"01:54.290 ","End":"01:58.220","Text":"Number 2, tool,"},{"Start":"01:58.220 ","End":"01:59.765","Text":"we haven\u0027t learned yet,"},{"Start":"01:59.765 ","End":"02:05.425","Text":"that\u0027s called the Law of Sines."},{"Start":"02:05.425 ","End":"02:07.780","Text":"Rule Number 3,"},{"Start":"02:07.780 ","End":"02:09.155","Text":"as you might guess,"},{"Start":"02:09.155 ","End":"02:12.270","Text":"is the Law of Cosine."},{"Start":"02:12.660 ","End":"02:16.125","Text":"I\u0027ll introduce these two in a moment."},{"Start":"02:16.125 ","End":"02:21.080","Text":"These two are practically the same because of tool Number 1."},{"Start":"02:21.080 ","End":"02:22.790","Text":"If we have two angles,"},{"Start":"02:22.790 ","End":"02:27.890","Text":"then we have three angles because you can always subtract from 180 to get the third."},{"Start":"02:27.890 ","End":"02:30.830","Text":"It\u0027s easy to see that these two are equivalent."},{"Start":"02:30.830 ","End":"02:33.855","Text":"They\u0027re both like three angles on a side."},{"Start":"02:33.855 ","End":"02:40.100","Text":"Now, the use of each one of them is that the law of"},{"Start":"02:40.100 ","End":"02:46.325","Text":"sines is used for the case of two angles on a side,"},{"Start":"02:46.325 ","End":"02:53.005","Text":"and also the case of two sides and not included angle."},{"Start":"02:53.005 ","End":"02:57.320","Text":"The Law of Cosine is used on the remaining cases,"},{"Start":"02:57.320 ","End":"02:59.000","Text":"when we have three sides,"},{"Start":"02:59.000 ","End":"03:02.524","Text":"we have two sides and an included angle."},{"Start":"03:02.524 ","End":"03:05.150","Text":"I\u0027m going to let you know something in advance that there is"},{"Start":"03:05.150 ","End":"03:07.640","Text":"one of these cases which is problematic,"},{"Start":"03:07.640 ","End":"03:11.420","Text":"and this is the problematic one or ambiguous."},{"Start":"03:11.420 ","End":"03:14.975","Text":"It turns out that in all the other cases we can"},{"Start":"03:14.975 ","End":"03:19.760","Text":"solve the triangle and get a unique solution."},{"Start":"03:19.760 ","End":"03:24.680","Text":"However, in the case of two sides and not included angle,"},{"Start":"03:24.680 ","End":"03:27.440","Text":"there might be one solution to the triangle,"},{"Start":"03:27.440 ","End":"03:29.330","Text":"but there could be no solution,"},{"Start":"03:29.330 ","End":"03:30.950","Text":"or there could be two solutions."},{"Start":"03:30.950 ","End":"03:33.345","Text":"In other words, we could have zero, one,"},{"Start":"03:33.345 ","End":"03:35.625","Text":"or two solutions,"},{"Start":"03:35.625 ","End":"03:38.510","Text":"we\u0027ll analyze that in more detail."},{"Start":"03:38.510 ","End":"03:41.350","Text":"I\u0027ll just letting you know in advance."},{"Start":"03:41.350 ","End":"03:44.010","Text":"What is the Law of Sines."},{"Start":"03:44.010 ","End":"03:47.300","Text":"The Law of Sines says is in any triangle,"},{"Start":"03:47.300 ","End":"03:49.475","Text":"it doesn\u0027t have to be oblique."},{"Start":"03:49.475 ","End":"03:56.750","Text":"The following double equation holds that a over sine Alpha"},{"Start":"03:56.750 ","End":"04:06.400","Text":"equals b over sine of Beta equals c over sine of Gamma."},{"Start":"04:06.400 ","End":"04:12.710","Text":"In other words, each side is proportional to the sine of the angle opposite."},{"Start":"04:12.710 ","End":"04:15.230","Text":"That\u0027s the law of sines."},{"Start":"04:15.230 ","End":"04:20.640","Text":"The Law of Cosine says that"},{"Start":"04:29.650 ","End":"04:35.905","Text":"c^2 equals a^2 plus b^2 minus 2ab Cos Gamma."},{"Start":"04:35.905 ","End":"04:39.160","Text":"The cosine is what gives it the name, the Cosine Law."},{"Start":"04:39.160 ","End":"04:42.995","Text":"Notice that if Gamma is 90 degrees,"},{"Start":"04:42.995 ","End":"04:45.605","Text":"cosine of Gamma is 0,"},{"Start":"04:45.605 ","End":"04:49.610","Text":"this bit drops out and we get Pythagoras theorem."},{"Start":"04:49.610 ","End":"04:51.005","Text":"Just thought I\u0027d mention it."},{"Start":"04:51.005 ","End":"04:53.420","Text":"Now, there\u0027s something not quite symmetric about this."},{"Start":"04:53.420 ","End":"04:55.910","Text":"We have C on one side and a and b on the other."},{"Start":"04:55.910 ","End":"05:00.420","Text":"Well, there\u0027s a thing called rotation of letters,"},{"Start":"05:00.420 ","End":"05:06.200","Text":"we can use any of the three letters here, it doesn\u0027t really matter."},{"Start":"05:06.200 ","End":"05:07.730","Text":"There\u0027s no special role of a, b,"},{"Start":"05:07.730 ","End":"05:11.069","Text":"and c. I could write, for example,"},{"Start":"05:11.069 ","End":"05:15.750","Text":"that a^2 and then it\u0027s the squares of the other 2,"},{"Start":"05:18.250 ","End":"05:22.855","Text":"b^2 plus c^2 minus 2bc Cosine Alpha."},{"Start":"05:22.855 ","End":"05:28.263","Text":"We could also write that b^2"},{"Start":"05:28.263 ","End":"05:34.350","Text":"equals a^2 minus c^2 minus 2ac Cosine of Beta."},{"Start":"05:34.350 ","End":"05:39.379","Text":"They are all equivalent is just rotation of letters."},{"Start":"05:39.379 ","End":"05:45.440","Text":"Now, this cosine rule is sometimes written in an alternate form."},{"Start":"05:45.440 ","End":"05:51.770","Text":"I\u0027ll just do it for the first form although it also exists for the other two."},{"Start":"05:51.770 ","End":"05:55.310","Text":"Is if we want the angle, if you think about it."},{"Start":"05:55.310 ","End":"06:00.095","Text":"If I bring this over to the left side and c^2 to the other side and then divide,"},{"Start":"06:00.095 ","End":"06:06.830","Text":"I can get from here that cosine of Gamma is equal"},{"Start":"06:06.830 ","End":"06:15.424","Text":"to a^2 plus b^2"},{"Start":"06:15.424 ","End":"06:19.975","Text":"minus c^2 over 2ab and you\u0027d get similar rules for this and this,"},{"Start":"06:19.975 ","End":"06:22.440","Text":"I just won\u0027t bother."},{"Start":"06:23.300 ","End":"06:28.760","Text":"I can tell you more that you see we have two cases for the Law of Cosine,"},{"Start":"06:28.760 ","End":"06:30.635","Text":"SAS, and SSS,"},{"Start":"06:30.635 ","End":"06:38.075","Text":"that this to the left of the arrow is the form that you would use for"},{"Start":"06:38.075 ","End":"06:47.735","Text":"SAS and this form on the right of the arrow is the SSS."},{"Start":"06:47.735 ","End":"06:49.430","Text":"Because you see then we would have a, b,"},{"Start":"06:49.430 ","End":"06:53.160","Text":"and c and we will be able to compute the angle from it."},{"Start":"06:53.350 ","End":"07:00.200","Text":"Next, I\u0027d like to return to this ambiguous case it\u0027s called."},{"Start":"07:00.200 ","End":"07:02.915","Text":"Here\u0027s a picture."},{"Start":"07:02.915 ","End":"07:07.580","Text":"The ambiguous case is when we\u0027re given two sides and"},{"Start":"07:07.580 ","End":"07:13.190","Text":"an angle and we\u0027re going to assume here that it\u0027s a,"},{"Start":"07:13.190 ","End":"07:22.190","Text":"b, and Alpha that are given two sides and an angle that\u0027s not included."},{"Start":"07:22.190 ","End":"07:27.785","Text":"We see that there\u0027s actually four general possibilities that could be,"},{"Start":"07:27.785 ","End":"07:34.710","Text":"we start by creating an angle of Alpha from the horizontal and going length b,"},{"Start":"07:34.710 ","End":"07:37.655","Text":"and then we have the length a,"},{"Start":"07:37.655 ","End":"07:38.690","Text":"but we don\u0027t know the angle,"},{"Start":"07:38.690 ","End":"07:42.350","Text":"so we make an arc and see if it hits here."},{"Start":"07:42.350 ","End":"07:44.405","Text":"Well, it might not hit at all."},{"Start":"07:44.405 ","End":"07:47.490","Text":"On the other hand, it might hit just once,"},{"Start":"07:47.490 ","End":"07:54.190","Text":"so that would be the case where this is 90 degrees."},{"Start":"07:54.190 ","End":"07:59.855","Text":"It might actually hit twice and this is the ambiguous case,"},{"Start":"07:59.855 ","End":"08:03.020","Text":"well, the case where there\u0027s two solutions."},{"Start":"08:03.020 ","End":"08:10.065","Text":"It could also be that it hits the extended line twice,"},{"Start":"08:10.065 ","End":"08:14.550","Text":"but only once on this side and that\u0027s also one solution."},{"Start":"08:14.550 ","End":"08:17.640","Text":"Like here there\u0027s zero solution, here,"},{"Start":"08:17.640 ","End":"08:18.880","Text":"there\u0027s one solution,"},{"Start":"08:18.880 ","End":"08:21.740","Text":"here there\u0027s two solutions, here\u0027s one solution."},{"Start":"08:21.740 ","End":"08:24.530","Text":"The question is what happens in practice?"},{"Start":"08:24.530 ","End":"08:30.935","Text":"Well, in practice, what we\u0027re going to do is look for the angle Beta."},{"Start":"08:30.935 ","End":"08:33.470","Text":"I\u0027ll just mark it here, here also,"},{"Start":"08:33.470 ","End":"08:36.635","Text":"and here there is no Beta here."},{"Start":"08:36.635 ","End":"08:38.120","Text":"What we do is,"},{"Start":"08:38.120 ","End":"08:42.795","Text":"we use the Law of Sines to say that"},{"Start":"08:42.795 ","End":"08:50.400","Text":"a/sine Alpha equals b/sine Beta."},{"Start":"08:50.400 ","End":"08:53.025","Text":"Remember, we\u0027re given a, b and Alpha."},{"Start":"08:53.025 ","End":"08:56.175","Text":"Beta or Beta is the unknown,"},{"Start":"08:56.175 ","End":"09:02.610","Text":"so we get that sine Beta equals this times this over this,"},{"Start":"09:02.610 ","End":"09:08.765","Text":"b times sine Alpha over a."},{"Start":"09:08.765 ","End":"09:16.560","Text":"Now, this quantity could be bigger than 1 equal to 1 or less than 1."},{"Start":"09:16.560 ","End":"09:21.375","Text":"If we get that sine Beta is bigger than 1,"},{"Start":"09:21.375 ","End":"09:30.690","Text":"then there is no solution because the sine is never bigger than 1."},{"Start":"09:30.690 ","End":"09:35.625","Text":"It\u0027s always between 1 and minus 1, inclusive."},{"Start":"09:35.625 ","End":"09:41.885","Text":"If we get the sine Beta equals exactly 1,"},{"Start":"09:41.885 ","End":"09:49.880","Text":"then we know that Beta is 90 degrees or Pi/2,"},{"Start":"09:49.880 ","End":"09:52.209","Text":"and that\u0027s this case."},{"Start":"09:52.209 ","End":"10:01.490","Text":"But if sine Beta is less than 1 then this two solutions."},{"Start":"10:01.490 ","End":"10:03.415","Text":"Let me give you an example."},{"Start":"10:03.415 ","End":"10:06.800","Text":"Suppose we say that sine Beta equals 1/2,"},{"Start":"10:07.100 ","End":"10:13.680","Text":"then Beta sine inverse or arc sine of 1/2,"},{"Start":"10:13.680 ","End":"10:17.855","Text":"and on the calculator will tell you 30 degrees if you do it in degrees."},{"Start":"10:17.855 ","End":"10:27.130","Text":"But it only gives you the principle branch which is from minus 90 to 90."},{"Start":"10:27.130 ","End":"10:32.975","Text":"What we have to do to get the other value is subtract from 180."},{"Start":"10:32.975 ","End":"10:39.560","Text":"If we do 180 minus 30 equals 150 degrees,"},{"Start":"10:39.560 ","End":"10:44.735","Text":"the sine of a 150 degrees is also 1/2."},{"Start":"10:44.735 ","End":"10:51.660","Text":"What we have in general is that we get two values of Beta."},{"Start":"10:51.660 ","End":"10:53.460","Text":"We get one value of Beta,"},{"Start":"10:53.460 ","End":"10:55.015","Text":"call it Beta_1,"},{"Start":"10:55.015 ","End":"11:00.185","Text":"which is the inverse sine of whatever it was here,"},{"Start":"11:00.185 ","End":"11:07.555","Text":"of b sine Alpha over a."},{"Start":"11:07.555 ","End":"11:13.205","Text":"We have Beta_2 is 180 degrees,"},{"Start":"11:13.205 ","End":"11:16.140","Text":"working in degrees otherwise Pi minus Beta_1."},{"Start":"11:18.290 ","End":"11:21.510","Text":"Now, it doesn\u0027t mean that we have two solutions,"},{"Start":"11:21.510 ","End":"11:24.110","Text":"there\u0027s more because think about Gamma."},{"Start":"11:24.110 ","End":"11:29.720","Text":"Now, we have Alpha given and we have two possible values of Beta,"},{"Start":"11:29.720 ","End":"11:39.330","Text":"but we have to take Gamma is equaling 180 degrees minus Alpha minus Beta,"},{"Start":"11:39.330 ","End":"11:42.360","Text":"this will be Beta_1 or Beta_2,"},{"Start":"11:42.360 ","End":"11:43.800","Text":"one time I\u0027ll take Beta_1,"},{"Start":"11:43.800 ","End":"11:45.615","Text":"and one time I\u0027ll take Beta_2,"},{"Start":"11:45.615 ","End":"11:50.415","Text":"and that will give us Gamma_1 and Gamma _2."},{"Start":"11:50.415 ","End":"11:53.330","Text":"I won\u0027t be lazy, I\u0027ll write it all out."},{"Start":"11:53.330 ","End":"11:57.200","Text":"Then each of these might be negative."},{"Start":"11:57.200 ","End":"11:58.850","Text":"If it\u0027s negative,"},{"Start":"11:58.850 ","End":"12:02.180","Text":"then we have to rule it out."},{"Start":"12:02.180 ","End":"12:05.449","Text":"It could be that from these two possibilities,"},{"Start":"12:05.449 ","End":"12:10.740","Text":"only 1 or 0 of them pan out."},{"Start":"12:10.740 ","End":"12:13.440","Text":"It could be 0, 1, or 2."},{"Start":"12:13.440 ","End":"12:20.350","Text":"It\u0027s still very theoretical and the tutorial is being long."},{"Start":"12:20.350 ","End":"12:29.015","Text":"I\u0027ll leave the solved examples for afterwards and not as part of the tutorial."},{"Start":"12:29.015 ","End":"12:30.920","Text":"I\u0027m going to take a break here."},{"Start":"12:30.920 ","End":"12:35.240","Text":"There is another topic or two on angles I\u0027d like to cover,"},{"Start":"12:35.240 ","End":"12:37.710","Text":"but after the break."}],"ID":10898},{"Watched":false,"Name":"Triangles - Part 3","Duration":"4m 30s","ChapterTopicVideoID":10479,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10479.jpeg","UploadDate":"2021-06-29T13:16:39.8030000","DurationForVideoObject":"PT4M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.495","Text":"This will be a short clip to conclude the chapter on triangles."},{"Start":"00:06.495 ","End":"00:11.610","Text":"Not directly related to triangles or solving them,"},{"Start":"00:11.610 ","End":"00:17.220","Text":"but related to angles and just had to put them somewhere."},{"Start":"00:17.220 ","End":"00:21.555","Text":"There\u0027ll be 2 topics. First of them is bearing,"},{"Start":"00:21.555 ","End":"00:23.855","Text":"also known as heading."},{"Start":"00:23.855 ","End":"00:30.525","Text":"It\u0027s very important to me to note that these definitions are not standard."},{"Start":"00:30.525 ","End":"00:32.715","Text":"I\u0027m following 1 book,"},{"Start":"00:32.715 ","End":"00:37.230","Text":"but other books have different definitions and so you"},{"Start":"00:37.230 ","End":"00:42.590","Text":"have to be careful find out from your professor which term you use."},{"Start":"00:42.590 ","End":"00:49.640","Text":"I\u0027m going to use the words bearing and heading to mean the same thing."},{"Start":"00:49.920 ","End":"00:59.020","Text":"A bearing is a term used in navigation to indicate a direction and there\u0027s 2 of them,"},{"Start":"00:59.020 ","End":"01:04.540","Text":"let me just focus on one of them first."},{"Start":"01:04.540 ","End":"01:12.930","Text":"One way of indicating a direction is to see where we are relative to north or south."},{"Start":"01:12.930 ","End":"01:17.355","Text":"If it\u0027s got a southern component,"},{"Start":"01:17.355 ","End":"01:25.015","Text":"then we say what the angle is from the south towards either the east or the west."},{"Start":"01:25.015 ","End":"01:29.545","Text":"In this case, this is the direction we\u0027re talking about."},{"Start":"01:29.545 ","End":"01:32.435","Text":"Since it\u0027s in this quadrant,"},{"Start":"01:32.435 ","End":"01:35.750","Text":"it\u0027s from the south towards the east and then you"},{"Start":"01:35.750 ","End":"01:39.440","Text":"measure this angle and say it comes out 34 degrees."},{"Start":"01:39.440 ","End":"01:43.475","Text":"This direction is called a bearing of,"},{"Start":"01:43.475 ","End":"01:44.960","Text":"I don\u0027t know how to pronounce it,"},{"Start":"01:44.960 ","End":"01:50.240","Text":"but you write it as S34 degrees E. From south,"},{"Start":"01:50.240 ","End":"01:53.465","Text":"34 degrees towards the east."},{"Start":"01:53.465 ","End":"01:55.880","Text":"These full combinations of letters,"},{"Start":"01:55.880 ","End":"01:57.310","Text":"it could be from south to east,"},{"Start":"01:57.310 ","End":"01:58.910","Text":"from south towards the west,"},{"Start":"01:58.910 ","End":"02:02.225","Text":"from north towards the east or north towards the west."},{"Start":"02:02.225 ","End":"02:05.285","Text":"That\u0027s one way of doing it."},{"Start":"02:05.285 ","End":"02:11.390","Text":"The other way is just to measure the angle from the north."},{"Start":"02:11.390 ","End":"02:16.345","Text":"The north is considered to be 0 degrees and clockwise,"},{"Start":"02:16.345 ","End":"02:19.670","Text":"we rotate clockwise until we get to the direction,"},{"Start":"02:19.670 ","End":"02:23.970","Text":"in this case, 241 degrees."},{"Start":"02:24.980 ","End":"02:28.910","Text":"Bearing in this format will always be between 0 and"},{"Start":"02:28.910 ","End":"02:35.210","Text":"360 and we just write a bearing of however many degrees it is."},{"Start":"02:35.210 ","End":"02:40.055","Text":"Notice that this is not the mathematical convention."},{"Start":"02:40.055 ","End":"02:44.060","Text":"In navigation we start from the north and turn clockwise."},{"Start":"02:44.060 ","End":"02:47.750","Text":"In mathematics, we start from the east and go counterclockwise,"},{"Start":"02:47.750 ","End":"02:50.160","Text":"so just bear that in mind."},{"Start":"02:50.160 ","End":"02:53.330","Text":"This is used in word problems and I won\u0027t say"},{"Start":"02:53.330 ","End":"02:56.945","Text":"anymore on this just you know the definition,"},{"Start":"02:56.945 ","End":"02:59.390","Text":"and I repeat this is not standard definitions,"},{"Start":"02:59.390 ","End":"03:01.685","Text":"they vary from book to book."},{"Start":"03:01.685 ","End":"03:11.045","Text":"The next term or pair of terms will be angles of elevation and depression."},{"Start":"03:11.045 ","End":"03:17.210","Text":"Separately there\u0027s elevation and there\u0027s depression and here I put both pictures in."},{"Start":"03:17.210 ","End":"03:25.335","Text":"An angle of elevation is usually relative to an observer and"},{"Start":"03:25.335 ","End":"03:30.720","Text":"it\u0027s what angle the object forms"},{"Start":"03:30.720 ","End":"03:36.025","Text":"with the horizontal line if it\u0027s above the horizontal."},{"Start":"03:36.025 ","End":"03:38.815","Text":"So if we have an object here and the observer here,"},{"Start":"03:38.815 ","End":"03:42.400","Text":"this is the angle of elevation."},{"Start":"03:42.400 ","End":"03:44.530","Text":"Elevation because it\u0027s elevated,"},{"Start":"03:44.530 ","End":"03:45.955","Text":"it\u0027s above the horizontal."},{"Start":"03:45.955 ","End":"03:52.930","Text":"Similarly for doing maybe some measurements in this valley or something like that,"},{"Start":"03:52.930 ","End":"03:57.520","Text":"or you\u0027re from the top of the building looking down and you see an object,"},{"Start":"03:57.520 ","End":"04:01.650","Text":"and it\u0027s below the horizontal line,"},{"Start":"04:01.650 ","End":"04:08.240","Text":"then this angle from the observer to the object,"},{"Start":"04:08.240 ","End":"04:13.275","Text":"this angle that it makes with the horizontal is called the angle of depression."},{"Start":"04:13.275 ","End":"04:17.360","Text":"They\u0027re both positive acute angles."},{"Start":"04:18.770 ","End":"04:22.100","Text":"This might also be used in word problems."},{"Start":"04:22.100 ","End":"04:24.365","Text":"I just wanted to introduce you to the terms."},{"Start":"04:24.365 ","End":"04:26.435","Text":"That\u0027s all I have to say."},{"Start":"04:26.435 ","End":"04:29.760","Text":"That concludes the chapter."}],"ID":10899},{"Watched":false,"Name":"Exercise 1","Duration":"3m 54s","ChapterTopicVideoID":10404,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10404.jpeg","UploadDate":"2017-11-02T16:06:42.8270000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, we have a right triangle, A, B,"},{"Start":"00:05.160 ","End":"00:08.010","Text":"C. Well, obviously,"},{"Start":"00:08.010 ","End":"00:11.775","Text":"this is the right angle and we don\u0027t usually bother to label it."},{"Start":"00:11.775 ","End":"00:15.210","Text":"This would be Gamma and it\u0027s 90 degrees."},{"Start":"00:15.210 ","End":"00:20.430","Text":"Now, we\u0027re given Alpha and we\u0027re given the side a here."},{"Start":"00:20.430 ","End":"00:21.960","Text":"We have to solve the triangle,"},{"Start":"00:21.960 ","End":"00:26.685","Text":"which means find everything else except the right angle, which is obvious."},{"Start":"00:26.685 ","End":"00:30.475","Text":"I mean, you could just repeat Gamma equals 90 if you want."},{"Start":"00:30.475 ","End":"00:34.190","Text":"Also, note that we are working in degrees."},{"Start":"00:34.190 ","End":"00:38.635","Text":"So that be sure to set your calculator to degree mode."},{"Start":"00:38.635 ","End":"00:43.010","Text":"If you pedantic you could write Gamma equals 90 degrees."},{"Start":"00:43.010 ","End":"00:44.630","Text":"We usually don\u0027t require that."},{"Start":"00:44.630 ","End":"00:46.625","Text":"It\u0027s given a right angle triangle."},{"Start":"00:46.625 ","End":"00:48.650","Text":"Let\u0027s see what we want to go for next."},{"Start":"00:48.650 ","End":"00:50.720","Text":"We have to find all the unknown quantities."},{"Start":"00:50.720 ","End":"00:54.070","Text":"We have Alpha and a. Let\u0027s go for B."},{"Start":"00:54.070 ","End":"00:57.390","Text":"Now, you know what,"},{"Start":"00:57.390 ","End":"01:00.150","Text":"we will go for Beta first, it\u0027s easier."},{"Start":"01:00.150 ","End":"01:04.560","Text":"Alpha plus Beta equals 90 degrees."},{"Start":"01:04.560 ","End":"01:06.960","Text":"Alpha and Beta are complementary angles."},{"Start":"01:06.960 ","End":"01:09.645","Text":"This is true in any right-angle triangle."},{"Start":"01:09.645 ","End":"01:17.900","Text":"We can get that Beta is equal to 90 minus 34.5,"},{"Start":"01:17.900 ","End":"01:24.990","Text":"which is Alpha, and that is equal to 55.5 degrees."},{"Start":"01:24.990 ","End":"01:27.875","Text":"We found Beta. We have a,"},{"Start":"01:27.875 ","End":"01:31.385","Text":"so we need to find b and c. Let\u0027s go for b."},{"Start":"01:31.385 ","End":"01:35.120","Text":"Now, we can connect b, the unknown,"},{"Start":"01:35.120 ","End":"01:36.500","Text":"with two known quantities,"},{"Start":"01:36.500 ","End":"01:39.350","Text":"a and Alpha by using the tangent."},{"Start":"01:39.350 ","End":"01:47.435","Text":"In fact, tangent of Alpha is a/b opposite over adjacent."},{"Start":"01:47.435 ","End":"01:50.990","Text":"Since we want b and we extract it from here,"},{"Start":"01:50.990 ","End":"01:57.610","Text":"we get b is a over tangent Alpha."},{"Start":"01:58.160 ","End":"02:03.350","Text":"We can do this on the calculator and say b is a,"},{"Start":"02:03.350 ","End":"02:09.050","Text":"which is 67.8 units over"},{"Start":"02:09.050 ","End":"02:15.770","Text":"the tangent of 34.5 degrees."},{"Start":"02:15.770 ","End":"02:20.735","Text":"Remember to set to degrees on the calculator,"},{"Start":"02:20.735 ","End":"02:22.880","Text":"and in my calculator,"},{"Start":"02:22.880 ","End":"02:31.115","Text":"it\u0027s 98.649 and a few more digits degrees."},{"Start":"02:31.115 ","End":"02:36.920","Text":"Now, the only thing remaining is c. You could,"},{"Start":"02:36.920 ","End":"02:41.330","Text":"in principle, use Pythagoras\u0027 theorem using a and b."},{"Start":"02:41.330 ","End":"02:44.210","Text":"But b is not original, fresh data,"},{"Start":"02:44.210 ","End":"02:48.500","Text":"it\u0027s computed data and errors tend to accumulate so it\u0027s better"},{"Start":"02:48.500 ","End":"02:53.000","Text":"if we can just use the original a and Alpha and we can,"},{"Start":"02:53.000 ","End":"02:56.330","Text":"just like we use the tangent to find b,"},{"Start":"02:56.330 ","End":"02:59.500","Text":"because tangent of Alpha as a/b,"},{"Start":"02:59.500 ","End":"03:03.315","Text":"the sine of Alpha is a/c opposite over hypotenuse."},{"Start":"03:03.315 ","End":"03:11.820","Text":"Let\u0027s write that. The sine of Alpha is a/c."},{"Start":"03:11.820 ","End":"03:17.995","Text":"If we extract c it\u0027s a over sine Alpha."},{"Start":"03:17.995 ","End":"03:22.895","Text":"That gives us 67.8"},{"Start":"03:22.895 ","End":"03:29.780","Text":"over sine of 34.5 degrees and once again,"},{"Start":"03:29.780 ","End":"03:32.815","Text":"on the calculator set for degrees."},{"Start":"03:32.815 ","End":"03:38.160","Text":"I make it 119.702,"},{"Start":"03:38.160 ","End":"03:44.315","Text":"etc., degrees, and that\u0027s it."},{"Start":"03:44.315 ","End":"03:45.620","Text":"Unless as I said,"},{"Start":"03:45.620 ","End":"03:49.310","Text":"you want to write Gamma equals 90 degrees,"},{"Start":"03:49.310 ","End":"03:50.825","Text":"but I wouldn\u0027t bother."},{"Start":"03:50.825 ","End":"03:54.990","Text":"All the quantities have been found and we are done."}],"ID":10764},{"Watched":false,"Name":"Exercise 2","Duration":"5m 34s","ChapterTopicVideoID":10405,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10405.jpeg","UploadDate":"2017-11-02T16:07:02.6070000","DurationForVideoObject":"PT5M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.269","Text":"In this exercise, we\u0027re given a right triangle ABC."},{"Start":"00:04.269 ","End":"00:06.225","Text":"It\u0027s not necessarily to scale."},{"Start":"00:06.225 ","End":"00:08.580","Text":"It doesn\u0027t say here,"},{"Start":"00:08.580 ","End":"00:13.110","Text":"but this is the 90 degrees,"},{"Start":"00:13.110 ","End":"00:15.015","Text":"and this is Gamma."},{"Start":"00:15.015 ","End":"00:20.340","Text":"We don\u0027t bother saying Gamma equals 90 degrees, so that\u0027s assumed."},{"Start":"00:20.340 ","End":"00:28.350","Text":"What we\u0027re given here are the 2 sides, c and a."},{"Start":"00:28.350 ","End":"00:30.750","Text":"We\u0027re not given any of the angles."},{"Start":"00:30.750 ","End":"00:33.480","Text":"Our task is to compute the unknown."},{"Start":"00:33.480 ","End":"00:34.965","Text":"In other words, you have to find b,"},{"Start":"00:34.965 ","End":"00:37.005","Text":"and Alpha, and Beta."},{"Start":"00:37.005 ","End":"00:40.299","Text":"That\u0027s solving the triangle."},{"Start":"00:40.299 ","End":"00:48.965","Text":"The extra thing is that we want to convert the Alpha and Beta into degrees and minutes,"},{"Start":"00:48.965 ","End":"00:51.694","Text":"as well as in decimal."},{"Start":"00:51.694 ","End":"00:58.985","Text":"The most obvious thing to do is to somehow use a and c to find 1 of these angles."},{"Start":"00:58.985 ","End":"01:00.860","Text":"Now, we could do either."},{"Start":"01:00.860 ","End":"01:05.855","Text":"We could say that sine Alpha is a over c, opposite over hypotenuse."},{"Start":"01:05.855 ","End":"01:07.430","Text":"But if you\u0027re looking from here,"},{"Start":"01:07.430 ","End":"01:10.550","Text":"you could also say cosine Beta is adjacent over"},{"Start":"01:10.550 ","End":"01:13.790","Text":"hypotenuse is a over c. Doesn\u0027t really matter."},{"Start":"01:13.790 ","End":"01:23.600","Text":"I\u0027ll go with sine Alpha is a over c. Therefore,"},{"Start":"01:23.600 ","End":"01:30.480","Text":"this is equal to, what is a?"},{"Start":"01:30.480 ","End":"01:38.640","Text":"5.43 over 7.65."},{"Start":"01:38.640 ","End":"01:41.475","Text":"The way I would get Alpha,"},{"Start":"01:41.475 ","End":"01:42.900","Text":"I won\u0027t compute this just yet,"},{"Start":"01:42.900 ","End":"01:44.285","Text":"I\u0027ll do 1 computation,"},{"Start":"01:44.285 ","End":"01:47.390","Text":"is to take the inverse sine, arcsine,"},{"Start":"01:47.390 ","End":"01:52.440","Text":"or sine minus 1 of this,"},{"Start":"01:52.440 ","End":"01:54.395","Text":"which I copy-pasted,"},{"Start":"01:54.395 ","End":"01:57.530","Text":"and then do it on the calculator."},{"Start":"01:57.530 ","End":"02:00.470","Text":"Remember to set the calculator to degrees."},{"Start":"02:00.470 ","End":"02:04.950","Text":"Comes out to 45.219."},{"Start":"02:08.510 ","End":"02:11.160","Text":"I rounded to 3 decimal places,"},{"Start":"02:11.160 ","End":"02:14.230","Text":"it\u0027s actually 0.89 something."},{"Start":"02:14.230 ","End":"02:18.020","Text":"Then we want to convert that to degrees and minutes."},{"Start":"02:18.020 ","End":"02:20.000","Text":"To convert to minutes,"},{"Start":"02:20.000 ","End":"02:24.500","Text":"I take this 219 or the"},{"Start":"02:24.500 ","End":"02:30.930","Text":"0.219 and write it as 219 over 1000,"},{"Start":"02:30.930 ","End":"02:37.820","Text":"and I have to take this portion of 60 minutes because 1 degree is 60 minutes."},{"Start":"02:40.490 ","End":"02:45.400","Text":"This comes out to 13.14."},{"Start":"02:45.650 ","End":"02:49.050","Text":"But when they say degrees and minutes,"},{"Start":"02:49.050 ","End":"02:51.600","Text":"round off to a whole number of minutes."},{"Start":"02:51.600 ","End":"03:01.135","Text":"I would write this as 45 degrees and just take the 13 minutes, nearest minute."},{"Start":"03:01.135 ","End":"03:03.330","Text":"What are we left with?"},{"Start":"03:03.330 ","End":"03:11.670","Text":"We have Alpha."},{"Start":"03:11.670 ","End":"03:13.755","Text":"That\u0027s all we have actually."},{"Start":"03:13.755 ","End":"03:16.395","Text":"We need to find Beta."},{"Start":"03:16.395 ","End":"03:22.110","Text":"That\u0027s easy because Alpha plus Beta is 90 degrees."},{"Start":"03:22.110 ","End":"03:25.500","Text":"These 2 are complementary angles in a right triangle."},{"Start":"03:25.500 ","End":"03:33.030","Text":"We can get that Beta is 90 minus,"},{"Start":"03:33.030 ","End":"03:35.820","Text":"we\u0027ll just continue working with degrees,"},{"Start":"03:35.820 ","End":"03:40.605","Text":"45 degrees and 13 minutes."},{"Start":"03:40.605 ","End":"03:43.845","Text":"That comes out to be,"},{"Start":"03:43.845 ","End":"03:50.275","Text":"do it on the calculator, but I mentally say this is 89 degrees and 60 minutes."},{"Start":"03:50.275 ","End":"03:58.905","Text":"We get 44 degrees and 47 minutes."},{"Start":"03:58.905 ","End":"04:01.650","Text":"That\u0027s Alpha, that\u0027s Beta."},{"Start":"04:01.650 ","End":"04:04.695","Text":"We just have to find b."},{"Start":"04:04.695 ","End":"04:07.365","Text":"Now, how are we going to find b?"},{"Start":"04:07.365 ","End":"04:16.170","Text":"I prefer to use original data rather than computed data. Let\u0027s see."},{"Start":"04:16.540 ","End":"04:18.979","Text":"The angles were computed,"},{"Start":"04:18.979 ","End":"04:20.480","Text":"the sides were given."},{"Start":"04:20.480 ","End":"04:23.665","Text":"These 3 are related by Pythagoras\u0027s theorem."},{"Start":"04:23.665 ","End":"04:25.265","Text":"I could say, yes,"},{"Start":"04:25.265 ","End":"04:27.890","Text":"a^2 plus b^2 is c^2."},{"Start":"04:27.890 ","End":"04:30.710","Text":"In other words, b is equal to,"},{"Start":"04:30.710 ","End":"04:32.390","Text":"if I extract just b,"},{"Start":"04:32.390 ","End":"04:37.265","Text":"the square root of c^2 minus a^2,"},{"Start":"04:37.265 ","End":"04:42.324","Text":"which is the square root of,"},{"Start":"04:42.324 ","End":"04:44.720","Text":"where is it now?"},{"Start":"04:44.720 ","End":"04:50.090","Text":"7.65^2 minus"},{"Start":"04:50.090 ","End":"04:55.950","Text":"5.43^2."},{"Start":"04:56.710 ","End":"05:00.200","Text":"Again, the calculator will do that."},{"Start":"05:00.200 ","End":"05:03.570","Text":"That will give us,"},{"Start":"05:03.880 ","End":"05:13.130","Text":"let\u0027s see, 5.3886 something."},{"Start":"05:13.130 ","End":"05:16.730","Text":"I think since the original data was given to 2 places,"},{"Start":"05:16.730 ","End":"05:23.010","Text":"we\u0027ll just approximate it and say 5.38."},{"Start":"05:23.030 ","End":"05:26.130","Text":"We found Alpha here,"},{"Start":"05:26.130 ","End":"05:28.530","Text":"we found Beta here,"},{"Start":"05:28.530 ","End":"05:31.890","Text":"and we found b here."},{"Start":"05:31.890 ","End":"05:34.450","Text":"That completes the picture."}],"ID":10765},{"Watched":false,"Name":"Exercise 3","Duration":"2m 34s","ChapterTopicVideoID":10406,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10406.jpeg","UploadDate":"2017-11-02T16:07:11.1300000","DurationForVideoObject":"PT2M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.350","Text":"Here we have a word problem involving the concept of angle of elevation of the sun."},{"Start":"00:07.350 ","End":"00:09.825","Text":"A pole casts a shadow."},{"Start":"00:09.825 ","End":"00:12.840","Text":"Here\u0027s a little diagram of the pole and it\u0027s"},{"Start":"00:12.840 ","End":"00:16.620","Text":"casting a shadow when the sun is coming from this direction."},{"Start":"00:16.620 ","End":"00:20.955","Text":"The angle of elevation is the one here."},{"Start":"00:20.955 ","End":"00:23.730","Text":"It\u0027s 28 degrees,"},{"Start":"00:23.730 ","End":"00:25.680","Text":"this may not be to scale."},{"Start":"00:25.680 ","End":"00:28.680","Text":"We have to find the height of the pole."},{"Start":"00:28.680 ","End":"00:30.390","Text":"As usual in these problems,"},{"Start":"00:30.390 ","End":"00:32.490","Text":"we assume the ground is level,"},{"Start":"00:32.490 ","End":"00:35.500","Text":"not on a slope and so on."},{"Start":"00:35.980 ","End":"00:40.040","Text":"I think it\u0027ll be easier if we make this a bit abstract,"},{"Start":"00:40.040 ","End":"00:44.300","Text":"and they brought this more general right triangle."},{"Start":"00:44.300 ","End":"00:47.765","Text":"This of course is the 90 degree one gamma."},{"Start":"00:47.765 ","End":"00:57.480","Text":"What we\u0027re given is Alpha here is equal to 28 degrees,"},{"Start":"00:57.480 ","End":"01:04.390","Text":"and we\u0027re given that b=12,"},{"Start":"01:05.000 ","End":"01:12.035","Text":"and we have to find out what a is equal to."},{"Start":"01:12.035 ","End":"01:14.270","Text":"The simplest thing to do,"},{"Start":"01:14.270 ","End":"01:17.180","Text":"is to find the formula involving a,"},{"Start":"01:17.180 ","End":"01:18.320","Text":"b, and Alpha,"},{"Start":"01:18.320 ","End":"01:22.430","Text":"and I think you\u0027ll quickly realize it\u0027s the tangent."},{"Start":"01:22.430 ","End":"01:27.495","Text":"The tangent of Alpha will provide the connection."},{"Start":"01:27.495 ","End":"01:33.140","Text":"Tangent Alpha opposite over adjacent is a over b."},{"Start":"01:33.140 ","End":"01:40.040","Text":"Which means that if we want a isolated,"},{"Start":"01:40.040 ","End":"01:44.570","Text":"it\u0027s equal to b times tangent alpha."},{"Start":"01:44.570 ","End":"01:47.585","Text":"Just multiply both sides by b and switch sides."},{"Start":"01:47.585 ","End":"01:52.690","Text":"B, we have Alpha we have, so a=12,"},{"Start":"01:52.690 ","End":"01:55.785","Text":"that\u0027s the 12 here,"},{"Start":"01:55.785 ","End":"02:03.580","Text":"times tangent of 28 degrees."},{"Start":"02:03.800 ","End":"02:06.965","Text":"Brackets are not necessary here."},{"Start":"02:06.965 ","End":"02:09.785","Text":"Then using the calculator,"},{"Start":"02:09.785 ","End":"02:12.965","Text":"remembering to set it to degree mode,"},{"Start":"02:12.965 ","End":"02:19.340","Text":"make it 6.380 something."},{"Start":"02:19.340 ","End":"02:21.920","Text":"I think 3 decimal places is enough."},{"Start":"02:21.920 ","End":"02:24.780","Text":"That\u0027s down to millimeters."},{"Start":"02:25.020 ","End":"02:28.090","Text":"That\u0027s it, we\u0027re done."},{"Start":"02:28.090 ","End":"02:34.220","Text":"Maybe I should just say meters. Now we\u0027re done."}],"ID":10766},{"Watched":false,"Name":"Exercise 4","Duration":"4m 58s","ChapterTopicVideoID":10407,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10407.jpeg","UploadDate":"2017-11-02T16:07:31.2870000","DurationForVideoObject":"PT4M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"Here we have an exercise which is a bit of a hybrid."},{"Start":"00:03.720 ","End":"00:10.793","Text":"It\u0027s distance, speed, time word problem combined with the concept,"},{"Start":"00:10.793 ","End":"00:13.530","Text":"and this is the important one, of bearing."},{"Start":"00:13.530 ","End":"00:15.030","Text":"Let\u0027s see."},{"Start":"00:15.030 ","End":"00:17.175","Text":"A ship leaves the port,"},{"Start":"00:17.175 ","End":"00:19.590","Text":"let\u0027s say this is the port,"},{"Start":"00:19.590 ","End":"00:28.350","Text":"and it sails at an average speed of 70 kilometers per hour at a bearing of,"},{"Start":"00:28.350 ","End":"00:30.405","Text":"how do we say this?"},{"Start":"00:30.405 ","End":"00:31.680","Text":"I\u0027m not sure how you pronounce it,"},{"Start":"00:31.680 ","End":"00:35.685","Text":"but it\u0027s you go from the south direction,"},{"Start":"00:35.685 ","End":"00:39.494","Text":"20 degrees towards the east."},{"Start":"00:39.494 ","End":"00:43.060","Text":"Something like this."},{"Start":"00:43.740 ","End":"00:49.480","Text":"So the ship goes in this direction where this would"},{"Start":"00:49.480 ","End":"00:56.390","Text":"be 20 degrees from the south towards the east."},{"Start":"00:56.390 ","End":"01:00.490","Text":"It goes along at this speed for this many hours."},{"Start":"01:00.490 ","End":"01:06.370","Text":"That\u0027s easy to find the distance, which will do first."},{"Start":"01:06.370 ","End":"01:10.480","Text":"Later we\u0027ll worry about the how far south and how far east."},{"Start":"01:10.480 ","End":"01:12.550","Text":"So the distance, let\u0027s call it D,"},{"Start":"01:12.550 ","End":"01:15.940","Text":"is how far the ship is from the port."},{"Start":"01:15.940 ","End":"01:19.690","Text":"Distance is speed times time."},{"Start":"01:19.690 ","End":"01:20.560","Text":"So it\u0027s going to be"},{"Start":"01:20.560 ","End":"01:30.450","Text":"70 kilometers per hour"},{"Start":"01:30.450 ","End":"01:35.510","Text":"times 4 hours."},{"Start":"01:35.510 ","End":"01:41.635","Text":"That will equal 280 kilometers."},{"Start":"01:41.635 ","End":"01:43.120","Text":"That\u0027s the first part."},{"Start":"01:43.120 ","End":"01:46.180","Text":"Now we get to the trigonometry."},{"Start":"01:46.180 ","End":"01:49.360","Text":"I just made this a bit larger,"},{"Start":"01:49.360 ","End":"01:51.370","Text":"the north, south, east,"},{"Start":"01:51.370 ","End":"01:54.385","Text":"west thing, the axis, so to speak."},{"Start":"01:54.385 ","End":"01:56.620","Text":"What I\u0027m going to do is make a right angle triangle."},{"Start":"01:56.620 ","End":"02:00.010","Text":"The ship would be here."},{"Start":"02:00.010 ","End":"02:04.915","Text":"This distance is the 280."},{"Start":"02:04.915 ","End":"02:07.885","Text":"To know how much east and how much south,"},{"Start":"02:07.885 ","End":"02:13.900","Text":"I just need to make a right angle triangle"},{"Start":"02:13.900 ","End":"02:21.285","Text":"and mark at this 90 degrees."},{"Start":"02:21.285 ","End":"02:26.810","Text":"Then the amount that it\u0027s south would be from here to here,"},{"Start":"02:26.810 ","End":"02:29.615","Text":"and the east would be from here to here."},{"Start":"02:29.615 ","End":"02:35.000","Text":"Of course, I could\u0027ve drop the perpendicular here and said from here to here,"},{"Start":"02:35.000 ","End":"02:36.955","Text":"but that\u0027s the same as from here to here."},{"Start":"02:36.955 ","End":"02:40.340","Text":"What we need to find out is what this is,"},{"Start":"02:40.340 ","End":"02:43.695","Text":"and that answers the question on the south."},{"Start":"02:43.695 ","End":"02:49.360","Text":"This length here answers the question on the east."},{"Start":"02:50.960 ","End":"02:54.020","Text":"Perhaps I will give them labels."},{"Start":"02:54.020 ","End":"02:56.795","Text":"This length here, I\u0027ll call it s for south,"},{"Start":"02:56.795 ","End":"03:00.540","Text":"and from here to here I\u0027ll call it e for east."},{"Start":"03:00.850 ","End":"03:05.615","Text":"Now we can use properties of the right-angled triangle."},{"Start":"03:05.615 ","End":"03:10.970","Text":"For example, we know that the sine of 20 degrees"},{"Start":"03:10.970 ","End":"03:16.670","Text":"is opposite over hypotenuse will be e over 280 and the cosine will be s over 280."},{"Start":"03:16.670 ","End":"03:18.800","Text":"Using that, we can find s and a."},{"Start":"03:18.800 ","End":"03:23.480","Text":"Look, the cosine of"},{"Start":"03:23.480 ","End":"03:33.780","Text":"20 degrees is adjacent over hypotenuse is s over 280."},{"Start":"03:33.780 ","End":"03:39.150","Text":"That makes s equals"},{"Start":"03:39.150 ","End":"03:45.245","Text":"280 times cosine of 20 degrees."},{"Start":"03:45.245 ","End":"03:47.240","Text":"If we do the computation,"},{"Start":"03:47.240 ","End":"03:50.945","Text":"I\u0027ll just do the second one in principle and the theory part."},{"Start":"03:50.945 ","End":"03:52.835","Text":"Sine of 20 degrees,"},{"Start":"03:52.835 ","End":"03:56.130","Text":"sine is opposite over hypotenuse,"},{"Start":"03:56.130 ","End":"04:00.590","Text":"would equal e over 280."},{"Start":"04:00.590 ","End":"04:02.450","Text":"That will give us that e is equal"},{"Start":"04:02.450 ","End":"04:11.750","Text":"to 280 times sine"},{"Start":"04:11.750 ","End":"04:13.805","Text":"of 20 degrees."},{"Start":"04:13.805 ","End":"04:19.085","Text":"Now all we need is the calculator to do the actual computations."},{"Start":"04:19.085 ","End":"04:23.600","Text":"Don\u0027t forget to set it to degree mode, not to radians."},{"Start":"04:23.600 ","End":"04:32.450","Text":"I make this 247.246, something, something."},{"Start":"04:32.450 ","End":"04:35.080","Text":"We\u0027ll leave it at 3 places."},{"Start":"04:35.080 ","End":"04:39.950","Text":"That would be in kilometers and that answers how far south."},{"Start":"04:39.950 ","End":"04:50.110","Text":"This comes out to be 95.765,"},{"Start":"04:50.110 ","End":"04:52.540","Text":"etc, leave it there,"},{"Start":"04:52.540 ","End":"04:54.860","Text":"3 places, kilometers."},{"Start":"04:54.860 ","End":"04:57.290","Text":"That\u0027s how far east it is,"},{"Start":"04:57.290 ","End":"04:59.430","Text":"and that\u0027s the end."}],"ID":10767},{"Watched":false,"Name":"Exercise 5","Duration":"6m 47s","ChapterTopicVideoID":10408,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10408.jpeg","UploadDate":"2017-11-02T16:07:57.5930000","DurationForVideoObject":"PT6M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.415","Text":"Here we have another exercise involving angle of elevation,"},{"Start":"00:05.415 ","End":"00:09.255","Text":"and that\u0027s read it and then I\u0027ll draw a diagram."},{"Start":"00:09.255 ","End":"00:13.800","Text":"There\u0027s a tower and the top of the tower is seen at an angle of"},{"Start":"00:13.800 ","End":"00:18.120","Text":"elevation of 32.1 degrees from the ground,"},{"Start":"00:18.120 ","End":"00:21.660","Text":"which we assume to be level horizontal."},{"Start":"00:21.660 ","End":"00:26.070","Text":"If we get 50 meters closer to the tower, well,"},{"Start":"00:26.070 ","End":"00:33.880","Text":"the top is going to appear steeper and the angle of elevation is now 54.3 degrees."},{"Start":"00:35.030 ","End":"00:38.534","Text":"Did I say was 50 meters closer?"},{"Start":"00:38.534 ","End":"00:41.935","Text":"We have to find the height of the tower from all this."},{"Start":"00:41.935 ","End":"00:47.705","Text":"What I\u0027m going to do is stop from a generic right-angled triangle."},{"Start":"00:47.705 ","End":"00:52.835","Text":"Let\u0027s say that this is the tower."},{"Start":"00:52.835 ","End":"00:55.160","Text":"That\u0027s the length a,"},{"Start":"00:55.160 ","End":"00:56.720","Text":"which would be the height of the tower."},{"Start":"00:56.720 ","End":"00:58.790","Text":"CB is the tower,"},{"Start":"00:58.790 ","End":"01:00.905","Text":"B is the top of the tower."},{"Start":"01:00.905 ","End":"01:04.100","Text":"A would be our point on the ground initially."},{"Start":"01:04.100 ","End":"01:09.350","Text":"Now we need another point on the ground that\u0027s closer to the tower,"},{"Start":"01:09.350 ","End":"01:15.495","Text":"where this bit from here to here is 50 meters."},{"Start":"01:15.495 ","End":"01:17.945","Text":"Maybe I\u0027ll give this a name,"},{"Start":"01:17.945 ","End":"01:25.405","Text":"abc call this D. DA is the 50 meters."},{"Start":"01:25.405 ","End":"01:28.575","Text":"Now, I want to connect B with"},{"Start":"01:28.575 ","End":"01:36.150","Text":"D. This is the angle"},{"Start":"01:36.150 ","End":"01:39.540","Text":"from the closer Point C. This is Alpha,"},{"Start":"01:39.540 ","End":"01:41.910","Text":"this is Beta, this would be Gamma,"},{"Start":"01:41.910 ","End":"01:45.375","Text":"the 90 degrees, Let\u0027s call this one Delta."},{"Start":"01:45.375 ","End":"01:52.410","Text":"Little Delta. This is what we are given that Alpha is the"},{"Start":"01:52.410 ","End":"02:01.605","Text":"32.1 degrees and Delta is 54.3 degrees."},{"Start":"02:01.605 ","End":"02:08.200","Text":"We want to know what a equals."},{"Start":"02:09.590 ","End":"02:15.380","Text":"Let\u0027s see now, there\u0027s more than one way of doing this,"},{"Start":"02:15.380 ","End":"02:17.585","Text":"but here\u0027s the method I propose."},{"Start":"02:17.585 ","End":"02:21.590","Text":"First of all, we\u0027ll look at the big triangle ABC."},{"Start":"02:21.590 ","End":"02:26.900","Text":"Then we can get the distance AC in terms"},{"Start":"02:26.900 ","End":"02:33.855","Text":"of BC using the tangent of Alpha."},{"Start":"02:33.855 ","End":"02:39.845","Text":"That will give us an equation that will express in terms of a."},{"Start":"02:39.845 ","End":"02:48.765","Text":"Then we\u0027ll do the same thing for a triangle BDC and get CD in terms of a."},{"Start":"02:48.765 ","End":"02:54.844","Text":"Then we\u0027ll introduce the equation that AC is DC plus 50 meters."},{"Start":"02:54.844 ","End":"02:58.205","Text":"That\u0027s one approach. Let\u0027s do that."},{"Start":"02:58.205 ","End":"03:03.200","Text":"What I can say is that the tangent"},{"Start":"03:03.200 ","End":"03:17.785","Text":"of Alpha is a/b."},{"Start":"03:17.785 ","End":"03:28.220","Text":"That means that b=a/tangent Alpha."},{"Start":"03:28.220 ","End":"03:33.245","Text":"Now, looking at the triangle here,"},{"Start":"03:33.245 ","End":"03:35.780","Text":"I\u0027ve slightly revised my proposal."},{"Start":"03:35.780 ","End":"03:38.960","Text":"Let\u0027s just say that we know what CD is."},{"Start":"03:38.960 ","End":"03:43.900","Text":"This is just b minus 50."},{"Start":"03:43.900 ","End":"03:53.420","Text":"B minus 50 meters is the distance CD because altogether we have b and this part is 50,"},{"Start":"03:53.420 ","End":"03:54.875","Text":"this is b minus 50."},{"Start":"03:54.875 ","End":"03:58.920","Text":"I\u0027ll do the same thing here with the tangent only with Delta."},{"Start":"03:59.090 ","End":"04:09.890","Text":"Tangent of Delta is a over this time not b but b minus 50,"},{"Start":"04:09.890 ","End":"04:19.288","Text":"which means that b"},{"Start":"04:19.288 ","End":"04:26.820","Text":"minus 50=a over tangent Delta."},{"Start":"04:27.230 ","End":"04:30.690","Text":"What I can do is plug in b,"},{"Start":"04:30.690 ","End":"04:35.900","Text":"this b I could replace by a/tangent Alpha."},{"Start":"04:35.900 ","End":"04:42.980","Text":"Then I can write this as a/tangent Alpha."},{"Start":"04:42.980 ","End":"04:50.430","Text":"But instead of Alpha, I\u0027ll put what it equals 32.1 degrees."},{"Start":"04:50.920 ","End":"05:01.340","Text":"Then minus 50 equals a/tangent of Delta,"},{"Start":"05:01.340 ","End":"05:05.220","Text":"which is 54.3 degrees."},{"Start":"05:05.560 ","End":"05:09.890","Text":"Now, the calculator will tell us what these 2 are,"},{"Start":"05:09.890 ","End":"05:14.700","Text":"and we should be able to figure a out from that."},{"Start":"05:15.010 ","End":"05:22.730","Text":"I suppose we could maybe rearrange and get a closed form."},{"Start":"05:22.730 ","End":"05:26.555","Text":"What a equals, I preferred to do that rather than stop computing."},{"Start":"05:26.555 ","End":"05:30.500","Text":"If I bring this over to the left and the 50 over to the right,"},{"Start":"05:30.500 ","End":"05:34.940","Text":"I will get a and then I\u0027ll take a out the brackets on the left,"},{"Start":"05:34.940 ","End":"05:43.265","Text":"I\u0027ve got a(1/tangent 32.1 degrees minus"},{"Start":"05:43.265 ","End":"05:53.959","Text":"1/tangent of 54.3 degrees)=50."},{"Start":"05:53.959 ","End":"05:57.560","Text":"Then I can do 50 divided by this."},{"Start":"05:57.560 ","End":"06:00.985","Text":"But I can also do something else."},{"Start":"06:00.985 ","End":"06:07.433","Text":"I can say that 1/tangent is cotangent,"},{"Start":"06:07.433 ","End":"06:15.920","Text":"so a=50 divided by this difference,"},{"Start":"06:15.920 ","End":"06:20.090","Text":"which is cotangent 32.1"},{"Start":"06:20.090 ","End":"06:26.960","Text":"degrees minus cotangent 54.3 degrees."},{"Start":"06:26.960 ","End":"06:32.810","Text":"Then we can just do this in one computation on the calculator."},{"Start":"06:32.810 ","End":"06:43.260","Text":"I make the answer to be 57.106 something meters."},{"Start":"06:43.390 ","End":"06:47.820","Text":"That\u0027s the answer. We\u0027re done."}],"ID":10768},{"Watched":false,"Name":"Exercise 6","Duration":"4m 4s","ChapterTopicVideoID":10409,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10409.jpeg","UploadDate":"2018-10-19T05:45:42.2730000","DurationForVideoObject":"PT4M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:05.205","Text":"In this exercise, we have to solve a triangle."},{"Start":"00:05.205 ","End":"00:09.689","Text":"We are given ABC as a triangle and we\u0027re given 3 pieces of information."},{"Start":"00:09.689 ","End":"00:11.445","Text":"You always need 3 bits."},{"Start":"00:11.445 ","End":"00:15.090","Text":"We\u0027re given C, which is this,"},{"Start":"00:15.090 ","End":"00:20.175","Text":"we\u0027re given Alpha and we\u0027re given Beta."},{"Start":"00:20.175 ","End":"00:23.460","Text":"To solve it means to find the other unknown quantities."},{"Start":"00:23.460 ","End":"00:26.130","Text":"We\u0027ll need Gamma a and b."},{"Start":"00:26.130 ","End":"00:28.410","Text":"Now the easiest, let\u0027s see,"},{"Start":"00:28.410 ","End":"00:31.530","Text":"obvious place to start is Gamma because we know that"},{"Start":"00:31.530 ","End":"00:35.655","Text":"the sum of the angles in a triangle is 180 degrees."},{"Start":"00:35.655 ","End":"00:38.155","Text":"That being the case,"},{"Start":"00:38.155 ","End":"00:48.895","Text":"we can say that Gamma is 180 degrees minus Alpha minus Beta."},{"Start":"00:48.895 ","End":"00:51.090","Text":"Do this on the calculator,"},{"Start":"00:51.090 ","End":"00:53.480","Text":"maybe we could do it in our heads."},{"Start":"00:53.480 ","End":"00:59.855","Text":"We could say Alpha plus Beta is 136,"},{"Start":"00:59.855 ","End":"01:07.940","Text":"and then subtract that from 180 and we get 44 degrees."},{"Start":"01:07.940 ","End":"01:10.175","Text":"That part is straightforward."},{"Start":"01:10.175 ","End":"01:12.470","Text":"Now we have all 3 angles."},{"Start":"01:12.470 ","End":"01:17.900","Text":"We also have a side c. At this point we\u0027re going to use the sine rule,"},{"Start":"01:17.900 ","End":"01:20.425","Text":"the law of sines,"},{"Start":"01:20.425 ","End":"01:26.880","Text":"that says that c over Sin Gamma=b over Sin Beta,"},{"Start":"01:26.880 ","End":"01:28.405","Text":"a over Sin Alpha."},{"Start":"01:28.405 ","End":"01:30.050","Text":"We\u0027ll just take them a pair at a time."},{"Start":"01:30.050 ","End":"01:32.240","Text":"Let\u0027s say we go for a first."},{"Start":"01:32.240 ","End":"01:35.365","Text":"We say that c over"},{"Start":"01:35.365 ","End":"01:43.395","Text":"Sin Gamma=a over Sine of Alpha."},{"Start":"01:43.395 ","End":"01:47.235","Text":"We have Alpha and Gamma,"},{"Start":"01:47.235 ","End":"01:51.730","Text":"and we also have c over missing is a."},{"Start":"01:51.730 ","End":"01:56.150","Text":"We can extract a and say that a equals this times this over"},{"Start":"01:56.150 ","End":"02:05.795","Text":"this c sin Alpha over sine of Gamma."},{"Start":"02:05.795 ","End":"02:13.910","Text":"Then we can just plug in the numbers and say that a equals 78.9,"},{"Start":"02:13.910 ","End":"02:22.100","Text":"which is c times sin of 26 degrees over"},{"Start":"02:22.100 ","End":"02:32.400","Text":"sin of 44 degrees."},{"Start":"02:32.830 ","End":"02:35.510","Text":"Then we just have to do it on the calculator,"},{"Start":"02:35.510 ","End":"02:37.270","Text":"will do that in the moment."},{"Start":"02:37.270 ","End":"02:46.205","Text":"The second part would be to say that c over sin Gamma equals b over"},{"Start":"02:46.205 ","End":"02:56.130","Text":"sin Beta and that would give us b is equal to c sin Beta"},{"Start":"02:56.130 ","End":"03:05.261","Text":"over sin Gamma and so"},{"Start":"03:05.261 ","End":"03:09.275","Text":"b=78.9 sin of Beta is"},{"Start":"03:09.275 ","End":"03:18.210","Text":"110 degrees and all this over sin 44 degrees."},{"Start":"03:18.210 ","End":"03:21.935","Text":"Now we just have 2 calculator computations to make."},{"Start":"03:21.935 ","End":"03:27.420","Text":"Don\u0027t forget to set your calculator to degrees before you take the sine."},{"Start":"03:27.730 ","End":"03:30.200","Text":"According to my calculator,"},{"Start":"03:30.200 ","End":"03:31.895","Text":"this comes out to"},{"Start":"03:31.895 ","End":"03:41.645","Text":"49.79 something units is not given just units."},{"Start":"03:41.645 ","End":"03:43.250","Text":"This one comes out"},{"Start":"03:43.250 ","End":"03:51.560","Text":"to 106.73"},{"Start":"03:51.560 ","End":"03:55.225","Text":"something also units."},{"Start":"03:55.225 ","End":"03:57.895","Text":"We have found Gamma,"},{"Start":"03:57.895 ","End":"04:04.860","Text":"we have found a and we have found b and that\u0027s it."}],"ID":10769},{"Watched":false,"Name":"Exercise 7","Duration":"8m 57s","ChapterTopicVideoID":10410,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10410.jpeg","UploadDate":"2017-11-02T16:08:35.7400000","DurationForVideoObject":"PT8M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:04.260","Text":"In this exercise, we have observation point number 2"},{"Start":"00:04.260 ","End":"00:08.190","Text":"situated 5 miles due East of observation point number 1."},{"Start":"00:08.190 ","End":"00:10.470","Text":"Let me just stop here and sketch that."},{"Start":"00:10.470 ","End":"00:16.310","Text":"Let\u0027s call this one number 1 and this one number 2,"},{"Start":"00:16.310 ","End":"00:17.750","Text":"which is East of it."},{"Start":"00:17.750 ","End":"00:21.110","Text":"Let\u0027s say the observation point is here and here."},{"Start":"00:21.110 ","End":"00:27.940","Text":"I have these degrees mark because I see that we\u0027re talking about headings."},{"Start":"00:27.940 ","End":"00:30.325","Text":"Let\u0027s connect the 2."},{"Start":"00:30.325 ","End":"00:34.349","Text":"Let\u0027s see, I\u0027ll take a line from here to here."},{"Start":"00:35.350 ","End":"00:41.570","Text":"I\u0027ll make a note that this is 5 miles."},{"Start":"00:41.570 ","End":"00:46.565","Text":"Now we have to locate this truck."},{"Start":"00:46.565 ","End":"00:48.170","Text":"Oh, I didn\u0027t continue reading."},{"Start":"00:48.170 ","End":"00:53.675","Text":"A truck is posted on the heading of so-and-so from 0.1 and so-and-so from 0.2,"},{"Start":"00:53.675 ","End":"00:56.660","Text":"we have to find the distance of the truck for each observation point."},{"Start":"00:56.660 ","End":"01:02.915","Text":"Let\u0027s start with point number 1 and see what a heading of 148 something means."},{"Start":"01:02.915 ","End":"01:04.985","Text":"Well, 0 is North,"},{"Start":"01:04.985 ","End":"01:07.745","Text":"90 degrees is East,"},{"Start":"01:07.745 ","End":"01:10.280","Text":"and here it\u0027s 180."},{"Start":"01:10.280 ","End":"01:19.030","Text":"We have to go 58 degrees past the 90 degrees."},{"Start":"01:21.590 ","End":"01:24.469","Text":"This doesn\u0027t have to be to scale."},{"Start":"01:24.469 ","End":"01:28.565","Text":"Let\u0027s go something like this, and from here,"},{"Start":"01:28.565 ","End":"01:34.350","Text":"216 degrees is more than 180,"},{"Start":"01:34.350 ","End":"01:38.930","Text":"it\u0027s 36 something past it."},{"Start":"01:38.930 ","End":"01:41.575","Text":"Let\u0027s say this,"},{"Start":"01:41.575 ","End":"01:45.820","Text":"and now we have a triangle."},{"Start":"01:45.820 ","End":"01:51.290","Text":"We can compute this angle and this angle. Let\u0027s see."},{"Start":"01:51.290 ","End":"01:53.180","Text":"This angle, like we said,"},{"Start":"01:53.180 ","End":"01:55.865","Text":"is this minus 90 degrees,"},{"Start":"01:55.865 ","End":"02:04.640","Text":"so this would be"},{"Start":"02:04.640 ","End":"02:11.860","Text":"58 degrees and 24 minutes."},{"Start":"02:11.860 ","End":"02:21.425","Text":"Here, what we have to do is subtract 270 degrees minus this."},{"Start":"02:21.425 ","End":"02:24.930","Text":"What we get is,"},{"Start":"02:24.930 ","End":"02:32.135","Text":"I\u0027ll make it 53 degrees and 15 minutes."},{"Start":"02:32.135 ","End":"02:35.280","Text":"Now all this is not to scale."},{"Start":"02:36.100 ","End":"02:42.150","Text":"This is the point where the truck is."},{"Start":"02:44.180 ","End":"02:51.765","Text":"We have a triangle with 1 side and 2 angles,"},{"Start":"02:51.765 ","End":"02:56.920","Text":"and we just have to solve the triangle."},{"Start":"02:57.160 ","End":"03:00.335","Text":"I would make it easier, I\u0027ll label things."},{"Start":"03:00.335 ","End":"03:06.045","Text":"Let\u0027s make this point A and this point B,"},{"Start":"03:06.045 ","End":"03:07.740","Text":"like number 1, number 2,"},{"Start":"03:07.740 ","End":"03:11.325","Text":"and this will be point C for the truck."},{"Start":"03:11.325 ","End":"03:15.630","Text":"Opposite point A,"},{"Start":"03:15.630 ","End":"03:16.920","Text":"we have side A,"},{"Start":"03:16.920 ","End":"03:18.290","Text":"this we\u0027ll have side B,"},{"Start":"03:18.290 ","End":"03:23.130","Text":"and here we\u0027ll have side c."},{"Start":"03:23.800 ","End":"03:33.210","Text":"This angle would be angle c Beta opposite b and a,"},{"Start":"03:33.210 ","End":"03:34.980","Text":"this one will be Alpha,"},{"Start":"03:34.980 ","End":"03:38.740","Text":"and this one will be Gamma."},{"Start":"03:39.200 ","End":"03:42.720","Text":"We have Alpha, Beta,"},{"Start":"03:42.720 ","End":"03:47.105","Text":"and c. I\u0027ll go for Gamma next,"},{"Start":"03:47.105 ","End":"03:49.520","Text":"because all the angles in a triangle,"},{"Start":"03:49.520 ","End":"03:51.665","Text":"the 3 of them add up to 180."},{"Start":"03:51.665 ","End":"03:58.730","Text":"I can say that Gamma is 180 minus the sum of these 2,"},{"Start":"03:58.730 ","End":"04:02.465","Text":"which is 58 degrees,"},{"Start":"04:02.465 ","End":"04:08.405","Text":"24 minutes plus 53 degrees,"},{"Start":"04:08.405 ","End":"04:18.030","Text":"15 minutes, and this comes out to be 180 degrees minus."},{"Start":"04:18.030 ","End":"04:19.635","Text":"If I add these up,"},{"Start":"04:19.635 ","End":"04:25.605","Text":"58 and 53 is 111 degrees,"},{"Start":"04:25.605 ","End":"04:30.185","Text":"24 and 15 is 39 minutes,"},{"Start":"04:30.185 ","End":"04:32.450","Text":"and this is equal to"},{"Start":"04:32.450 ","End":"04:42.130","Text":"68 degrees and 21 minutes."},{"Start":"04:42.130 ","End":"04:43.890","Text":"I did this mentally."},{"Start":"04:43.890 ","End":"04:48.860","Text":"I thought of 180 degrees is 179 degrees and 60 minutes."},{"Start":"04:48.860 ","End":"04:55.280","Text":"60 minus 39 is 21 and 179 minus 111 is this."},{"Start":"04:55.280 ","End":"04:57.070","Text":"We have Gamma."},{"Start":"04:57.070 ","End":"05:01.840","Text":"Now we can use the law of sines."},{"Start":"05:05.030 ","End":"05:08.610","Text":"Well, first of all, we\u0027ll go for a,"},{"Start":"05:08.610 ","End":"05:18.285","Text":"so c over sine Gamma= a over sine Alpha."},{"Start":"05:18.285 ","End":"05:19.770","Text":"We have Alpha, Gamma,"},{"Start":"05:19.770 ","End":"05:22.460","Text":"and c, so we just need a."},{"Start":"05:22.460 ","End":"05:24.525","Text":"We can write that a=,"},{"Start":"05:24.525 ","End":"05:26.235","Text":"this times this over this,"},{"Start":"05:26.235 ","End":"05:31.305","Text":"c times sine Alpha"},{"Start":"05:31.305 ","End":"05:39.645","Text":"over sine Gamma."},{"Start":"05:39.645 ","End":"05:41.800","Text":"If we spell it out,"},{"Start":"05:41.800 ","End":"05:47.155","Text":"a is equal to c is 5 sine Alpha"},{"Start":"05:47.155 ","End":"05:53.760","Text":"is 58 degrees,"},{"Start":"05:53.760 ","End":"06:00.795","Text":"24 minutes and divided by sine."},{"Start":"06:00.795 ","End":"06:07.810","Text":"Now where is our Gamma? Here it is 68 degrees 21 minutes."},{"Start":"06:08.720 ","End":"06:12.700","Text":"Just have to calculate this."},{"Start":"06:12.720 ","End":"06:17.650","Text":"If you don\u0027t and you probably don\u0027t have degrees on your calculator,"},{"Start":"06:17.650 ","End":"06:18.875","Text":"what you do is,"},{"Start":"06:18.875 ","End":"06:21.960","Text":"instead of 58 and 24,"},{"Start":"06:21.960 ","End":"06:27.645","Text":"we can write 58 and 24 over 60."},{"Start":"06:27.645 ","End":"06:29.580","Text":"Similarly, for the Gamma,"},{"Start":"06:29.580 ","End":"06:35.220","Text":"we could write it as 68 and 21 over 60."},{"Start":"06:35.220 ","End":"06:44.570","Text":"Anyway, I won\u0027t bore you with those details and I\u0027ll just give you the answer."},{"Start":"06:44.570 ","End":"06:50.205","Text":"I make it 4.582 something."},{"Start":"06:50.205 ","End":"06:52.875","Text":"That will just take 3 decimal places,"},{"Start":"06:52.875 ","End":"06:55.710","Text":"and the answer is in miles."},{"Start":"06:55.710 ","End":"06:59.420","Text":"We should say the distance from the first observation point to"},{"Start":"06:59.420 ","End":"07:03.710","Text":"the truck is this number of miles."},{"Start":"07:03.710 ","End":"07:06.870","Text":"Now let\u0027s go for b."},{"Start":"07:10.610 ","End":"07:18.590","Text":"Yes, I should say that a is the distance of the second observation point from the truck."},{"Start":"07:18.590 ","End":"07:20.720","Text":"Yes. That\u0027s right."},{"Start":"07:20.720 ","End":"07:22.130","Text":"It belongs to here."},{"Start":"07:22.130 ","End":"07:25.275","Text":"That\u0027s the second distance."},{"Start":"07:25.275 ","End":"07:27.525","Text":"Now we\u0027ll get the first distance,"},{"Start":"07:27.525 ","End":"07:29.370","Text":"and that will be b."},{"Start":"07:29.370 ","End":"07:32.000","Text":"We\u0027ll use the similar formula to this,"},{"Start":"07:32.000 ","End":"07:37.940","Text":"that c over sine"},{"Start":"07:37.940 ","End":"07:43.065","Text":"Gamma= b over sine Beta,"},{"Start":"07:43.065 ","End":"07:47.055","Text":"and this will give us b in terms of the others,"},{"Start":"07:47.055 ","End":"07:54.820","Text":"it will be c times sine Beta over sine Gamma."},{"Start":"07:54.820 ","End":"07:58.640","Text":"Then we just have to write what everything is."},{"Start":"07:58.640 ","End":"08:00.305","Text":"This is 5,"},{"Start":"08:00.305 ","End":"08:07.920","Text":"this is sin(53 degrees"},{"Start":"08:07.920 ","End":"08:13.980","Text":"and 15 minutes over) same thing,"},{"Start":"08:13.980 ","End":"08:22.410","Text":"sin(68 degrees and 21 minutes),"},{"Start":"08:22.410 ","End":"08:30.915","Text":"and I\u0027ll make this 4.310 something miles."},{"Start":"08:30.915 ","End":"08:37.325","Text":"This is the distance of point number 1. Just make a note here."},{"Start":"08:37.325 ","End":"08:41.870","Text":"This is the distance number 1 and distance number 2."},{"Start":"08:41.870 ","End":"08:45.710","Text":"That\u0027s before we convert the minutes to percentages."},{"Start":"08:45.710 ","End":"08:48.005","Text":"This is 53.25,"},{"Start":"08:48.005 ","End":"08:51.340","Text":"this is 68.35,"},{"Start":"08:51.340 ","End":"08:54.255","Text":"just take 21 over 60,"},{"Start":"08:54.255 ","End":"08:58.270","Text":"and that\u0027s the answer."}],"ID":10770},{"Watched":false,"Name":"Exercise 8","Duration":"5m 31s","ChapterTopicVideoID":10411,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10411.jpeg","UploadDate":"2017-11-02T16:08:56.1530000","DurationForVideoObject":"PT5M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we\u0027re given a triangle A, B, C,"},{"Start":"00:03.930 ","End":"00:07.380","Text":"and we\u0027re given sides a and b,"},{"Start":"00:07.380 ","End":"00:09.374","Text":"and also angle Alpha."},{"Start":"00:09.374 ","End":"00:10.980","Text":"This is just a generic picture."},{"Start":"00:10.980 ","End":"00:12.164","Text":"It\u0027s not to scale."},{"Start":"00:12.164 ","End":"00:14.070","Text":"Let\u0027s see. We\u0027re given a."},{"Start":"00:14.070 ","End":"00:19.395","Text":"We\u0027re given b and we\u0027re given Alpha,"},{"Start":"00:19.395 ","End":"00:24.420","Text":"2 sides and an angle which is not the included angle."},{"Start":"00:24.420 ","End":"00:29.820","Text":"We want to go for the law of sines and we\u0027re going to use it on a,"},{"Start":"00:29.820 ","End":"00:34.380","Text":"Alpha and b, Beta first."},{"Start":"00:34.380 ","End":"00:41.340","Text":"We have a over sine Alpha equals b over"},{"Start":"00:41.340 ","End":"00:49.620","Text":"sine Beta and that gives us the sine Beta equals,"},{"Start":"00:49.620 ","End":"00:52.590","Text":"let see, this times this over this,"},{"Start":"00:52.590 ","End":"00:55.635","Text":"b times"},{"Start":"00:55.635 ","End":"01:04.500","Text":"sine Alpha over a."},{"Start":"01:04.500 ","End":"01:07.695","Text":"If I spell it out, B is,"},{"Start":"01:07.695 ","End":"01:13.145","Text":"let\u0027s see, 40.3 times,"},{"Start":"01:13.145 ","End":"01:19.125","Text":"I\u0027ll put an extra times, sine Alpha,"},{"Start":"01:19.125 ","End":"01:28.540","Text":"which is sine of 24.6."},{"Start":"01:29.120 ","End":"01:42.470","Text":"Alpha is 24.6 degrees and this is over a, which is 32.4."},{"Start":"01:42.470 ","End":"01:51.140","Text":"I make this equal to 0.51 something."},{"Start":"01:51.140 ","End":"01:52.775","Text":"It doesn\u0027t really matter."},{"Start":"01:52.775 ","End":"02:00.935","Text":"The point is leave it on the calculator because we need to take the inverse sine of this,"},{"Start":"02:00.935 ","End":"02:08.070","Text":"because Beta is the inverse sine of the above."},{"Start":"02:08.680 ","End":"02:13.070","Text":"I just put the 0.51,"},{"Start":"02:13.070 ","End":"02:15.845","Text":"meaning leave it on the calculator,"},{"Start":"02:15.845 ","End":"02:23.130","Text":"then do an inverse sine and I round it off to 1 decimal place like this."},{"Start":"02:23.130 ","End":"02:26.129","Text":"It closest to 31.2 degrees."},{"Start":"02:26.129 ","End":"02:30.430","Text":"It\u0027s 30.1 something but closer to 2."},{"Start":"02:31.400 ","End":"02:39.470","Text":"Now we have Beta and now we can compute Gamma because you have 2 angles,"},{"Start":"02:39.470 ","End":"02:41.119","Text":"you can get the third."},{"Start":"02:41.119 ","End":"02:45.290","Text":"Gamma is going to be 180 degrees."},{"Start":"02:45.290 ","End":"02:51.685","Text":"That\u0027s the total less than some of the other two minus Alpha plus Beta,"},{"Start":"02:51.685 ","End":"02:56.840","Text":"which is equal to 180 minus."},{"Start":"02:56.840 ","End":"03:01.130","Text":"Let\u0027s see, Alpha less 24.6 plus"},{"Start":"03:01.130 ","End":"03:09.435","Text":"31.2 gives me 55.8 degrees."},{"Start":"03:09.435 ","End":"03:15.850","Text":"If I do the subtraction, I get 124.2."},{"Start":"03:18.380 ","End":"03:22.210","Text":"That\u0027s the third one."},{"Start":"03:22.930 ","End":"03:27.860","Text":"Now we can use the sine rule again because we have"},{"Start":"03:27.860 ","End":"03:32.930","Text":"Gamma and we can get c from a and Alpha."},{"Start":"03:32.930 ","End":"03:36.745","Text":"I should say that if you are going to be accurate,"},{"Start":"03:36.745 ","End":"03:43.910","Text":"you wouldn\u0027t round off so much here to 1 decimal place."},{"Start":"03:43.910 ","End":"03:51.623","Text":"This was in fact 31.1835 and so on,"},{"Start":"03:51.623 ","End":"03:56.705","Text":"something degrees and you take as many decimal places as you can."},{"Start":"03:56.705 ","End":"04:01.280","Text":"But I\u0027m going to use the rounded data."},{"Start":"04:01.280 ","End":"04:03.980","Text":"We\u0027re going to expect an inaccuracy."},{"Start":"04:03.980 ","End":"04:07.115","Text":"But the idea is what matters here."},{"Start":"04:07.115 ","End":"04:08.570","Text":"Now that we have Gamma,"},{"Start":"04:08.570 ","End":"04:11.540","Text":"we go with the sine rule."},{"Start":"04:11.540 ","End":"04:19.135","Text":"But again, but a over sine Alpha this time c over sine Gamma."},{"Start":"04:19.135 ","End":"04:29.695","Text":"So c equals a sine Gamma over sine Alpha,"},{"Start":"04:29.695 ","End":"04:35.240","Text":"which is equal to a was from the beginning,"},{"Start":"04:35.240 ","End":"04:42.655","Text":"32.4 times sine of Gamma."},{"Start":"04:42.655 ","End":"04:48.170","Text":"As I said, we could get it more accurate if we didn\u0027t do the rounding."},{"Start":"04:48.170 ","End":"04:51.470","Text":"It all depends how many decimal places you need,"},{"Start":"04:51.470 ","End":"04:52.750","Text":"how important it is,"},{"Start":"04:52.750 ","End":"04:59.155","Text":"over sine of 24.6 degrees."},{"Start":"04:59.155 ","End":"05:01.805","Text":"This will give us,"},{"Start":"05:01.805 ","End":"05:08.695","Text":"I make it 64.30 something."},{"Start":"05:08.695 ","End":"05:16.340","Text":"Let\u0027s just take one decimal place and actually we\u0027re done."},{"Start":"05:16.340 ","End":"05:17.420","Text":"Let\u0027s just summarize."},{"Start":"05:17.420 ","End":"05:21.485","Text":"We got that Beta was 31.2."},{"Start":"05:21.485 ","End":"05:24.904","Text":"We got Gamma as this,"},{"Start":"05:24.904 ","End":"05:28.490","Text":"and we got c as this."},{"Start":"05:28.490 ","End":"05:32.820","Text":"We got all the 3 unknown quantities and we\u0027re done."}],"ID":10771},{"Watched":false,"Name":"Exercise 9","Duration":"9m 28s","ChapterTopicVideoID":10412,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10412.jpeg","UploadDate":"2017-11-02T16:09:35.5570000","DurationForVideoObject":"PT9M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this exercise, we have to solve a triangle."},{"Start":"00:03.540 ","End":"00:07.380","Text":"We are given 3 pieces of information and you always need 3."},{"Start":"00:07.380 ","End":"00:10.890","Text":"This time it\u0027s side a,"},{"Start":"00:10.890 ","End":"00:13.455","Text":"we\u0027re given side c,"},{"Start":"00:13.455 ","End":"00:21.930","Text":"and we\u0027re given the angle Beta which notices the included angle between the 2 sides."},{"Start":"00:21.930 ","End":"00:25.020","Text":"When this scenario occurs,"},{"Start":"00:25.020 ","End":"00:28.300","Text":"then we need to use the law of cosine."},{"Start":"00:28.490 ","End":"00:37.980","Text":"In this case, we can find b using one of the forms of the law of cosines,"},{"Start":"00:37.980 ","End":"00:45.590","Text":"and we get that b^2 is equal to a^2 plus c^2."},{"Start":"00:45.590 ","End":"00:49.465","Text":"It actually starts off like Pythagoras theorem."},{"Start":"00:49.465 ","End":"00:54.380","Text":"Sometimes the law of cosine is called the extended Pythagoras\u0027 theorem."},{"Start":"00:54.380 ","End":"00:57.080","Text":"But we have to subtract something which is twice"},{"Start":"00:57.080 ","End":"00:59.915","Text":"this times this times the cosine of this,"},{"Start":"00:59.915 ","End":"01:05.980","Text":"so 2ac cosine Beta."},{"Start":"01:05.980 ","End":"01:08.970","Text":"Now we have all these quantities,"},{"Start":"01:08.970 ","End":"01:15.180","Text":"so we can say that this is equal to, let\u0027s see,"},{"Start":"01:15.820 ","End":"01:22.085","Text":"34.56^2 plus"},{"Start":"01:22.085 ","End":"01:28.050","Text":"81.23^2 minus,"},{"Start":"01:28.940 ","End":"01:33.215","Text":"maybe to distinguish my decimal points I\u0027ll write it like this,"},{"Start":"01:33.215 ","End":"01:42.600","Text":"twice 34.56 times 81.23"},{"Start":"01:42.910 ","End":"01:52.115","Text":"times cosine of 16.78 degrees."},{"Start":"01:52.115 ","End":"01:53.899","Text":"Here we got it in decimal,"},{"Start":"01:53.899 ","End":"01:56.825","Text":"sometimes it will be given in degrees and minutes,"},{"Start":"01:56.825 ","End":"02:01.750","Text":"you know how to convert back and forth. What do we get?"},{"Start":"02:01.750 ","End":"02:05.400","Text":"This equal, it doesn\u0027t really matter,"},{"Start":"02:05.400 ","End":"02:13.965","Text":"2417 point something."},{"Start":"02:13.965 ","End":"02:15.600","Text":"This one it doesn\u0027t matter,"},{"Start":"02:15.600 ","End":"02:17.810","Text":"just leave it in the calculator,"},{"Start":"02:17.810 ","End":"02:22.340","Text":"in the display and then just take the square root of that,"},{"Start":"02:22.340 ","End":"02:29.500","Text":"so b is the square root of 2417 point whatever."},{"Start":"02:29.500 ","End":"02:36.360","Text":"That comes out to be 49.17,"},{"Start":"02:36.360 ","End":"02:39.420","Text":"if you take it to 2 decimal places."},{"Start":"02:39.420 ","End":"02:42.135","Text":"That\u0027s given as b."},{"Start":"02:42.135 ","End":"02:47.235","Text":"Now we have the 3 sides and we can get Alpha and Gamma"},{"Start":"02:47.235 ","End":"02:53.745","Text":"using the law of sines."},{"Start":"02:53.745 ","End":"02:55.770","Text":"What should we do first?"},{"Start":"02:55.770 ","End":"02:58.530","Text":"Well, we have b and Beta."},{"Start":"02:58.530 ","End":"03:02.290","Text":"Let\u0027s go for Alpha."},{"Start":"03:04.160 ","End":"03:09.360","Text":"a over sine Alpha"},{"Start":"03:09.360 ","End":"03:19.215","Text":"equals b over sine Beta,"},{"Start":"03:19.215 ","End":"03:21.510","Text":"we\u0027re looking for b,"},{"Start":"03:21.510 ","End":"03:24.980","Text":"so b if we isolate it is"},{"Start":"03:24.980 ","End":"03:35.675","Text":"a sine Beta over sine Alpha,"},{"Start":"03:35.675 ","End":"03:39.260","Text":"which is equal to,"},{"Start":"03:39.260 ","End":"03:41.180","Text":"just plug in the numbers,"},{"Start":"03:41.180 ","End":"03:46.670","Text":"34.56 times"},{"Start":"03:46.670 ","End":"03:54.000","Text":"sine of 16.78 degrees."},{"Start":"04:01.150 ","End":"04:04.170","Text":"Where are we?"},{"Start":"04:06.890 ","End":"04:08.995","Text":"We\u0027re looking for Alpha."},{"Start":"04:08.995 ","End":"04:12.445","Text":"Let\u0027s first of all see what sine Alpha is."},{"Start":"04:12.445 ","End":"04:20.665","Text":"It\u0027s a sine Beta over b,"},{"Start":"04:20.665 ","End":"04:27.985","Text":"which is equal to 34.56"},{"Start":"04:27.985 ","End":"04:37.150","Text":"times sine of 16.78 degrees over b,"},{"Start":"04:37.150 ","End":"04:44.485","Text":"which is what we just found here, 49.17."},{"Start":"04:44.485 ","End":"04:49.725","Text":"If you want to take more places for more accuracy then do so,"},{"Start":"04:49.725 ","End":"04:52.540","Text":"or you could save this in the memory of"},{"Start":"04:52.540 ","End":"04:57.410","Text":"the calculator and divide by the recall of that memory."},{"Start":"04:58.290 ","End":"05:01.835","Text":"What does it come out to?"},{"Start":"05:01.835 ","End":"05:10.220","Text":"Well, Alpha is going to equal the inverse sine of whatever it comes out to."},{"Start":"05:10.320 ","End":"05:21.205","Text":"This computation comes out to be 0.202887. I don\u0027t know."},{"Start":"05:21.205 ","End":"05:23.020","Text":"Anyway the point is,"},{"Start":"05:23.020 ","End":"05:24.820","Text":"you don\u0027t have to actually write this,"},{"Start":"05:24.820 ","End":"05:33.125","Text":"just leave it in the display and then do the inverse sine function on that."},{"Start":"05:33.125 ","End":"05:38.310","Text":"It comes out to about 11.7 degrees,"},{"Start":"05:38.310 ","End":"05:43.975","Text":"but here you have to be careful because the inverse sine"},{"Start":"05:43.975 ","End":"05:50.020","Text":"of a positive quantity could also be in the second quadrant."},{"Start":"05:50.020 ","End":"05:53.485","Text":"I could take the supplement of this,"},{"Start":"05:53.485 ","End":"06:00.985","Text":"which would be something like 178.3 degrees,"},{"Start":"06:00.985 ","End":"06:06.370","Text":"but that can\u0027t b because we already have"},{"Start":"06:06.370 ","End":"06:11.685","Text":"an angle of 16.78 degrees."},{"Start":"06:11.685 ","End":"06:16.480","Text":"If you add that to whatever I just said, it exceeds 180."},{"Start":"06:16.480 ","End":"06:22.595","Text":"This is it, but in principle it might\u0027ve been the supplement of this angle."},{"Start":"06:22.595 ","End":"06:26.300","Text":"Now we have Alpha,"},{"Start":"06:26.300 ","End":"06:31.095","Text":"and we just computed b,"},{"Start":"06:31.095 ","End":"06:34.800","Text":"so what remains is Gamma."},{"Start":"06:34.800 ","End":"06:38.310","Text":"I need a bit more space here."},{"Start":"06:38.310 ","End":"06:41.690","Text":"Again, we\u0027ll do by the law of sines,"},{"Start":"06:41.690 ","End":"06:44.140","Text":"and we\u0027ll get that"},{"Start":"06:44.140 ","End":"06:54.425","Text":"a over sine Alpha equals c over sine Gamma."},{"Start":"06:54.425 ","End":"06:59.280","Text":"What we\u0027re looking for is Gamma."},{"Start":"07:04.520 ","End":"07:06.960","Text":"You know what? silly me,"},{"Start":"07:06.960 ","End":"07:09.523","Text":"but this is instructive."},{"Start":"07:09.523 ","End":"07:11.110","Text":"We could have gone this way,"},{"Start":"07:11.110 ","End":"07:12.685","Text":"but there is an easier way."},{"Start":"07:12.685 ","End":"07:15.115","Text":"I just realized we have 2 angles,"},{"Start":"07:15.115 ","End":"07:17.260","Text":"we can find the third."},{"Start":"07:17.260 ","End":"07:25.035","Text":"Gamma is 180 degrees minus Alpha plus Beta,"},{"Start":"07:25.035 ","End":"07:27.990","Text":"because all 3 angles add up to 180,"},{"Start":"07:27.990 ","End":"07:32.560","Text":"so it\u0027s a 180 minus."},{"Start":"07:34.700 ","End":"07:37.140","Text":"Where is my Alpha?"},{"Start":"07:37.140 ","End":"07:40.725","Text":"There\u0027s Beta 16.78,"},{"Start":"07:40.725 ","End":"07:44.795","Text":"and the Alpha is here."},{"Start":"07:44.795 ","End":"07:55.580","Text":"Let\u0027s just do a side calculation,16.78 plus 11.7."},{"Start":"07:55.580 ","End":"08:00.560","Text":"I think it was 11.70 degrees."},{"Start":"08:00.560 ","End":"08:04.355","Text":"Make them both the same number of places."},{"Start":"08:04.355 ","End":"08:10.505","Text":"That is equal to, let\u0027s see,"},{"Start":"08:10.505 ","End":"08:17.480","Text":"78 plus 70 is 1.48,"},{"Start":"08:17.480 ","End":"08:26.135","Text":"carry 1, 16 and 11 is 27 so that is 28 degrees."},{"Start":"08:26.135 ","End":"08:32.600","Text":"I\u0027ll write that here. 28.48 degrees,"},{"Start":"08:32.600 ","End":"08:43.330","Text":"and so this is equal to 151.52"},{"Start":"08:47.650 ","End":"08:56.420","Text":"degrees. Quick check 52"},{"Start":"08:56.420 ","End":"08:59.270","Text":"and 48 is 1."},{"Start":"08:59.270 ","End":"09:04.485","Text":"That\u0027s right. We have all the quantities."},{"Start":"09:04.485 ","End":"09:13.950","Text":"We have b which is this,"},{"Start":"09:13.950 ","End":"09:18.720","Text":"we found Alpha to be this,"},{"Start":"09:18.720 ","End":"09:22.650","Text":"and we found Gamma to be this,"},{"Start":"09:22.650 ","End":"09:28.260","Text":"and so we\u0027ve got all the 3 missing quantities and we\u0027re done."}],"ID":10772},{"Watched":false,"Name":"Exercise 10","Duration":"6m 15s","ChapterTopicVideoID":10413,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10413.jpeg","UploadDate":"2017-11-02T16:09:57.8930000","DurationForVideoObject":"PT6M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.635","Text":"In this exercise, we have to solve a triangle where we\u0027re given all 3 sides,"},{"Start":"00:07.635 ","End":"00:11.385","Text":"and then we have to find the 3 angles,"},{"Start":"00:11.385 ","End":"00:13.605","Text":"and the details are here."},{"Start":"00:13.605 ","End":"00:16.765","Text":"Now, in the case of 3 sides,"},{"Start":"00:16.765 ","End":"00:26.315","Text":"I recommend going for the longest side first because if there is an obtuse angle,"},{"Start":"00:26.315 ","End":"00:30.755","Text":"it has to be opposite the longest side."},{"Start":"00:30.755 ","End":"00:32.240","Text":"In this case, well,"},{"Start":"00:32.240 ","End":"00:33.680","Text":"this figure is not to scale,"},{"Start":"00:33.680 ","End":"00:38.080","Text":"but we can see that the biggest one is b"},{"Start":"00:38.080 ","End":"00:43.400","Text":"and so we are going to compute Beta using the cosine rule or one of"},{"Start":"00:43.400 ","End":"00:49.820","Text":"the variants of it is that cosine of Beta is equal"},{"Start":"00:49.820 ","End":"00:59.170","Text":"to a^2 plus c^2 minus b^2,"},{"Start":"00:59.170 ","End":"01:02.280","Text":"the odd one out, the Beta belongs to b, and that\u0027s the odd one out,"},{"Start":"01:02.280 ","End":"01:06.330","Text":"so it\u0027s a minus over 2ac,"},{"Start":"01:06.330 ","End":"01:08.895","Text":"it\u0027s from here and here."},{"Start":"01:08.895 ","End":"01:11.800","Text":"If we get something negative,"},{"Start":"01:11.800 ","End":"01:15.130","Text":"then Beta is obtuse,"},{"Start":"01:15.130 ","End":"01:16.960","Text":"otherwise it\u0027s acute and in any event,"},{"Start":"01:16.960 ","End":"01:19.330","Text":"the other two will be acute."},{"Start":"01:19.460 ","End":"01:23.080","Text":"This is equal to,"},{"Start":"01:23.660 ","End":"01:32.590","Text":"with numbers we get 2.83^2 plus from here 2.37^2"},{"Start":"01:33.740 ","End":"01:40.110","Text":"minus 4.53^2"},{"Start":"01:40.110 ","End":"01:46.800","Text":"over twice a,"},{"Start":"01:46.800 ","End":"01:54.070","Text":"2.83 and then c, 2.37."},{"Start":"01:57.880 ","End":"02:00.505","Text":"I won\u0027t give the intermediate detail,"},{"Start":"02:00.505 ","End":"02:03.470","Text":"let\u0027s just do it on the calculator."},{"Start":"02:03.470 ","End":"02:08.030","Text":"I might get minus 0.5125"},{"Start":"02:08.030 ","End":"02:17.165","Text":", and so on."},{"Start":"02:17.165 ","End":"02:26.225","Text":"Just leave this on the calculator as is and then Beta is the inverse cosine of this,"},{"Start":"02:26.225 ","End":"02:32.975","Text":"which will be in any event in first or second quadrants."},{"Start":"02:32.975 ","End":"02:38.900","Text":"It comes out to 120.83 degrees,"},{"Start":"02:38.900 ","End":"02:40.590","Text":"going to work in decimals,"},{"Start":"02:40.590 ","End":"02:45.005","Text":"won\u0027t bother to convert this to degrees or in minutes or something."},{"Start":"02:45.005 ","End":"02:50.569","Text":"There we have Beta."},{"Start":"02:50.569 ","End":"02:57.260","Text":"We have an angle opposite a side or a side opposite an angle,"},{"Start":"02:57.260 ","End":"02:59.510","Text":"now we\u0027ll go for one of the others."},{"Start":"02:59.510 ","End":"03:06.650","Text":"Let\u0027s say we go for a and we can use the sine rule for that."},{"Start":"03:06.650 ","End":"03:12.107","Text":"We can say that a/sine"},{"Start":"03:12.107 ","End":"03:19.660","Text":"Alpha=b/sine Beta."},{"Start":"03:19.760 ","End":"03:23.610","Text":"Now, we already have this,"},{"Start":"03:23.610 ","End":"03:30.870","Text":"so all we\u0027re missing is the Alpha."},{"Start":"03:30.870 ","End":"03:39.320","Text":"That gives us the sine Alpha is just this diagonal divided by this,"},{"Start":"03:39.320 ","End":"03:45.955","Text":"a sine Beta/b,"},{"Start":"03:45.955 ","End":"03:49.895","Text":"which comes out to be what?"},{"Start":"03:49.895 ","End":"03:58.350","Text":"A is 2.83 times sine Beta,"},{"Start":"03:59.450 ","End":"04:04.950","Text":"sine of 120.83,"},{"Start":"04:04.950 ","End":"04:07.190","Text":"I\u0027m using rounded data, you could, of course,"},{"Start":"04:07.190 ","End":"04:10.310","Text":"leave more decimal places for more accuracy,"},{"Start":"04:10.310 ","End":"04:14.185","Text":"divided by b is"},{"Start":"04:14.185 ","End":"04:24.020","Text":"4.53 and I get 0.536."},{"Start":"04:24.020 ","End":"04:27.440","Text":"What I mean is this doesn\u0027t matter,"},{"Start":"04:27.440 ","End":"04:33.800","Text":"just leave it on the display of the calculator and then we need to take the inverse sine."},{"Start":"04:33.800 ","End":"04:42.290","Text":"Alpha is sine inverse of this thing here, 0.536,"},{"Start":"04:42.290 ","End":"04:46.940","Text":"etc., and if I do the inverse sign"},{"Start":"04:46.940 ","End":"04:53.225","Text":"comes out about 32.44 degrees."},{"Start":"04:53.225 ","End":"04:58.340","Text":"Now, the easiest way to find the last one,"},{"Start":"04:58.340 ","End":"04:59.970","Text":"which is Gamma,"},{"Start":"04:59.970 ","End":"05:02.745","Text":"is once you have 2 angles,"},{"Start":"05:02.745 ","End":"05:05.470","Text":"you can find the third."},{"Start":"05:06.980 ","End":"05:11.780","Text":"Let\u0027s see where do I have space, I\u0027ll just scroll."},{"Start":"05:11.780 ","End":"05:18.845","Text":"Gamma is equal to 180 minus the other two."},{"Start":"05:18.845 ","End":"05:28.685","Text":"Beta was given, or rather we computed it as 120.83 degrees,"},{"Start":"05:28.685 ","End":"05:34.490","Text":"and Alpha here is 32.44 degrees."},{"Start":"05:34.490 ","End":"05:37.830","Text":"If I add these two together, I will get 1,"},{"Start":"05:37.830 ","End":"05:47.320","Text":"I will just do the thing in the calculator and give you the answer,"},{"Start":"05:47.320 ","End":"05:55.570","Text":"and this comes out 26.73 degrees."},{"Start":"05:55.760 ","End":"05:57.930","Text":"We found everything,"},{"Start":"05:57.930 ","End":"06:05.190","Text":"here\u0027s Beta, I\u0027ll highlight it on these two."},{"Start":"06:05.190 ","End":"06:08.715","Text":"Here we have Alpha,"},{"Start":"06:08.715 ","End":"06:10.350","Text":"which is this,"},{"Start":"06:10.350 ","End":"06:12.165","Text":"and here we have Gamma,"},{"Start":"06:12.165 ","End":"06:16.150","Text":"which is this, so let\u0027s solve the triangle then."}],"ID":10773},{"Watched":false,"Name":"Exercise 11","Duration":"4m 14s","ChapterTopicVideoID":10414,"CourseChapterTopicPlaylistID":257206,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10414.jpeg","UploadDate":"2017-11-02T16:10:14.1970000","DurationForVideoObject":"PT4M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"A word problem with time,"},{"Start":"00:03.030 ","End":"00:06.420","Text":"distance, and so on and also trigonometry."},{"Start":"00:06.420 ","End":"00:09.630","Text":"A man leaves his house for work on a straight road,"},{"Start":"00:09.630 ","End":"00:12.570","Text":"and drives at 52 miles per hour,"},{"Start":"00:12.570 ","End":"00:14.535","Text":"that\u0027s his average speed."},{"Start":"00:14.535 ","End":"00:16.080","Text":"5 minutes later,"},{"Start":"00:16.080 ","End":"00:18.345","Text":"his wife leaves the house."},{"Start":"00:18.345 ","End":"00:21.700","Text":"This will be the house."},{"Start":"00:22.220 ","End":"00:27.850","Text":"Maybe I\u0027ll call it letter h for house."},{"Start":"00:28.760 ","End":"00:34.050","Text":"Now, his wife leaves on a different straight road at a different speed,"},{"Start":"00:34.050 ","End":"00:40.235","Text":"but we know that the angle between the roads is 108°."},{"Start":"00:40.235 ","End":"00:42.425","Text":"This is 108."},{"Start":"00:42.425 ","End":"00:53.090","Text":"Let\u0027s say the man drives in this direction and his wife drives in this direction."},{"Start":"00:53.090 ","End":"00:56.845","Text":"Well, he\u0027s driving faster and for longer so obviously,"},{"Start":"00:56.845 ","End":"01:01.000","Text":"the longer side is this 1. It\u0027s the man."},{"Start":"01:01.000 ","End":"01:07.000","Text":"Let\u0027s call this one m for man and w for his wife."},{"Start":"01:08.380 ","End":"01:14.330","Text":"The information can be used to compute this and this,"},{"Start":"01:14.330 ","End":"01:17.635","Text":"and then we\u0027ll have 2 sides and an included angle."},{"Start":"01:17.635 ","End":"01:26.100","Text":"Then we can figure out how far they are apart from that."},{"Start":"01:27.230 ","End":"01:29.670","Text":"Suppose I could label them,"},{"Start":"01:29.670 ","End":"01:32.480","Text":"let\u0027s call this distance a."},{"Start":"01:32.480 ","End":"01:35.015","Text":"Let\u0027s call this distance b."},{"Start":"01:35.015 ","End":"01:37.715","Text":"What we\u0027re looking for, we\u0027ll call c,"},{"Start":"01:37.715 ","End":"01:40.650","Text":"and this will be Gamma."},{"Start":"01:44.320 ","End":"01:55.660","Text":"We can compute a where he goes at 52 miles per hour,"},{"Start":"01:55.790 ","End":"02:02.595","Text":"and he goes for 16 minutes."},{"Start":"02:02.595 ","End":"02:08.390","Text":"We have to compute that because we\u0027re working in miles per hour."},{"Start":"02:08.390 ","End":"02:14.120","Text":"We need it in hours 16/60,"},{"Start":"02:14.120 ","End":"02:21.785","Text":"and that comes out 13.866,"},{"Start":"02:21.785 ","End":"02:25.760","Text":"and the 6 repeats endlessly."},{"Start":"02:25.760 ","End":"02:36.425","Text":"Then we have b=45 miles an hour,"},{"Start":"02:36.425 ","End":"02:43.465","Text":"and 5 minutes later she only went for 11 minutes."},{"Start":"02:43.465 ","End":"02:49.840","Text":"I didn\u0027t write down that, 16-5=11."},{"Start":"02:49.840 ","End":"02:53.500","Text":"It\u0027s obvious, 11/60,"},{"Start":"02:53.500 ","End":"02:58.080","Text":"and that comes out to"},{"Start":"02:58.080 ","End":"03:04.230","Text":"be 8.25."},{"Start":"03:04.230 ","End":"03:09.045","Text":"The way we\u0027ll find c is using the cosine law."},{"Start":"03:09.045 ","End":"03:19.570","Text":"We have that c^2=a^2+b^2-2ab"},{"Start":"03:24.200 ","End":"03:30.960","Text":"cos of Gamma."},{"Start":"03:30.960 ","End":"03:40.270","Text":"I\u0027ll skip the stage of plugging in the numbers from a and b and cos Gamma,"},{"Start":"03:41.180 ","End":"03:45.770","Text":"and just tell you this part comes out to be"},{"Start":"03:45.770 ","End":"03:53.710","Text":"7986.3 something I think."},{"Start":"03:53.990 ","End":"04:00.810","Text":"Then we take the square root of this and"},{"Start":"04:00.810 ","End":"04:08.580","Text":"it comes out about 89.366."},{"Start":"04:08.580 ","End":"04:11.345","Text":"This is the answer,"},{"Start":"04:11.345 ","End":"04:14.430","Text":"and we are done."}],"ID":10774}],"Thumbnail":null,"ID":257206},{"Name":"Trigonometric Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Trigonometric Equations - Part 1","Duration":"19m 28s","ChapterTopicVideoID":13606,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13606.jpeg","UploadDate":"2021-06-29T13:59:01.7500000","DurationForVideoObject":"PT19M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"Now we\u0027re starting a new subject, trigonometric equations."},{"Start":"00:04.755 ","End":"00:08.655","Text":"It\u0027ll be in 3 parts. Here\u0027s part 1."},{"Start":"00:08.655 ","End":"00:13.230","Text":"Part 1 will be subdivided into 3 cases."},{"Start":"00:13.230 ","End":"00:20.145","Text":"Case A will be equations of the form sin of something,"},{"Start":"00:20.145 ","End":"00:22.575","Text":"let\u0027s say square,"},{"Start":"00:22.575 ","End":"00:25.620","Text":"equal sin of something else,"},{"Start":"00:25.620 ","End":"00:30.075","Text":"say a triangle, or reducible to this form."},{"Start":"00:30.075 ","End":"00:32.805","Text":"Now these will have a variable in them,"},{"Start":"00:32.805 ","End":"00:40.140","Text":"might be Alpha or Theta or x. I\u0027ll give an example right away."},{"Start":"00:40.140 ","End":"00:42.300","Text":"There\u0027s a basic example,"},{"Start":"00:42.300 ","End":"00:51.070","Text":"we\u0027ll take sin x = sin of 60 degrees."},{"Start":"00:51.130 ","End":"00:58.070","Text":"Let me say right away that in this course we\u0027ll use degrees."},{"Start":"00:58.070 ","End":"01:00.905","Text":"Although everything works with radians,"},{"Start":"01:00.905 ","End":"01:03.710","Text":"also with the appropriate changes."},{"Start":"01:03.710 ","End":"01:07.610","Text":"Now this does indeed look like this form."},{"Start":"01:07.610 ","End":"01:12.385","Text":"There\u0027s a formula for how to get the solution to this."},{"Start":"01:12.385 ","End":"01:18.305","Text":"Here it is, that when you have sin of one thing equals sin of another thing,"},{"Start":"01:18.305 ","End":"01:20.950","Text":"and I call them Theta and Alpha here."},{"Start":"01:20.950 ","End":"01:28.720","Text":"Then Theta can be one of 2 things and each one of them is a family of solutions."},{"Start":"01:28.720 ","End":"01:34.240","Text":"It could be the Alpha here plus multiples of 360."},{"Start":"01:34.240 ","End":"01:40.599","Text":"Because obviously multiples of whole circles don\u0027t change the sign and"},{"Start":"01:40.599 ","End":"01:48.370","Text":"the other family of solutions is by taking the supplement of the angle Alpha."},{"Start":"01:48.370 ","End":"01:51.520","Text":"Supplement means 180 minus."},{"Start":"01:51.520 ","End":"01:56.710","Text":"Again, adding multiples of 360 degrees arbitrarily,"},{"Start":"01:56.710 ","End":"01:59.500","Text":"I should say that n is an integer,"},{"Start":"01:59.500 ","End":"02:04.880","Text":"which means a whole number positive, negative or zero."},{"Start":"02:04.910 ","End":"02:08.120","Text":"In our case, let\u0027s see what we have."},{"Start":"02:08.120 ","End":"02:09.560","Text":"According to this,"},{"Start":"02:09.560 ","End":"02:15.050","Text":"we have that x has got to equal either Alpha,"},{"Start":"02:15.050 ","End":"02:22.050","Text":"which is 60, plus n times 360 degrees."},{"Start":"02:22.050 ","End":"02:26.800","Text":"The other family of solutions is 180 minus Alpha."},{"Start":"02:26.800 ","End":"02:34.470","Text":"That would be 120 degrees plus multiples of 360."},{"Start":"02:34.730 ","End":"02:41.765","Text":"I want to say, sometimes we like to write a few of them out just to get an idea."},{"Start":"02:41.765 ","End":"02:42.920","Text":"This is a bit abstract,"},{"Start":"02:42.920 ","End":"02:47.765","Text":"so you could start with n = zero and get 60 degrees."},{"Start":"02:47.765 ","End":"02:53.210","Text":"You could let n = 1 and you\u0027d get 420 degrees,"},{"Start":"02:53.210 ","End":"02:57.050","Text":"n = minus 1,"},{"Start":"02:57.050 ","End":"03:00.260","Text":"you\u0027d get minus 300 degrees."},{"Start":"03:00.260 ","End":"03:02.780","Text":"Maybe I should write n = zero,"},{"Start":"03:02.780 ","End":"03:08.305","Text":"n = 1, n = minus 1,"},{"Start":"03:08.305 ","End":"03:11.310","Text":"and so on and so on this way."},{"Start":"03:11.310 ","End":"03:14.420","Text":"You can also get another family of solutions."},{"Start":"03:14.420 ","End":"03:16.490","Text":"We also have a 120 degrees."},{"Start":"03:16.490 ","End":"03:18.320","Text":"Again you could add 360,"},{"Start":"03:18.320 ","End":"03:22.320","Text":"which makes it 480 and so on,"},{"Start":"03:22.320 ","End":"03:30.390","Text":"or n could be minus 1 and then we\u0027d get minus 240 degrees and so on."},{"Start":"03:30.410 ","End":"03:33.380","Text":"If you start writing a few hours,"},{"Start":"03:33.380 ","End":"03:35.150","Text":"which we sometimes do."},{"Start":"03:35.150 ","End":"03:39.800","Text":"That was the very first example now let\u0027s take a variation,"},{"Start":"03:39.800 ","End":"03:42.530","Text":"this time the variable is Theta,"},{"Start":"03:42.530 ","End":"03:48.920","Text":"and it doesn\u0027t look like sin something equals sin something but that case,"},{"Start":"03:48.920 ","End":"03:55.610","Text":"we can bring it to this form because 1.5 is one of those special values of sin."},{"Start":"03:55.610 ","End":"03:58.609","Text":"You should remember some special angles,"},{"Start":"03:58.609 ","End":"04:04.115","Text":"and you should know that 1.5 is sin of 30 degrees."},{"Start":"04:04.115 ","End":"04:06.935","Text":"We\u0027ll look it up in the table of special angles."},{"Start":"04:06.935 ","End":"04:10.325","Text":"Now, we have sin of something equals sin of something,"},{"Start":"04:10.325 ","End":"04:13.860","Text":"so we can write the solution just by following."},{"Start":"04:13.860 ","End":"04:19.325","Text":"This, is going to equal either 30 degrees plus multiples of"},{"Start":"04:19.325 ","End":"04:27.545","Text":"360 or the supplement of the angle 180 minus 30,"},{"Start":"04:27.545 ","End":"04:32.790","Text":"which is 150 plus multiples of 360."},{"Start":"04:34.070 ","End":"04:36.585","Text":"Another example."},{"Start":"04:36.585 ","End":"04:40.500","Text":"We have 2 sin 3x = root 3."},{"Start":"04:40.500 ","End":"04:50.625","Text":"What we can do is first of all get rid of the 2 here by saying sin 3x = root 3 over 2."},{"Start":"04:50.625 ","End":"04:59.930","Text":"Then we remember the special angles that root 3 over 2 is the sin of 60 degrees."},{"Start":"04:59.930 ","End":"05:06.110","Text":"Then we apply the formula here with Theta 3x Alpha 60,"},{"Start":"05:06.110 ","End":"05:11.210","Text":"and we get that 3x is equal to,"},{"Start":"05:11.210 ","End":"05:13.250","Text":"and we have 2 families,"},{"Start":"05:13.250 ","End":"05:19.550","Text":"either 60 plus multiples of 360,"},{"Start":"05:19.550 ","End":"05:20.825","Text":"where n is an integer,"},{"Start":"05:20.825 ","End":"05:22.370","Text":"or the supplement to this,"},{"Start":"05:22.370 ","End":"05:25.535","Text":"which is 120 degrees, subtract from 180."},{"Start":"05:25.535 ","End":"05:30.065","Text":"Also plus n times 360 degrees."},{"Start":"05:30.065 ","End":"05:34.115","Text":"But that\u0027s not the end of it because want x, so from here,"},{"Start":"05:34.115 ","End":"05:37.280","Text":"we get that x is equal to"},{"Start":"05:37.280 ","End":"05:45.995","Text":"either 20 degrees plus multiples of 120 degrees this time,"},{"Start":"05:45.995 ","End":"05:48.260","Text":"or this divided by 3,"},{"Start":"05:48.260 ","End":"05:53.480","Text":"which is 40 plus multiples,"},{"Start":"05:53.480 ","End":"05:57.450","Text":"also over 120 degrees."},{"Start":"05:57.490 ","End":"06:02.435","Text":"Now, sometimes in the problem,"},{"Start":"06:02.435 ","End":"06:07.350","Text":"they give you an extra condition. I\u0027ll do that here."},{"Start":"06:07.350 ","End":"06:09.275","Text":"Well I just copied this."},{"Start":"06:09.275 ","End":"06:12.320","Text":"But sometimes you\u0027re given an extra condition."},{"Start":"06:12.320 ","End":"06:19.550","Text":"For example, we require that x is between something and something,"},{"Start":"06:19.550 ","End":"06:23.345","Text":"let\u0027s say between 0 and 360 degrees."},{"Start":"06:23.345 ","End":"06:26.705","Text":"Let\u0027s even say less than or equal to."},{"Start":"06:26.705 ","End":"06:33.200","Text":"Then what we do is we have this solution and I copy of the solution over here."},{"Start":"06:33.200 ","End":"06:38.495","Text":"One way to add this restriction is just to write a few values out."},{"Start":"06:38.495 ","End":"06:41.450","Text":"For example, if n is 0,"},{"Start":"06:41.450 ","End":"06:43.325","Text":"we have 20 degrees."},{"Start":"06:43.325 ","End":"06:45.305","Text":"If n is 1,"},{"Start":"06:45.305 ","End":"06:51.935","Text":"then we have here 140 degrees and that\u0027s still within the range."},{"Start":"06:51.935 ","End":"06:53.915","Text":"If n is 2,"},{"Start":"06:53.915 ","End":"07:03.680","Text":"then we have 240 plus 20 is 260 degrees but if n is 3,"},{"Start":"07:03.680 ","End":"07:09.110","Text":"then we have 360 plus 20 is 380,"},{"Start":"07:09.110 ","End":"07:12.425","Text":"and that\u0027s already not in the range,"},{"Start":"07:12.425 ","End":"07:13.865","Text":"but let\u0027s try the other way."},{"Start":"07:13.865 ","End":"07:16.145","Text":"Let\u0027s try n = minus 1."},{"Start":"07:16.145 ","End":"07:20.475","Text":"Then we get 20 minus 120,"},{"Start":"07:20.475 ","End":"07:22.710","Text":"which is minus 100 degrees."},{"Start":"07:22.710 ","End":"07:25.550","Text":"That\u0027s also out of range."},{"Start":"07:25.550 ","End":"07:27.470","Text":"We have just these 3,"},{"Start":"07:27.470 ","End":"07:29.915","Text":"but that\u0027s from this family of solutions."},{"Start":"07:29.915 ","End":"07:35.420","Text":"Then we try the same thing with 40 and we\u0027ll find that 40 is okay."},{"Start":"07:35.420 ","End":"07:38.975","Text":"If n is 1, then it\u0027s a 160,"},{"Start":"07:38.975 ","End":"07:40.565","Text":"and that\u0027s within range,"},{"Start":"07:40.565 ","End":"07:42.160","Text":"and if n is 2,"},{"Start":"07:42.160 ","End":"07:45.150","Text":"then we get 280 degrees."},{"Start":"07:45.150 ","End":"07:47.765","Text":"That\u0027s also within range, but more than that,"},{"Start":"07:47.765 ","End":"07:49.775","Text":"it will be out of range and below that,"},{"Start":"07:49.775 ","End":"07:52.085","Text":"it will also be out of range."},{"Start":"07:52.085 ","End":"07:57.700","Text":"Actually we have 6 solutions."},{"Start":"07:57.700 ","End":"07:59.780","Text":"Put them in a little box here."},{"Start":"07:59.780 ","End":"08:05.315","Text":"These are the 6 solutions to this equation with this restriction."},{"Start":"08:05.315 ","End":"08:08.300","Text":"That\u0027ll do for the sign."},{"Start":"08:08.300 ","End":"08:10.010","Text":"Now let\u0027s go on to part B,"},{"Start":"08:10.010 ","End":"08:12.060","Text":"which will be the cosine."},{"Start":"08:12.170 ","End":"08:15.760","Text":"This will be part B of part 1,"},{"Start":"08:15.760 ","End":"08:18.890","Text":"where we have cosine of something,"},{"Start":"08:18.890 ","End":"08:24.370","Text":"let\u0027s say square equals cosine of something else, let\u0027s say triangle."},{"Start":"08:24.370 ","End":"08:25.955","Text":"Here\u0027s an example."},{"Start":"08:25.955 ","End":"08:27.695","Text":"It\u0027s already is in this form."},{"Start":"08:27.695 ","End":"08:33.110","Text":"I need to give you the general formula for this equation."},{"Start":"08:33.110 ","End":"08:39.695","Text":"Here it is rather similar to the one with the sin."},{"Start":"08:39.695 ","End":"08:44.845","Text":"We also have 2 infinite families of solutions."},{"Start":"08:44.845 ","End":"08:48.470","Text":"The difference is that in the case of sin here we had"},{"Start":"08:48.470 ","End":"08:55.380","Text":"180 minus. Alpha. In the case of cosine is just minus Alpha. Very similar."},{"Start":"08:55.380 ","End":"08:57.720","Text":"Let\u0027s get back to this example."},{"Start":"08:57.720 ","End":"09:02.360","Text":"Using this formula, we have 2 possibilities."},{"Start":"09:02.360 ","End":"09:10.955","Text":"We either have the 3x minus 40 using this is equal to x plus multiples"},{"Start":"09:10.955 ","End":"09:19.720","Text":"of 360 or we have the 3x minus 40 is minus Alpha,"},{"Start":"09:19.720 ","End":"09:21.115","Text":"which is minus x,"},{"Start":"09:21.115 ","End":"09:25.090","Text":"also plus multiples of 360."},{"Start":"09:25.090 ","End":"09:27.520","Text":"From the top row,"},{"Start":"09:27.520 ","End":"09:31.270","Text":"this gives us that,"},{"Start":"09:31.270 ","End":"09:37.330","Text":"if you bring the x to this side and the 40 to that side,"},{"Start":"09:37.330 ","End":"09:47.688","Text":"we get that 2x is equal to 40 plus multiples of 360."},{"Start":"09:47.688 ","End":"09:50.620","Text":"The other one gives me"},{"Start":"09:50.620 ","End":"10:00.910","Text":"that 4x equals 40 plus n times 360."},{"Start":"10:00.910 ","End":"10:06.925","Text":"After we divide the top one by 2 and the bottom one by 4,"},{"Start":"10:06.925 ","End":"10:12.130","Text":"then we will get that from these 2,"},{"Start":"10:12.130 ","End":"10:17.155","Text":"x equals from the top row dividing by 2,"},{"Start":"10:17.155 ","End":"10:23.860","Text":"we get 20 plus n times 180."},{"Start":"10:23.860 ","End":"10:26.275","Text":"From the second one,"},{"Start":"10:26.275 ","End":"10:28.615","Text":"we get that x is,"},{"Start":"10:28.615 ","End":"10:30.655","Text":"dividing it by 4,"},{"Start":"10:30.655 ","End":"10:37.045","Text":"10 degrees plus n times 90 degrees,"},{"Start":"10:37.045 ","End":"10:39.580","Text":"and that would be the answer for that."},{"Start":"10:39.580 ","End":"10:42.820","Text":"You know what? Instead of going to the next example,"},{"Start":"10:42.820 ","End":"10:48.835","Text":"I\u0027ll modify this example and add a condition that x has to be"},{"Start":"10:48.835 ","End":"10:56.365","Text":"between 0 and 360 degrees."},{"Start":"10:56.365 ","End":"10:59.530","Text":"Let\u0027s see how that will change things."},{"Start":"10:59.530 ","End":"11:00.850","Text":"When I have a condition like this,"},{"Start":"11:00.850 ","End":"11:06.055","Text":"I like to write a few of these out."},{"Start":"11:06.055 ","End":"11:09.460","Text":"If n is 0,"},{"Start":"11:09.460 ","End":"11:12.280","Text":"I\u0027d get 20 degrees,"},{"Start":"11:12.280 ","End":"11:14.950","Text":"if n is minus 1,"},{"Start":"11:14.950 ","End":"11:20.380","Text":"then I get minus 160 so that\u0027s too low."},{"Start":"11:20.380 ","End":"11:23.275","Text":"I can try the next one, n is 1,"},{"Start":"11:23.275 ","End":"11:28.150","Text":"20 plus 180 is 200."},{"Start":"11:28.150 ","End":"11:32.740","Text":"That\u0027s still in range but if I take n equals 2,"},{"Start":"11:32.740 ","End":"11:34.360","Text":"then they\u0027ll be out of range,"},{"Start":"11:34.360 ","End":"11:37.750","Text":"so I\u0027ll get 2 solutions from this family."},{"Start":"11:37.750 ","End":"11:40.480","Text":"Now, from here, if n is 0,"},{"Start":"11:40.480 ","End":"11:42.205","Text":"I\u0027ve got 10 degrees."},{"Start":"11:42.205 ","End":"11:48.070","Text":"If n is negative and below 0, no good."},{"Start":"11:48.070 ","End":"11:54.299","Text":"At n equals 1 is still good, so 100 degrees."},{"Start":"11:54.299 ","End":"11:55.830","Text":"If n is 2,"},{"Start":"11:55.830 ","End":"12:00.195","Text":"I\u0027ve got 190, that\u0027s good."},{"Start":"12:00.195 ","End":"12:02.795","Text":"If n is 3,"},{"Start":"12:02.795 ","End":"12:07.180","Text":"3 times 90 plus 10 is 280."},{"Start":"12:07.180 ","End":"12:10.045","Text":"That\u0027s good but n equals 4,"},{"Start":"12:10.045 ","End":"12:14.065","Text":"then I\u0027m already over the 360, that\u0027s not."},{"Start":"12:14.065 ","End":"12:23.570","Text":"So there was actually 6 solutions that fit this equation in this range."},{"Start":"12:23.580 ","End":"12:26.245","Text":"Here\u0027s another example,"},{"Start":"12:26.245 ","End":"12:31.870","Text":"cosine of x minus 60 is minus cosine of x minus 30."},{"Start":"12:31.870 ","End":"12:34.660","Text":"Here, the problem is the minus here."},{"Start":"12:34.660 ","End":"12:36.295","Text":"How do we get it to be cosine,"},{"Start":"12:36.295 ","End":"12:38.125","Text":"something equals cosine of something."},{"Start":"12:38.125 ","End":"12:40.135","Text":"When you have a minus cosine,"},{"Start":"12:40.135 ","End":"12:46.990","Text":"the standard trick is to use the formula that minus cosine of,"},{"Start":"12:46.990 ","End":"12:52.420","Text":"let\u0027s say Alpha is cosine of the supplement of Alpha,"},{"Start":"12:52.420 ","End":"12:55.375","Text":"which is 180 minus Alpha."},{"Start":"12:55.375 ","End":"12:59.830","Text":"We can get rid of the minus if you replace the angle by its supplement."},{"Start":"12:59.830 ","End":"13:09.850","Text":"In this case, we get cosine of x minus 60 equals cosine of,"},{"Start":"13:09.850 ","End":"13:20.604","Text":"now a 180 minus this would be a 180 plus 30 minus x."},{"Start":"13:20.604 ","End":"13:25.105","Text":"Think about it to be 210 minus x."},{"Start":"13:25.105 ","End":"13:29.485","Text":"Now we\u0027re at the ordinary case of cosine equals cosine,"},{"Start":"13:29.485 ","End":"13:37.915","Text":"and so we have possibilities that x minus 60 is either equal to what\u0027s written here,"},{"Start":"13:37.915 ","End":"13:44.335","Text":"210 minus x plus a multiple of 360 degrees,"},{"Start":"13:44.335 ","End":"13:47.680","Text":"or minus what\u0027s here,"},{"Start":"13:47.680 ","End":"13:49.000","Text":"not make it minus,"},{"Start":"13:49.000 ","End":"13:52.810","Text":"just write it in the other way round,"},{"Start":"13:52.810 ","End":"13:57.595","Text":"plus n times 360 degrees."},{"Start":"13:57.595 ","End":"13:59.260","Text":"What will this give us?"},{"Start":"13:59.260 ","End":"14:03.400","Text":"The first case will be,"},{"Start":"14:03.400 ","End":"14:08.320","Text":"if I make this equals this and bring the x to this side,"},{"Start":"14:08.320 ","End":"14:09.580","Text":"and the other to that side,"},{"Start":"14:09.580 ","End":"14:15.925","Text":"I get 2x and then the 60 to the other side,"},{"Start":"14:15.925 ","End":"14:23.830","Text":"so that will be 270 degrees plus n times 360."},{"Start":"14:23.830 ","End":"14:25.885","Text":"From here, well,"},{"Start":"14:25.885 ","End":"14:34.600","Text":"we get something funny because the x cancels and what we get is you bring the 210 over,"},{"Start":"14:34.600 ","End":"14:39.220","Text":"you get 150 degrees"},{"Start":"14:39.220 ","End":"14:45.064","Text":"is equal to n times 360 degrees,"},{"Start":"14:45.064 ","End":"14:48.370","Text":"and there is no x here at all."},{"Start":"14:48.370 ","End":"14:51.925","Text":"You just have to check if this is possible."},{"Start":"14:51.925 ","End":"14:55.420","Text":"Now, this is never possible because no whole multiple of"},{"Start":"14:55.420 ","End":"15:01.465","Text":"360 is going to be equal to 150."},{"Start":"15:01.465 ","End":"15:04.075","Text":"Since this is impossible throughout,"},{"Start":"15:04.075 ","End":"15:07.240","Text":"there is no value of x that\u0027s going to make this equal to this,"},{"Start":"15:07.240 ","End":"15:08.740","Text":"because this has nothing to do with x."},{"Start":"15:08.740 ","End":"15:12.445","Text":"Just this, and so our solution is,"},{"Start":"15:12.445 ","End":"15:14.230","Text":"just rewrite this,"},{"Start":"15:14.230 ","End":"15:17.350","Text":"is that x is equal to"},{"Start":"15:17.350 ","End":"15:25.660","Text":"135 degrees plus n times 180 degrees."},{"Start":"15:25.660 ","End":"15:29.545","Text":"The second set yields nothing."},{"Start":"15:29.545 ","End":"15:33.020","Text":"That\u0027s for part B."},{"Start":"15:33.020 ","End":"15:37.210","Text":"Part C, as you probably guessed,"},{"Start":"15:37.210 ","End":"15:38.800","Text":"is going to be the tangent,"},{"Start":"15:38.800 ","End":"15:40.750","Text":"we already did sine and cosine,"},{"Start":"15:40.750 ","End":"15:46.225","Text":"is tangent of something equals tangent of something."},{"Start":"15:46.225 ","End":"15:52.510","Text":"There\u0027s a variable in 1 or both of these, square and triangle."},{"Start":"15:52.510 ","End":"15:55.810","Text":"Here\u0027s an example,"},{"Start":"15:55.810 ","End":"15:57.310","Text":"let\u0027s make it a good one. You know what?"},{"Start":"15:57.310 ","End":"16:05.935","Text":"I\u0027ll also add an extra condition that x has to be between minus 90 and 90,"},{"Start":"16:05.935 ","End":"16:08.650","Text":"this together with this."},{"Start":"16:08.650 ","End":"16:14.815","Text":"Now, we need a general formula for how to solve tangent equals tangent."},{"Start":"16:14.815 ","End":"16:20.785","Text":"Here it is, much simpler than for sine and cosine."},{"Start":"16:20.785 ","End":"16:23.410","Text":"If the tangents are equal,"},{"Start":"16:23.410 ","End":"16:30.850","Text":"then this one is equal to this one plus multiples of 180 degrees."},{"Start":"16:30.850 ","End":"16:33.760","Text":"There\u0027s one more thing. We need a trick for how to"},{"Start":"16:33.760 ","End":"16:36.820","Text":"get rid of a minus in front of a tangent."},{"Start":"16:36.820 ","End":"16:42.265","Text":"The trick is just to remember that tangent is an odd function so that"},{"Start":"16:42.265 ","End":"16:50.140","Text":"minus tangent of an angle Theta is tangent of minus Theta,"},{"Start":"16:50.140 ","End":"16:54.235","Text":"just like putting the minus inside."},{"Start":"16:54.235 ","End":"16:57.820","Text":"In our case, what we get is"},{"Start":"16:57.820 ","End":"17:05.585","Text":"that tangent of 5x plus 20 equals tangent of,"},{"Start":"17:05.585 ","End":"17:10.135","Text":"minus of this is reversing it, x minus 80."},{"Start":"17:10.135 ","End":"17:18.160","Text":"Now, I can use this formula and say that 5x plus 20 is"},{"Start":"17:18.160 ","End":"17:26.455","Text":"equal to x minus 80 plus multiples of not 360, 180."},{"Start":"17:26.455 ","End":"17:30.235","Text":"Then if I bring this to the other side,"},{"Start":"17:30.235 ","End":"17:34.165","Text":"I\u0027ll get that 4x,"},{"Start":"17:34.165 ","End":"17:36.580","Text":"I\u0027ll bring the 20 to this side,"},{"Start":"17:36.580 ","End":"17:46.360","Text":"is equal to minus 100 plus n times 180 degrees."},{"Start":"17:46.360 ","End":"17:48.000","Text":"Then finally, I\u0027ll divide by 4,"},{"Start":"17:48.000 ","End":"17:57.790","Text":"so we\u0027ve got x equals minus 25 plus multiples of 45 degrees."},{"Start":"17:57.790 ","End":"18:00.220","Text":"That would be the general solution."},{"Start":"18:00.220 ","End":"18:05.770","Text":"Now I have to take care of this inequality, this restriction."},{"Start":"18:05.770 ","End":"18:08.185","Text":"We just do it by,"},{"Start":"18:08.185 ","End":"18:09.625","Text":"and I call it trial and error,"},{"Start":"18:09.625 ","End":"18:13.210","Text":"just plug in some values of n. Let\u0027s try."},{"Start":"18:13.210 ","End":"18:18.700","Text":"If n is 0, then we have a minus 25,"},{"Start":"18:18.700 ","End":"18:22.480","Text":"which is okay, and that\u0027s if n is 0."},{"Start":"18:22.480 ","End":"18:23.905","Text":"I\u0027ll write that here."},{"Start":"18:23.905 ","End":"18:26.605","Text":"Let\u0027s try n equals minus 1."},{"Start":"18:26.605 ","End":"18:28.390","Text":"If it\u0027s minus 1,"},{"Start":"18:28.390 ","End":"18:38.050","Text":"then I\u0027ve got minus 45 minus 25 is minus 70 degrees."},{"Start":"18:38.050 ","End":"18:40.840","Text":"That\u0027s also good. Obviously,"},{"Start":"18:40.840 ","End":"18:46.010","Text":"minus 2 is going to be below the lower limit."},{"Start":"18:46.410 ","End":"18:48.670","Text":"Let\u0027s try increasing."},{"Start":"18:48.670 ","End":"18:50.500","Text":"Let\u0027s try n equals 1,"},{"Start":"18:50.500 ","End":"18:54.310","Text":"and then we have 45 minus 25 is 20."},{"Start":"18:54.310 ","End":"18:57.295","Text":"That\u0027s also good, n is 2,"},{"Start":"18:57.295 ","End":"19:06.985","Text":"then we have 90 minus 25, is 65."},{"Start":"19:06.985 ","End":"19:10.360","Text":"When n is 3, it\u0027s going to be too much,"},{"Start":"19:10.360 ","End":"19:14.580","Text":"it\u0027s going to be 135 minus 25, obviously over 90."},{"Start":"19:14.580 ","End":"19:18.150","Text":"These are the 4 solutions."},{"Start":"19:18.270 ","End":"19:22.280","Text":"I\u0027ll settle for this one example because there are"},{"Start":"19:22.280 ","End":"19:27.510","Text":"many solved examples following this tutorial."}],"ID":14347},{"Watched":false,"Name":"Trigonometric Equations - Part 2","Duration":"18m 22s","ChapterTopicVideoID":13607,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13607.jpeg","UploadDate":"2021-06-29T13:56:51.3570000","DurationForVideoObject":"PT18M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.505","Text":"Now we come to part 2 of trigonometric equations altogether those 3 parts."},{"Start":"00:05.505 ","End":"00:10.935","Text":"This part involves a technique called substitution,"},{"Start":"00:10.935 ","End":"00:16.140","Text":"which is best explained through the use of an example."},{"Start":"00:16.140 ","End":"00:22.530","Text":"Let\u0027s take 2 sine squared x minus"},{"Start":"00:22.530 ","End":"00:29.160","Text":"7 sine x plus 3 equals 0."},{"Start":"00:29.160 ","End":"00:30.225","Text":"If you look at it,"},{"Start":"00:30.225 ","End":"00:35.160","Text":"you\u0027ll see that x only appears through sine x,"},{"Start":"00:35.160 ","End":"00:37.278","Text":"so what we do is substitute,"},{"Start":"00:37.278 ","End":"00:38.970","Text":"it could be any letter,"},{"Start":"00:38.970 ","End":"00:40.605","Text":"but t is the common letter,"},{"Start":"00:40.605 ","End":"00:43.020","Text":"we let t equals sine x,"},{"Start":"00:43.020 ","End":"00:48.820","Text":"and then everything becomes in terms of t. So what we get is 2t"},{"Start":"00:48.820 ","End":"00:57.610","Text":"squared minus 7t plus 3 equals 0."},{"Start":"00:57.610 ","End":"00:59.970","Text":"That\u0027s a quadratic equation."},{"Start":"00:59.970 ","End":"01:01.670","Text":"Let\u0027s quickly solve that,"},{"Start":"01:01.670 ","End":"01:06.260","Text":"and use the formula t equals minus b plus,"},{"Start":"01:06.260 ","End":"01:12.615","Text":"or minus the square root of b squared minus 4ac,"},{"Start":"01:12.615 ","End":"01:15.120","Text":"4 times 2 times 3,"},{"Start":"01:15.120 ","End":"01:19.125","Text":"all this over 2a, which is 4."},{"Start":"01:19.125 ","End":"01:25.260","Text":"Now, 4 times 2 times 3 is 8 times 3 is 24,"},{"Start":"01:25.260 ","End":"01:27.735","Text":"from 49 is 25."},{"Start":"01:27.735 ","End":"01:31.220","Text":"So what we get is 7 plus,"},{"Start":"01:31.220 ","End":"01:32.960","Text":"or minus square root of 25,"},{"Start":"01:32.960 ","End":"01:35.970","Text":"which is 5 over 4."},{"Start":"01:35.970 ","End":"01:38.249","Text":"Let\u0027s take the 2 cases for the plus,"},{"Start":"01:38.249 ","End":"01:39.960","Text":"and for the minus,"},{"Start":"01:39.960 ","End":"01:42.150","Text":"7 plus 5 is 12,"},{"Start":"01:42.150 ","End":"01:44.460","Text":"over 4 is 3,"},{"Start":"01:44.460 ","End":"01:47.445","Text":"and 7 minus 5 is 2,"},{"Start":"01:47.445 ","End":"01:51.015","Text":"over 4, which is 1/5."},{"Start":"01:51.015 ","End":"01:54.255","Text":"So that\u0027s for t. Now,"},{"Start":"01:54.255 ","End":"01:56.590","Text":"if t equals 3,"},{"Start":"01:56.590 ","End":"01:59.105","Text":"since t equals sine x,"},{"Start":"01:59.105 ","End":"02:03.260","Text":"we get sine x equals 3."},{"Start":"02:03.260 ","End":"02:10.535","Text":"But, that is impossible because the sign is always between minus 1, and 1."},{"Start":"02:10.535 ","End":"02:13.310","Text":"There\u0027s nothing whose sine is equal to 3,"},{"Start":"02:13.310 ","End":"02:14.900","Text":"it\u0027s outside the range."},{"Start":"02:14.900 ","End":"02:17.300","Text":"So we get no solutions from here,"},{"Start":"02:17.300 ","End":"02:19.000","Text":"so then we try,"},{"Start":"02:19.000 ","End":"02:21.200","Text":"we can get some solutions from here."},{"Start":"02:21.200 ","End":"02:24.905","Text":"If t equals 1/ 5, t is sine x."},{"Start":"02:24.905 ","End":"02:28.768","Text":"So sin x equals 1/5,"},{"Start":"02:28.768 ","End":"02:33.370","Text":"and that brings us to the technique of sine something equals sign something."},{"Start":"02:33.370 ","End":"02:37.640","Text":"Now, 1/2 is a sine of a special angle,"},{"Start":"02:37.640 ","End":"02:42.095","Text":"if you remember, it\u0027s the sine of 30 degrees."},{"Start":"02:42.095 ","End":"02:48.230","Text":"You\u0027re expected to know this, or at least have the tables handy of special angles."},{"Start":"02:48.230 ","End":"02:51.005","Text":"Sin x is sine 30,"},{"Start":"02:51.005 ","End":"02:55.180","Text":"and now we use the usual technique when sine equals sine,"},{"Start":"02:55.180 ","End":"02:59.730","Text":"and that is that x is equal to 1 of 2 things,"},{"Start":"02:59.730 ","End":"03:05.900","Text":"it\u0027s either the 30 degrees plus multiples of a whole circle,"},{"Start":"03:05.900 ","End":"03:09.320","Text":"n times 360 degrees,"},{"Start":"03:09.320 ","End":"03:13.204","Text":"or the supplement of this angle,"},{"Start":"03:13.204 ","End":"03:16.595","Text":"supplement is 180 minus it,"},{"Start":"03:16.595 ","End":"03:19.700","Text":"which is 150, again,"},{"Start":"03:19.700 ","End":"03:22.760","Text":"plus multiples of a whole circle,"},{"Start":"03:22.760 ","End":"03:24.740","Text":"where n is any integer,"},{"Start":"03:24.740 ","End":"03:27.890","Text":"positive or negative or 0."},{"Start":"03:27.890 ","End":"03:33.320","Text":"That\u0027s the solution. Now,"},{"Start":"03:33.320 ","End":"03:39.710","Text":"another example, and I\u0027m going to base it on this example to save some work."},{"Start":"03:39.710 ","End":"03:45.500","Text":"Suppose I have 2 cosine squared"},{"Start":"03:45.500 ","End":"03:55.130","Text":"2x minus 7 cosine 2x plus 3 equals 0."},{"Start":"03:55.130 ","End":"03:57.275","Text":"I\u0027ve already used the quadratic equation."},{"Start":"03:57.275 ","End":"04:00.350","Text":"You can see that this is also a substitution,"},{"Start":"04:00.350 ","End":"04:07.140","Text":"and this time we would substitute t equals cosine of 2x."},{"Start":"04:08.710 ","End":"04:14.220","Text":"I\u0027m going to reuse the result of the,"},{"Start":"04:14.220 ","End":"04:16.910","Text":"I mean, we already did the quadratic equation,"},{"Start":"04:16.910 ","End":"04:19.230","Text":"so we know,"},{"Start":"04:21.770 ","End":"04:25.245","Text":"let\u0027s start from here, instead of sin x,"},{"Start":"04:25.245 ","End":"04:26.760","Text":"we have cosine 2x."},{"Start":"04:26.760 ","End":"04:32.205","Text":"So we know that cosine 2x is equal to 3,"},{"Start":"04:32.205 ","End":"04:40.250","Text":"or cosine 2x is equal to 1/2."},{"Start":"04:40.250 ","End":"04:44.750","Text":"Now, again, the cosine of something, just like sine,"},{"Start":"04:44.750 ","End":"04:47.975","Text":"something can\u0027t be outside the range from minus 1 to 1,"},{"Start":"04:47.975 ","End":"04:50.285","Text":"so this can be ruled out."},{"Start":"04:50.285 ","End":"04:56.000","Text":"We just have the 1/2 as a possibility, and once again,"},{"Start":"04:56.000 ","End":"05:03.470","Text":"using special angles, we know that 1/2 is cosine of 60 degrees."},{"Start":"05:03.470 ","End":"05:08.210","Text":"So we have cosine of something equals cosine of something."},{"Start":"05:08.210 ","End":"05:11.865","Text":"Then we get that 2x is,"},{"Start":"05:11.865 ","End":"05:14.585","Text":"remember the rule for cosine,"},{"Start":"05:14.585 ","End":"05:22.430","Text":"it\u0027s either equal to this angle here plus multiples of the whole circle,"},{"Start":"05:22.430 ","End":"05:25.785","Text":"n times 360 degrees,"},{"Start":"05:25.785 ","End":"05:28.940","Text":"or in the case of sine,"},{"Start":"05:28.940 ","End":"05:30.350","Text":"it was a supplement of the angle,"},{"Start":"05:30.350 ","End":"05:33.275","Text":"in the case of cosine is just the negative of this,"},{"Start":"05:33.275 ","End":"05:39.480","Text":"minus 60 plus n times 360."},{"Start":"05:40.100 ","End":"05:45.995","Text":"All we have to do now is divide by 2 to get what x is."},{"Start":"05:45.995 ","End":"05:48.930","Text":"So x is either,"},{"Start":"05:50.990 ","End":"05:55.650","Text":"I can just save time and write instead of 30,"},{"Start":"05:55.650 ","End":"05:58.445","Text":"or minus 30 is sometimes we could do it this way,"},{"Start":"05:58.445 ","End":"06:01.370","Text":"plus or minus 30 degrees,"},{"Start":"06:01.370 ","End":"06:03.835","Text":"I guess I don\u0027t need this,"},{"Start":"06:03.835 ","End":"06:05.605","Text":"and then plus,"},{"Start":"06:05.605 ","End":"06:12.350","Text":"in each case it\u0027s going to be n times 180 degrees."},{"Start":"06:12.350 ","End":"06:14.465","Text":"That\u0027s a variation."},{"Start":"06:14.465 ","End":"06:18.610","Text":"Now let\u0027s go on to do another example."},{"Start":"06:18.610 ","End":"06:21.450","Text":"For the next example,"},{"Start":"06:21.450 ","End":"06:22.850","Text":"let\u0027s see what we\u0027ll do."},{"Start":"06:22.850 ","End":"06:32.110","Text":"We\u0027ll take square root of 2 cosine squared x plus,"},{"Start":"06:32.110 ","End":"06:33.840","Text":"let\u0027s make it matter,"},{"Start":"06:33.840 ","End":"06:40.020","Text":"plus sin x equals 0."},{"Start":"06:40.020 ","End":"06:43.880","Text":"Here it\u0027s not immediately clear what the substitute,"},{"Start":"06:43.880 ","End":"06:47.465","Text":"because we have sin x and we have cosine x."},{"Start":"06:47.465 ","End":"06:51.695","Text":"This is where we need to use some trigonometric identities."},{"Start":"06:51.695 ","End":"06:54.830","Text":"For example, we know that cosine squared,"},{"Start":"06:54.830 ","End":"06:56.915","Text":"and sine squared are related,"},{"Start":"06:56.915 ","End":"07:00.140","Text":"and maybe we can convert cosine squared to sine squared."},{"Start":"07:00.140 ","End":"07:09.110","Text":"In fact, we have that cosine squared alpha is 1 minus sin squared Alpha,"},{"Start":"07:09.110 ","End":"07:11.380","Text":"because cosine squared plus sine squared is 1,"},{"Start":"07:11.380 ","End":"07:14.945","Text":"that\u0027s one of the most basic Pythagorean identities."},{"Start":"07:14.945 ","End":"07:17.680","Text":"So in this case, we\u0027ll use this,"},{"Start":"07:17.680 ","End":"07:21.960","Text":"and if we substitute this as 1 minus sine squared"},{"Start":"07:21.960 ","End":"07:29.610","Text":"x then what we get is square root of 2."},{"Start":"07:29.820 ","End":"07:32.905","Text":"Let me put the minus here,"},{"Start":"07:32.905 ","End":"07:37.940","Text":"minus square root of 2 sine^2x."},{"Start":"07:38.700 ","End":"07:44.740","Text":"Then I have plus square root of 2,"},{"Start":"07:44.740 ","End":"07:47.740","Text":"but I\u0027ll put that here plus square root of 2."},{"Start":"07:47.740 ","End":"07:55.120","Text":"Then I want the sine x= 0."},{"Start":"07:55.120 ","End":"07:57.790","Text":"Now we can make the substitution."},{"Start":"07:57.790 ","End":"08:05.660","Text":"We can substitute t = sine x so if we plug that in here,"},{"Start":"08:06.360 ","End":"08:10.375","Text":"I\u0027d rather change the signs and make,"},{"Start":"08:10.375 ","End":"08:11.920","Text":"I like to start with a plus."},{"Start":"08:11.920 ","End":"08:16.510","Text":"Let\u0027s make this as plus,"},{"Start":"08:16.510 ","End":"08:19.930","Text":"then minus, then minus and we\u0027ll be okay."},{"Start":"08:19.930 ","End":"08:26.215","Text":"Root 2t^2 minus t"},{"Start":"08:26.215 ","End":"08:33.085","Text":"minus root 2 = 0 straightforward quadratic."},{"Start":"08:33.085 ","End":"08:39.490","Text":"Let\u0027s solve it, t = minus b plus or minus the square root"},{"Start":"08:39.490 ","End":"08:45.850","Text":"of b^2 minus 4ac that\u0027s minus with a minus,"},{"Start":"08:45.850 ","End":"08:54.318","Text":"minus is going to be be plus 4 times root 2 times root 2,"},{"Start":"08:54.318 ","End":"09:01.105","Text":"and all this over twice root 2 is 2a."},{"Start":"09:01.105 ","End":"09:05.440","Text":"Let\u0027s see, this equals 1 plus or minus, now look,"},{"Start":"09:05.440 ","End":"09:10.045","Text":"root 2 times root 2 is 2 times 4 is 8 plus 1 is 9,"},{"Start":"09:10.045 ","End":"09:12.535","Text":"square root of 9 is 3."},{"Start":"09:12.535 ","End":"09:14.829","Text":"This is what we get."},{"Start":"09:14.829 ","End":"09:16.840","Text":"Now we\u0027ll take 2 cases,"},{"Start":"09:16.840 ","End":"09:19.360","Text":"one is with the plus and one is with the minus."},{"Start":"09:19.360 ","End":"09:24.970","Text":"If we take the plus, we have 4 over 2 root 2,"},{"Start":"09:24.970 ","End":"09:33.565","Text":"which is 2 over root 2 and 2 over root 2 happens to equal root"},{"Start":"09:33.565 ","End":"09:44.540","Text":"2,1 minus 3 is minus 2 so we get minus 1 over root 2."},{"Start":"09:44.730 ","End":"09:50.175","Text":"Now, back to x,"},{"Start":"09:50.175 ","End":"09:53.010","Text":"we need to substitute t = sine x."},{"Start":"09:53.010 ","End":"10:03.415","Text":"So this would give us that sine x is equal to root 2 because that\u0027s t, and t is sine x."},{"Start":"10:03.415 ","End":"10:09.880","Text":"That\u0027s impossible because root 2 is like 1.4 something."},{"Start":"10:09.880 ","End":"10:15.745","Text":"Sine x can\u0027t be bigger than 1 or less than minus 1 so this is not possible."},{"Start":"10:15.745 ","End":"10:19.150","Text":"Now, this is possible."},{"Start":"10:19.150 ","End":"10:24.880","Text":"We have the sine x is minus 1 over root 2."},{"Start":"10:24.880 ","End":"10:31.300","Text":"Though sometimes it\u0027s written as root 2 over 2, it\u0027s the same thing."},{"Start":"10:31.300 ","End":"10:32.890","Text":"If I didn\u0027t have the minus,"},{"Start":"10:32.890 ","End":"10:35.875","Text":"it would be 45 degrees,"},{"Start":"10:35.875 ","End":"10:39.258","Text":"but since it does have the minus,"},{"Start":"10:39.258 ","End":"10:41.050","Text":"and sine is an odd function."},{"Start":"10:41.050 ","End":"10:44.305","Text":"It has to be the sine not of 45,"},{"Start":"10:44.305 ","End":"10:48.160","Text":"but of minus 45 degrees."},{"Start":"10:48.160 ","End":"10:56.440","Text":"Now we have that sine of something equals sine of something."},{"Start":"10:56.440 ","End":"11:05.770","Text":"There\u0027s the standard formula that x is going to equal either the angle"},{"Start":"11:05.770 ","End":"11:16.285","Text":"here minus 45 degrees plus multiples of 360,"},{"Start":"11:16.285 ","End":"11:19.345","Text":"or the supplement of this angle,"},{"Start":"11:19.345 ","End":"11:22.795","Text":"supplement means a 180 minus,"},{"Start":"11:22.795 ","End":"11:27.010","Text":"180 minus this is going to be"},{"Start":"11:27.010 ","End":"11:35.450","Text":"225 degrees plus n times 360 degrees."},{"Start":"11:36.210 ","End":"11:39.189","Text":"That\u0027s it for this example."},{"Start":"11:39.189 ","End":"11:43.000","Text":"Continuing with part two of trigonometric equations,"},{"Start":"11:43.000 ","End":"11:46.630","Text":"which is using the method of substitution."},{"Start":"11:46.630 ","End":"11:49.975","Text":"Let\u0027s take a more difficult example."},{"Start":"11:49.975 ","End":"11:57.550","Text":"For cosine^4x plus 3 cosine(2x) = 1."},{"Start":"11:57.550 ","End":"12:00.475","Text":"It\u0027s not clear at all what to substitute."},{"Start":"12:00.475 ","End":"12:05.200","Text":"You have to know your trigonometric identities pretty well."},{"Start":"12:05.200 ","End":"12:07.780","Text":"When you\u0027re solving trigonometric equations,"},{"Start":"12:07.780 ","End":"12:11.275","Text":"you should have a table of trigonometric identities."},{"Start":"12:11.275 ","End":"12:16.360","Text":"There\u0027s a trigonometric identity for cosine of twice an angle,"},{"Start":"12:16.360 ","End":"12:23.035","Text":"say twice Alpha is 1 minus 2 sine squared Alpha."},{"Start":"12:23.035 ","End":"12:29.575","Text":"The other formula I\u0027m going to use is I see that cosine^4 is (cosine^2)^2."},{"Start":"12:29.575 ","End":"12:34.480","Text":"We know that cosine^2 Alpha can also be written in terms of"},{"Start":"12:34.480 ","End":"12:40.000","Text":"sine Alpha is 1 minus sine squared alpha."},{"Start":"12:40.000 ","End":"12:42.805","Text":"Let\u0027s see what we get over here."},{"Start":"12:42.805 ","End":"12:51.925","Text":"We get 4 the cosine^4 is (cosine^2)^2 but cosine^2,"},{"Start":"12:51.925 ","End":"12:56.815","Text":"we said was 1 minus sine^2x."},{"Start":"12:56.815 ","End":"12:58.615","Text":"That\u0027s for this bit."},{"Start":"12:58.615 ","End":"13:08.530","Text":"For this bit we have 3 times the cosine of 2x will be 1 minus 2 sine^2x,"},{"Start":"13:08.530 ","End":"13:13.810","Text":"and then = 1."},{"Start":"13:13.810 ","End":"13:16.405","Text":"Now, here\u0027s where we substitute."},{"Start":"13:16.405 ","End":"13:18.460","Text":"You could be tempted to say, look,"},{"Start":"13:18.460 ","End":"13:24.070","Text":"everything\u0027s in terms of sine x and to say let t equals sine x."},{"Start":"13:24.070 ","End":"13:28.585","Text":"But then we\u0027ll get a degree four equation because sine squared squared."},{"Start":"13:28.585 ","End":"13:32.425","Text":"In fact, if you notice the sign only appears in sine^2."},{"Start":"13:32.425 ","End":"13:37.930","Text":"So what we\u0027re actually going to substitute is t = sine^2x,"},{"Start":"13:37.930 ","End":"13:39.595","Text":"and after we do that,"},{"Start":"13:39.595 ","End":"13:47.740","Text":"then we get 4(1 minus t^2 plus 3 times"},{"Start":"13:47.740 ","End":"13:56.995","Text":"1 minus 2t minus will bring this over, minus 1 =0."},{"Start":"13:56.995 ","End":"13:59.575","Text":"As for this bit,"},{"Start":"13:59.575 ","End":"14:02.110","Text":"just in case you forgot your algebra."},{"Start":"14:02.110 ","End":"14:05.755","Text":"A minus b^2 is a^2"},{"Start":"14:05.755 ","End":"14:12.340","Text":"minus 2ab plus b^2."},{"Start":"14:12.340 ","End":"14:14.860","Text":"So just write that underneath here."},{"Start":"14:14.860 ","End":"14:18.880","Text":"That\u0027s 1 minus 2t plus t^2,"},{"Start":"14:18.880 ","End":"14:26.170","Text":"4 times this is 4 minus 8t plus"},{"Start":"14:26.170 ","End":"14:35.230","Text":"4t^2 plus 3 minus 6t minus 1 equals 0."},{"Start":"14:35.230 ","End":"14:40.045","Text":"Now collecting t^2, we have 4t^2."},{"Start":"14:40.045 ","End":"14:42.790","Text":"How many t minus 8,"},{"Start":"14:42.790 ","End":"14:47.320","Text":"minus 6 minus 14t and then"},{"Start":"14:47.320 ","End":"14:54.100","Text":"4 plus 3 minus 1 is 6 equals 0."},{"Start":"14:54.100 ","End":"14:56.725","Text":"I like to simplify everything is even."},{"Start":"14:56.725 ","End":"14:58.615","Text":"Let\u0027s divide by 2,"},{"Start":"14:58.615 ","End":"15:05.960","Text":"and get 2t^2 minus 7t plus 3 equals 0,"},{"Start":"15:05.960 ","End":"15:10.385","Text":"a quadratic equation in t. Now,"},{"Start":"15:10.385 ","End":"15:13.910","Text":"I don\u0027t really want to practice quadratic equations too much."},{"Start":"15:13.910 ","End":"15:17.540","Text":"I\u0027m just going to give you the answer assuming you know how to solve quadratic equations,"},{"Start":"15:17.540 ","End":"15:19.205","Text":"we get two solutions."},{"Start":"15:19.205 ","End":"15:24.440","Text":"We get t equals 3 or t = 1.5."},{"Start":"15:24.440 ","End":"15:26.150","Text":"After we\u0027ve got that,"},{"Start":"15:26.150 ","End":"15:30.155","Text":"now we go back and substitute t = sine^2x."},{"Start":"15:30.155 ","End":"15:34.850","Text":"We either have sine^2x = 3,"},{"Start":"15:34.850 ","End":"15:41.740","Text":"or sine^2x = 1.5."},{"Start":"15:41.740 ","End":"15:48.445","Text":"Now, this is not possible because sine is between minus 1 and 1,"},{"Start":"15:48.445 ","End":"15:51.398","Text":"then the event that absolute value It\u0027s less than 1,"},{"Start":"15:51.398 ","End":"15:52.855","Text":"and if you square it,"},{"Start":"15:52.855 ","End":"15:56.680","Text":"that\u0027s still gonna be less than 1 in absolute value,"},{"Start":"15:56.680 ","End":"15:59.545","Text":"so sine^2 cannot be 3."},{"Start":"15:59.545 ","End":"16:04.360","Text":"So we\u0027re left with this but that gives us two possibilities."},{"Start":"16:04.360 ","End":"16:13.420","Text":"We could have sine x being 1 over root 2, or negative that."},{"Start":"16:13.420 ","End":"16:16.580","Text":"So I\u0027ll just put a plus, or a minus here."},{"Start":"16:16.800 ","End":"16:23.210","Text":"Now 1 over root 2 is a famous sine of an angle,"},{"Start":"16:23.210 ","End":"16:29.820","Text":"or it\u0027s a sign of a famous well-known angles,"},{"Start":"16:29.820 ","End":"16:32.575","Text":"in fact, 45 degrees."},{"Start":"16:32.575 ","End":"16:41.420","Text":"This is either equal to sine of 45 degrees if we take the plus and if we take the minus,"},{"Start":"16:41.420 ","End":"16:43.129","Text":"because sine is an odd function,"},{"Start":"16:43.129 ","End":"16:48.140","Text":"this will be the sine of minus 45 degrees."},{"Start":"16:48.140 ","End":"16:51.740","Text":"Now we have the case where sine something equals sign something,"},{"Start":"16:51.740 ","End":"16:53.765","Text":"and we know how to solve that."},{"Start":"16:53.765 ","End":"16:55.730","Text":"If we take the first case,"},{"Start":"16:55.730 ","End":"17:00.450","Text":"we have that x ="},{"Start":"17:00.450 ","End":"17:10.900","Text":"45 degrees plus multiples of 360 and the other possibility with the sign is to take"},{"Start":"17:10.900 ","End":"17:15.370","Text":"the supplement 180 minus 45 is"},{"Start":"17:15.370 ","End":"17:22.330","Text":"135 plus n times 360 degrees."},{"Start":"17:22.330 ","End":"17:27.290","Text":"That\u0027s just for the case of 45,"},{"Start":"17:27.600 ","End":"17:34.675","Text":"but we also have the case of minus 45 and that\u0027ll give us two other possibilities."},{"Start":"17:34.675 ","End":"17:37.060","Text":"Just to be rearranging here,"},{"Start":"17:37.060 ","End":"17:39.820","Text":"so we\u0027ll actually get 4 things."},{"Start":"17:39.820 ","End":"17:49.370","Text":"Now, here we\u0027ll get minus 45 degrees plus all multiples of 360,"},{"Start":"17:49.370 ","End":"17:55.670","Text":"and the supplement of this is a 180 minus this will give us"},{"Start":"17:55.670 ","End":"18:02.425","Text":"225 degrees plus n times 360 degrees."},{"Start":"18:02.425 ","End":"18:10.840","Text":"There\u0027s 4 families of solutions that concludes this example,"},{"Start":"18:10.840 ","End":"18:12.770","Text":"but also part two,"},{"Start":"18:12.770 ","End":"18:15.590","Text":"and I want to remind you that there are"},{"Start":"18:15.590 ","End":"18:22.590","Text":"more solved examples following the tutorial. Okay, That\u0027s it."}],"ID":14348},{"Watched":false,"Name":"Trigonometric Equations - Part 3","Duration":"12m 38s","ChapterTopicVideoID":13608,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13608.jpeg","UploadDate":"2021-06-29T13:57:41.6200000","DurationForVideoObject":"PT12M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.168","Text":"Now we come to the third,"},{"Start":"00:02.168 ","End":"00:06.810","Text":"and last part of trigonometric equations."},{"Start":"00:06.810 ","End":"00:09.645","Text":"I wasn\u0027t quite sure what to call this,"},{"Start":"00:09.645 ","End":"00:12.960","Text":"I\u0027ve called it factors of zero."},{"Start":"00:12.960 ","End":"00:16.185","Text":"What I mean by this,"},{"Start":"00:16.185 ","End":"00:18.420","Text":"is that if we have an equation,"},{"Start":"00:18.420 ","End":"00:21.390","Text":"and it\u0027s arranged that something equals 0,"},{"Start":"00:21.390 ","End":"00:24.840","Text":"and if we can factorize the left-hand side as to a product,"},{"Start":"00:24.840 ","End":"00:27.375","Text":"say a times b is 0,"},{"Start":"00:27.375 ","End":"00:32.955","Text":"then we can break it down as a=0 or b=0."},{"Start":"00:32.955 ","End":"00:35.560","Text":"This is the logical symbol for all."},{"Start":"00:37.130 ","End":"00:39.203","Text":"Let\u0027s start with an example,"},{"Start":"00:39.203 ","End":"00:41.025","Text":"and you\u0027ll see what I mean."},{"Start":"00:41.025 ","End":"00:46.590","Text":"Let\u0027s take 2 sine x cosine"},{"Start":"00:46.590 ","End":"00:54.400","Text":"x minus cosine x=0."},{"Start":"00:54.400 ","End":"00:57.755","Text":"It\u0027s already written as something equals 0."},{"Start":"00:57.755 ","End":"00:59.113","Text":"What we want to do is try,"},{"Start":"00:59.113 ","End":"01:01.700","Text":"and factorize the left-hand side."},{"Start":"01:01.700 ","End":"01:06.035","Text":"If we look at it, we see that cosine x comes outside the brackets."},{"Start":"01:06.035 ","End":"01:14.040","Text":"We get cosine x times 2 sine x minus 1=0."},{"Start":"01:14.040 ","End":"01:17.760","Text":"This gives us two possibilities."},{"Start":"01:17.760 ","End":"01:25.610","Text":"We either have that cosine x=0 or that this equals 0,"},{"Start":"01:25.610 ","End":"01:32.870","Text":"which gives us that sin x=1/2."},{"Start":"01:32.870 ","End":"01:36.035","Text":"Now, cosine x=0,"},{"Start":"01:36.035 ","End":"01:43.005","Text":"0 is equal to cosine of 90 degrees."},{"Start":"01:43.005 ","End":"01:50.430","Text":"The 1/2 is equal to sine of 30 degrees."},{"Start":"01:50.430 ","End":"01:56.220","Text":"Now we can use the techniques for sine equal sine and cosine equal cosine."},{"Start":"02:05.750 ","End":"02:09.330","Text":"Let\u0027s combine the two."},{"Start":"02:09.330 ","End":"02:11.083","Text":"We usually have 90,"},{"Start":"02:11.083 ","End":"02:12.590","Text":"and then minus 90."},{"Start":"02:12.590 ","End":"02:17.660","Text":"It\u0027s plus or minus 90 degrees plus multiples,"},{"Start":"02:17.660 ","End":"02:24.050","Text":"which means n times the whole circle, 360 degrees."},{"Start":"02:24.050 ","End":"02:27.305","Text":"That\u0027s for this one."},{"Start":"02:27.305 ","End":"02:29.480","Text":"For the other one,"},{"Start":"02:29.480 ","End":"02:34.985","Text":"we get that x is equal to"},{"Start":"02:34.985 ","End":"02:41.530","Text":"either 30 degrees plus whole circles."},{"Start":"02:41.530 ","End":"02:44.280","Text":"All the supplement of this angle,"},{"Start":"02:44.280 ","End":"02:52.740","Text":"180 minus 30 is 150 also plus n times 360 degrees."},{"Start":"02:52.740 ","End":"02:58.865","Text":"There\u0027s actually four families of solutions."},{"Start":"02:58.865 ","End":"03:03.470","Text":"We have 90 minus 90,"},{"Start":"03:03.470 ","End":"03:08.555","Text":"30 or 150 degrees."},{"Start":"03:08.555 ","End":"03:17.135","Text":"All these possibilities, and each of them we add n times 360 degrees."},{"Start":"03:17.135 ","End":"03:20.150","Text":"Four infinite families of solutions."},{"Start":"03:20.150 ","End":"03:22.160","Text":"I put the degree sign."},{"Start":"03:22.160 ","End":"03:25.125","Text":"That\u0027s the first example."},{"Start":"03:25.125 ","End":"03:29.380","Text":"Now let\u0027s jump to a more difficult example."},{"Start":"03:29.380 ","End":"03:33.415","Text":"Let\u0027s take sine x plus sine"},{"Start":"03:33.415 ","End":"03:41.445","Text":"5x=cosine x plus cosine 5x."},{"Start":"03:41.445 ","End":"03:45.700","Text":"The first doesn\u0027t look like any way of factorizing this."},{"Start":"03:45.700 ","End":"03:51.310","Text":"But, we need to bring out a formula sheet for trigonometric identities."},{"Start":"03:51.310 ","End":"03:53.778","Text":"There are formulas for sine plus sine,"},{"Start":"03:53.778 ","End":"03:56.195","Text":"and cosine plus cosine."},{"Start":"03:56.195 ","End":"04:05.800","Text":"The formula we need for this is sine Alpha plus sine Beta."},{"Start":"04:05.800 ","End":"04:13.680","Text":"This is equal to twice sine of"},{"Start":"04:13.680 ","End":"04:23.460","Text":"Alpha plus Beta over 2 cosine of Alpha minus Beta over 2,"},{"Start":"04:23.460 ","End":"04:26.320","Text":"from the formula sheet."},{"Start":"04:26.320 ","End":"04:33.635","Text":"The corresponding formula for cosine is that cosine Alpha plus cosine"},{"Start":"04:33.635 ","End":"04:41.930","Text":"Beta is twice cosine Alpha plus Beta over 2."},{"Start":"04:41.930 ","End":"04:43.535","Text":"The same second 1/2,"},{"Start":"04:43.535 ","End":"04:48.810","Text":"cosine Alpha minus Beta over 2."},{"Start":"04:50.070 ","End":"04:56.200","Text":"The left-hand side using this first formula where this is Alpha,"},{"Start":"04:56.200 ","End":"05:04.090","Text":"this is Beta will give us sine of this plus this over 2 gives us 3x."},{"Start":"05:04.090 ","End":"05:06.440","Text":"It\u0027s x over 2 twice."},{"Start":"05:08.340 ","End":"05:19.511","Text":"Then cosine of this minus this over 2 comes out to be minus 2x equals,"},{"Start":"05:19.511 ","End":"05:21.880","Text":"then using the second formula."},{"Start":"05:21.880 ","End":"05:23.980","Text":"The Alpha, and the Beta are the same,"},{"Start":"05:23.980 ","End":"05:25.900","Text":"so I can just copy them from here."},{"Start":"05:25.900 ","End":"05:28.870","Text":"But, I have cosine in both cases."},{"Start":"05:28.870 ","End":"05:31.340","Text":"It\u0027s cosine (3x)."},{"Start":"05:32.190 ","End":"05:38.120","Text":"Again, cosine (-2x)."},{"Start":"05:38.120 ","End":"05:42.140","Text":"Note that I can cancel the 2 on both sides."},{"Start":"05:42.140 ","End":"05:47.270","Text":"Now let me bring everything to the left-hand side."},{"Start":"05:47.270 ","End":"05:51.020","Text":"Also notice that since cosine is an even function,"},{"Start":"05:51.020 ","End":"05:53.765","Text":"I can forget about the minus here."},{"Start":"05:53.765 ","End":"05:58.835","Text":"I get sine 3x cosine"},{"Start":"05:58.835 ","End":"06:04.245","Text":"2x minus cosine"},{"Start":"06:04.245 ","End":"06:10.920","Text":"3x cosine 2x=0."},{"Start":"06:10.920 ","End":"06:14.115","Text":"Now, cosine 2x is a common factor."},{"Start":"06:14.115 ","End":"06:24.500","Text":"I can say cosine 2x times sine 3x minus cosine 3x=0."},{"Start":"06:24.500 ","End":"06:27.484","Text":"Now we have a product equals 0."},{"Start":"06:27.484 ","End":"06:32.240","Text":"We can say that this factor is 0 or the other factor is 0."},{"Start":"06:32.240 ","End":"06:34.115","Text":"Let\u0027s take first of all,"},{"Start":"06:34.115 ","End":"06:39.840","Text":"the case where cosine 2x=0."},{"Start":"06:39.840 ","End":"06:41.220","Text":"We\u0027ll deal with that one first,"},{"Start":"06:41.220 ","End":"06:43.330","Text":"then we\u0027ll come to this."},{"Start":"06:44.660 ","End":"06:50.595","Text":"Since 0 is cosine of 90 degrees,"},{"Start":"06:50.595 ","End":"06:56.055","Text":"so we get that 2x is equal to"},{"Start":"06:56.055 ","End":"07:05.075","Text":"90 plus multiples of 360,"},{"Start":"07:05.075 ","End":"07:07.865","Text":"or minus of this."},{"Start":"07:07.865 ","End":"07:13.950","Text":"Minus 90 plus multiples of 360."},{"Start":"07:20.690 ","End":"07:22.830","Text":"I could combine them."},{"Start":"07:22.830 ","End":"07:25.280","Text":"I could have put plus or minus 90 here."},{"Start":"07:25.280 ","End":"07:35.030","Text":"We\u0027ll get plus, or minus 45 degrees plus multiples of 180 degrees."},{"Start":"07:35.030 ","End":"07:37.294","Text":"That\u0027s for this part."},{"Start":"07:37.294 ","End":"07:39.185","Text":"Now the other part,"},{"Start":"07:39.185 ","End":"07:41.135","Text":"if this is 0."},{"Start":"07:41.135 ","End":"07:42.905","Text":"Then we get"},{"Start":"07:42.905 ","End":"07:51.835","Text":"that sine 3x=cosine 3x."},{"Start":"07:51.835 ","End":"07:53.450","Text":"Let me just maybe label this."},{"Start":"07:53.450 ","End":"07:56.855","Text":"This is part A, this is B."},{"Start":"07:56.855 ","End":"07:59.750","Text":"Then here we\u0027re working on part A,"},{"Start":"07:59.750 ","End":"08:02.465","Text":"and here we\u0027re working on part B."},{"Start":"08:02.465 ","End":"08:08.800","Text":"If this is 0, then sine 3x=cosine 3x."},{"Start":"08:08.800 ","End":"08:12.980","Text":"What I can do now is divide both sides by cosine 3x,"},{"Start":"08:12.980 ","End":"08:15.620","Text":"because cosine over cosine is tangent."},{"Start":"08:15.620 ","End":"08:20.693","Text":"I get that tangent 3x=1,"},{"Start":"08:20.693 ","End":"08:26.150","Text":"and 1 is tangent of 45 degrees."},{"Start":"08:26.150 ","End":"08:28.670","Text":"You should know all these special angles."},{"Start":"08:28.670 ","End":"08:30.875","Text":"The solution is,"},{"Start":"08:30.875 ","End":"08:32.060","Text":"for the tangent,"},{"Start":"08:32.060 ","End":"08:40.160","Text":"that 3x=45 plus multiples of 180."},{"Start":"08:40.160 ","End":"08:43.385","Text":"That\u0027s how we solve tangent something equals tangent something."},{"Start":"08:43.385 ","End":"08:44.870","Text":"But, there\u0027s a 3 here."},{"Start":"08:44.870 ","End":"08:54.980","Text":"We want just x. X is 15 degrees plus multiples of 60 degrees."},{"Start":"08:54.980 ","End":"08:57.680","Text":"That\u0027s it for this example."},{"Start":"08:57.680 ","End":"09:00.270","Text":"I think we\u0027ll do one more."},{"Start":"09:00.280 ","End":"09:04.075","Text":"We\u0027ll do this last example on a new page,"},{"Start":"09:04.075 ","End":"09:10.580","Text":"and the example will be sine"},{"Start":"09:11.240 ","End":"09:22.210","Text":"5x=sine 3x minus 2 sine x."},{"Start":"09:22.210 ","End":"09:26.720","Text":"We\u0027re going to use the formulas for difference of sines."},{"Start":"09:26.720 ","End":"09:28.550","Text":"I\u0027ll bring first of all,"},{"Start":"09:28.550 ","End":"09:34.345","Text":"this to the other side and we get sine 5x minus sine"},{"Start":"09:34.345 ","End":"09:41.060","Text":"3x= minus 2 sine x."},{"Start":"09:41.060 ","End":"09:45.385","Text":"Now we need the formula for sine minus sine."},{"Start":"09:45.385 ","End":"09:47.330","Text":"It goes as follows."},{"Start":"09:47.330 ","End":"09:52.490","Text":"Sine of Alpha minus sine of Beta is"},{"Start":"09:52.490 ","End":"09:59.350","Text":"2 sine of Alpha minus Beta over 2."},{"Start":"09:59.350 ","End":"10:06.865","Text":"Then cosine Alpha plus Beta over 2."},{"Start":"10:06.865 ","End":"10:10.810","Text":"Let\u0027s see what this does for our left-hand side."},{"Start":"10:10.810 ","End":"10:13.590","Text":"We get 2 from here,"},{"Start":"10:13.590 ","End":"10:16.380","Text":"sine of 1/2 the difference."},{"Start":"10:16.380 ","End":"10:18.675","Text":"Sine of this minus this over 2."},{"Start":"10:18.675 ","End":"10:24.075","Text":"It comes sine, 5x minus 3x over 2 is just x."},{"Start":"10:24.075 ","End":"10:28.920","Text":"Then cosine Alpha plus Beta over 2,"},{"Start":"10:28.920 ","End":"10:31.770","Text":"5x plus 3x over 2 is"},{"Start":"10:31.770 ","End":"10:40.630","Text":"4x= minus 2 sine x."},{"Start":"10:41.390 ","End":"10:48.650","Text":"Now, we\u0027re going to bring this to the other side and then we\u0027ll take out the sine."},{"Start":"10:48.650 ","End":"10:52.859","Text":"But, before that I can cancel the 2 already."},{"Start":"10:53.330 ","End":"10:55.935","Text":"You know what, I\u0027ll do two steps in one."},{"Start":"10:55.935 ","End":"10:57.790","Text":"Bringing the sine x to the other side,"},{"Start":"10:57.790 ","End":"11:03.198","Text":"so it\u0027s plus sine x and then we can take sine x outside the brackets,"},{"Start":"11:03.198 ","End":"11:10.100","Text":"and we get cosine 4x plus 1=0."},{"Start":"11:10.100 ","End":"11:12.640","Text":"Now we have our product."},{"Start":"11:12.640 ","End":"11:14.810","Text":"Something times something equals 0,"},{"Start":"11:14.810 ","End":"11:22.780","Text":"so w e have case A that this will be 0 and the case B, this will be 0."},{"Start":"11:22.910 ","End":"11:27.675","Text":"Let\u0027s take A first. Sine x=0."},{"Start":"11:27.675 ","End":"11:30.100","Text":"Well, this is well-known."},{"Start":"11:30.100 ","End":"11:38.950","Text":"It\u0027s known that this solution simplifies 2x equals multiples of 180 degrees."},{"Start":"11:38.950 ","End":"11:41.165","Text":"Let us see in B."},{"Start":"11:41.165 ","End":"11:52.080","Text":"We get that cosine 4x=minus 1."},{"Start":"11:52.490 ","End":"11:58.490","Text":"It\u0027s well-known that the solution for cosine equals minus 1,"},{"Start":"11:58.490 ","End":"12:01.055","Text":"I\u0027ll just spare you the details,"},{"Start":"12:01.055 ","End":"12:04.340","Text":"is that 4x is"},{"Start":"12:04.340 ","End":"12:12.355","Text":"180 degrees plus multiples of 360 degrees."},{"Start":"12:12.355 ","End":"12:17.540","Text":"That gives us that x is equal to"},{"Start":"12:17.540 ","End":"12:26.940","Text":"45 degrees plus n times 90 degrees."},{"Start":"12:26.940 ","End":"12:29.805","Text":"We have this solution,"},{"Start":"12:29.805 ","End":"12:37.870","Text":"and we have this solution. We\u0027re done."}],"ID":14349},{"Watched":false,"Name":"Exercise 1","Duration":"3m 49s","ChapterTopicVideoID":5394,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5394.jpeg","UploadDate":"2016-03-10T21:08:45.5230000","DurationForVideoObject":"PT3M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise, we have a pair of trigonometric equations to solve,"},{"Start":"00:06.000 ","End":"00:09.615","Text":"each of them involving the sine function."},{"Start":"00:09.615 ","End":"00:12.375","Text":"Let\u0027s start with the first."},{"Start":"00:12.375 ","End":"00:15.190","Text":"I\u0027ll scroll down a bit."},{"Start":"00:15.320 ","End":"00:22.590","Text":"In this one we\u0027re given the sine of x is sine of 30, and it\u0027s degrees."},{"Start":"00:22.590 ","End":"00:26.694","Text":"But I\u0027m not going to keep writing degrees"},{"Start":"00:26.694 ","End":"00:31.410","Text":"in all these exercises where the angles are in degrees."},{"Start":"00:31.660 ","End":"00:40.070","Text":"One solution that\u0027s the obvious one is that x is just 30 and that\u0027s what we start with."},{"Start":"00:40.070 ","End":"00:44.135","Text":"Then we extend it to find more solutions."},{"Start":"00:44.135 ","End":"00:47.990","Text":"One way of getting more solutions is adding whole circles."},{"Start":"00:47.990 ","End":"00:50.675","Text":"Each circle is 360 degrees,"},{"Start":"00:50.675 ","End":"00:55.190","Text":"and we add 360 times some whole number"},{"Start":"00:55.190 ","End":"00:58.190","Text":"k. k could be positive or"},{"Start":"00:58.190 ","End":"01:03.410","Text":"negative but a whole number or integer is the more precise term."},{"Start":"01:03.410 ","End":"01:08.370","Text":"That\u0027s one, let\u0027s call it a family of solutions."},{"Start":"01:08.620 ","End":"01:14.975","Text":"There\u0027s another family of solutions and what you do is instead of 30,"},{"Start":"01:14.975 ","End":"01:19.730","Text":"you take another angle that is subtracting this from 180."},{"Start":"01:19.730 ","End":"01:21.620","Text":"Actually, there\u0027s an old-fashioned word for it,"},{"Start":"01:21.620 ","End":"01:23.825","Text":"it\u0027s called the supplementary angle."},{"Start":"01:23.825 ","End":"01:31.322","Text":"What I do is 180 minus 30 and get 150 and that\u0027s called the supplement of 30,"},{"Start":"01:31.322 ","End":"01:33.890","Text":"but it\u0027s out of fashion."},{"Start":"01:33.890 ","End":"01:37.835","Text":"Just say 180 minus the angle."},{"Start":"01:37.835 ","End":"01:40.480","Text":"I get 150 and again,"},{"Start":"01:40.480 ","End":"01:47.330","Text":"add 360 times k. It\u0027s twice infinity solutions."},{"Start":"01:47.330 ","End":"01:50.735","Text":"But these are the two principle solutions"},{"Start":"01:50.735 ","End":"01:56.610","Text":"between 0 and 360 degrees and everything else is just k circles."},{"Start":"01:57.940 ","End":"02:01.790","Text":"Similar thing in part b,"},{"Start":"02:01.790 ","End":"02:07.250","Text":"except here we\u0027re not given that this is the sine of something."},{"Start":"02:07.250 ","End":"02:13.330","Text":"You can either look it up in the table of well-known angles."},{"Start":"02:13.330 ","End":"02:20.670","Text":"There\u0027s a table: 0,30,45,60 and 90 degrees,"},{"Start":"02:20.670 ","End":"02:25.610","Text":"and you should memorize these sines and cosines of those important angles."},{"Start":"02:25.610 ","End":"02:31.070","Text":"If you look it up you\u0027ll find that this is the sine of 60 degrees."},{"Start":"02:31.070 ","End":"02:35.178","Text":"But if you don\u0027t have the table handy and you\u0027ve forgotten and you have a calculator,"},{"Start":"02:35.178 ","End":"02:37.835","Text":"you can compute the square root of 3/2."},{"Start":"02:37.835 ","End":"02:42.830","Text":"Then depending on the calculator press something like shift sine or"},{"Start":"02:42.830 ","End":"02:48.230","Text":"inverse sine or according to the calculator and you\u0027ll get 60 degrees."},{"Start":"02:48.230 ","End":"02:52.020","Text":"Calculator only gives you one solution."},{"Start":"02:52.220 ","End":"02:58.610","Text":"We start off with x equals 60 as a possible solution and we"},{"Start":"02:58.610 ","End":"03:05.340","Text":"expand in one way by adding multiples of 360, that\u0027s the 360k."},{"Start":"03:06.050 ","End":"03:09.560","Text":"Here I didn\u0027t do it, but sometimes we want to add the k"},{"Start":"03:09.560 ","End":"03:12.488","Text":"as a whole number or k is an integer,"},{"Start":"03:12.488 ","End":"03:16.385","Text":"but that\u0027s pretty much understood I\u0027ll write it once."},{"Start":"03:16.385 ","End":"03:21.380","Text":"Then we take the other sets of solutions which is instead of 60,"},{"Start":"03:21.380 ","End":"03:24.240","Text":"to take 180 minus."},{"Start":"03:24.240 ","End":"03:28.040","Text":"I\u0027ll just do that at the side, 180 minus 60."},{"Start":"03:28.040 ","End":"03:29.336","Text":"Normally I wouldn\u0027t even write it,"},{"Start":"03:29.336 ","End":"03:32.780","Text":"you do it in your head. It\u0027s 120."},{"Start":"03:32.780 ","End":"03:35.900","Text":"We get the second family of solutions."},{"Start":"03:35.900 ","End":"03:39.950","Text":"120 again, plus any number,"},{"Start":"03:39.950 ","End":"03:49.200","Text":"any whole number k times 360 and this is the complete solution. We\u0027re done."}],"ID":5393},{"Watched":false,"Name":"Exercise 2","Duration":"4m 58s","ChapterTopicVideoID":5395,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5395.jpeg","UploadDate":"2016-03-10T21:09:21.8730000","DurationForVideoObject":"PT4M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"In this exercise, we have a pair of trigonometric equations specifically involving"},{"Start":"00:05.100 ","End":"00:10.470","Text":"the sin function and let\u0027s start with the first one."},{"Start":"00:10.470 ","End":"00:15.015","Text":"Sin 3x is square root of 2 over 2."},{"Start":"00:15.015 ","End":"00:22.470","Text":"The first thing to do is to identify this as the sin of some angle and you either use"},{"Start":"00:22.470 ","End":"00:27.425","Text":"the calculator or you just use the table"},{"Start":"00:27.425 ","End":"00:33.110","Text":"of well-known angles and their sins or you might just remember,"},{"Start":"00:33.110 ","End":"00:34.550","Text":"and hopefully you do,"},{"Start":"00:34.550 ","End":"00:39.230","Text":"that this is the sine of 45 degrees."},{"Start":"00:39.230 ","End":"00:45.860","Text":"Either way. The first possible equation I can"},{"Start":"00:45.860 ","End":"00:53.510","Text":"get without sin is that 3x could be 45."},{"Start":"00:53.510 ","End":"00:56.990","Text":"That\u0027s possible. Because of the sin,"},{"Start":"00:56.990 ","End":"00:59.540","Text":"we need to branch out in a couple of directions."},{"Start":"00:59.540 ","End":"01:03.160","Text":"One of them is to add multiples of 360,"},{"Start":"01:03.160 ","End":"01:06.170","Text":"so we add 360 k,"},{"Start":"01:06.170 ","End":"01:08.660","Text":"and I don\u0027t write it every time,"},{"Start":"01:08.660 ","End":"01:12.620","Text":"but K is a whole number positive or negative."},{"Start":"01:12.620 ","End":"01:16.440","Text":"Any K, 1, 2, 3, 4,"},{"Start":"01:16.440 ","End":"01:18.120","Text":"5 or minus 1,"},{"Start":"01:18.120 ","End":"01:20.165","Text":"minus 2, and so on."},{"Start":"01:20.165 ","End":"01:25.730","Text":"The other direction that we branch out in is to take a different angle,"},{"Start":"01:25.730 ","End":"01:31.440","Text":"not 45, by subtracting from 180."},{"Start":"01:31.440 ","End":"01:37.780","Text":"So I do 180 minus 45 is 135."},{"Start":"01:37.780 ","End":"01:40.280","Text":"There is an old-fashioned term for this,"},{"Start":"01:40.280 ","End":"01:42.080","Text":"it\u0027s called the supplement of the angle."},{"Start":"01:42.080 ","End":"01:44.330","Text":"The supplement of 45 is 135,"},{"Start":"01:44.330 ","End":"01:46.265","Text":"but I think it\u0027s not in use anymore,"},{"Start":"01:46.265 ","End":"01:49.390","Text":"so you just remember 180 minus the angle."},{"Start":"01:49.390 ","End":"01:53.790","Text":"Then we get the same 3x that was here,"},{"Start":"01:53.790 ","End":"01:57.270","Text":"could be also 135."},{"Start":"01:57.270 ","End":"02:02.515","Text":"Again, plus multiples of 360."},{"Start":"02:02.515 ","End":"02:05.840","Text":"Now when we solve an equation,"},{"Start":"02:05.840 ","End":"02:08.510","Text":"we don\u0027t want 3x, we want x,"},{"Start":"02:08.510 ","End":"02:14.115","Text":"so what I do is I divide everything by 3,"},{"Start":"02:14.115 ","End":"02:21.110","Text":"and so the first one would give me that x is 45 over 3 is 15,"},{"Start":"02:21.110 ","End":"02:25.290","Text":"360 over 3 is 120,"},{"Start":"02:25.290 ","End":"02:33.635","Text":"so I get 15 plus 120 K. And the other family of solutions gives me that x is"},{"Start":"02:33.635 ","End":"02:38.035","Text":"135 over 3 is 45"},{"Start":"02:38.035 ","End":"02:44.390","Text":"plus 120 K. I just might be able to make a note of this."},{"Start":"02:44.390 ","End":"02:50.825","Text":"I divided everything by 3 in order to get this."},{"Start":"02:50.825 ","End":"02:53.075","Text":"That\u0027s part a,"},{"Start":"02:53.075 ","End":"02:56.369","Text":"and in part b."},{"Start":"02:58.910 ","End":"03:03.390","Text":"We have 2sine2x=1."},{"Start":"03:03.390 ","End":"03:07.290","Text":"First thing is that I don\u0027t have sin,"},{"Start":"03:07.290 ","End":"03:08.790","Text":"I have 2sin,"},{"Start":"03:08.790 ","End":"03:12.270","Text":"not a problem, divide by 2 first of all."},{"Start":"03:12.270 ","End":"03:16.890","Text":"I would say sine2x equals 1/2"},{"Start":"03:16.890 ","End":"03:22.355","Text":"and only when I have sin of something do I start asking what angle does this belong to."},{"Start":"03:22.355 ","End":"03:25.865","Text":"As usual, calculator or table or from memory,"},{"Start":"03:25.865 ","End":"03:30.980","Text":"this is the sin of 30 degrees."},{"Start":"03:30.980 ","End":"03:33.290","Text":"Now that we have this,"},{"Start":"03:33.290 ","End":"03:36.980","Text":"I can say that the first possibility is that 2x is"},{"Start":"03:36.980 ","End":"03:40.940","Text":"30 degrees and then I branch out in two ways,"},{"Start":"03:40.940 ","End":"03:48.310","Text":"one is by adding multiples of 360 and K is a general integer."},{"Start":"03:48.310 ","End":"03:50.830","Text":"The other thing is to,"},{"Start":"03:50.830 ","End":"03:57.595","Text":"instead of 30, take the angle obtained by subtracting from 180."},{"Start":"03:57.595 ","End":"04:00.250","Text":"Usually we do it in our heads,"},{"Start":"04:00.250 ","End":"04:02.425","Text":"but I\u0027m just writing to show you what I\u0027ve done."},{"Start":"04:02.425 ","End":"04:05.890","Text":"180 minus 30 is 150,"},{"Start":"04:05.890 ","End":"04:15.490","Text":"so the other family of solutions is a 150 plus 360 K. We have twice infinity solutions."},{"Start":"04:15.490 ","End":"04:17.080","Text":"This is not the end though,"},{"Start":"04:17.080 ","End":"04:20.835","Text":"because this gives us what 2x is and we want x,"},{"Start":"04:20.835 ","End":"04:25.635","Text":"so just divide everything by 2,"},{"Start":"04:25.635 ","End":"04:29.565","Text":"and so x is either 30 over 2."},{"Start":"04:29.565 ","End":"04:32.850","Text":"We do this in our heads, is 15,"},{"Start":"04:32.850 ","End":"04:40.385","Text":"360 over 2 is 180 K. And the other family of solutions for x,"},{"Start":"04:40.385 ","End":"04:44.230","Text":"150 over 2 is 75,"},{"Start":"04:44.230 ","End":"04:49.260","Text":"360 over 2 is still a 180 K."},{"Start":"04:49.260 ","End":"04:59.320","Text":"This is the general solution for this trigonometric equation. Done."}],"ID":5394},{"Watched":false,"Name":"Exercise 3","Duration":"5m 54s","ChapterTopicVideoID":5396,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5396.jpeg","UploadDate":"2017-02-13T19:19:01.9370000","DurationForVideoObject":"PT5M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have a pair of trigonometric equations to"},{"Start":"00:03.780 ","End":"00:08.535","Text":"solve in building the sine function."},{"Start":"00:08.535 ","End":"00:13.740","Text":"Will start with part a and we"},{"Start":"00:13.740 ","End":"00:19.620","Text":"have familiar situation of sine something equals sine something."},{"Start":"00:19.620 ","End":"00:24.315","Text":"The standard thing to do is to say that we have 1 of 2 things."},{"Start":"00:24.315 ","End":"00:29.940","Text":"We either have that 2x plus 30=x,"},{"Start":"00:29.940 ","End":"00:31.590","Text":"we can pair these,"},{"Start":"00:31.590 ","End":"00:34.035","Text":"and then we branch out."},{"Start":"00:34.035 ","End":"00:42.385","Text":"1 possibility is to add 360 times a whole number, whole circles."},{"Start":"00:42.385 ","End":"00:48.095","Text":"The other thing is to say that, sorry,"},{"Start":"00:48.095 ","End":"00:57.165","Text":"plus instead of x I take 180 minus x and again,"},{"Start":"00:57.165 ","End":"01:02.800","Text":"add whole number 360."},{"Start":"01:03.550 ","End":"01:11.225","Text":"But this form is not good because we want just x on the left-hand side."},{"Start":"01:11.225 ","End":"01:17.745","Text":"Let\u0027s see if I simplify the first 1 we just"},{"Start":"01:17.745 ","End":"01:25.205","Text":"bring the x over to this side and what I get is,"},{"Start":"01:25.205 ","End":"01:29.480","Text":"let\u0027s see, working off this 1 now."},{"Start":"01:29.480 ","End":"01:37.955","Text":"X is equal to bring the 30 over to the other side,"},{"Start":"01:37.955 ","End":"01:45.545","Text":"I get x equals minus 30 plus 360K."},{"Start":"01:45.545 ","End":"01:48.175","Text":"That\u0027s 1 part."},{"Start":"01:48.175 ","End":"01:55.840","Text":"Now I\u0027ll continue with the second part over here."},{"Start":"01:55.940 ","End":"02:02.960","Text":"In this case, what we get is that I bring"},{"Start":"02:02.960 ","End":"02:10.679","Text":"the x over 2 here and get 3x and the 30 over here."},{"Start":"02:10.679 ","End":"02:18.950","Text":"Then I get 150 plus 360K."},{"Start":"02:18.950 ","End":"02:20.975","Text":"The previous case, we were fortunate,"},{"Start":"02:20.975 ","End":"02:23.030","Text":"just got x right away here."},{"Start":"02:23.030 ","End":"02:27.530","Text":"We just have to divide by 3. No problem."},{"Start":"02:27.530 ","End":"02:31.190","Text":"X is equal to 150 over 3 is 50,"},{"Start":"02:31.190 ","End":"02:35.870","Text":"360 over 3 is 120. This is it."},{"Start":"02:35.870 ","End":"02:39.165","Text":"I\u0027ll just highlight the important lines."},{"Start":"02:39.165 ","End":"02:47.330","Text":"This line, and this line give the complete solution for x,"},{"Start":"02:47.330 ","End":"02:51.210","Text":"2 infinite families of solutions."},{"Start":"02:51.850 ","End":"02:55.619","Text":"Now onto part b."},{"Start":"02:56.710 ","End":"03:02.015","Text":"Here we don\u0027t have it in the form of sine of something equals sine of something."},{"Start":"03:02.015 ","End":"03:04.970","Text":"But think you can see what we have to do."},{"Start":"03:04.970 ","End":"03:07.475","Text":"We just bring this over to the other side."},{"Start":"03:07.475 ","End":"03:11.550","Text":"Then we get sine of 2x."},{"Start":"03:11.750 ","End":"03:19.090","Text":"Sine of 2x equals sine of 4x."},{"Start":"03:19.390 ","End":"03:25.280","Text":"Here were unfamiliar territory where sine of something equals sine of something."},{"Start":"03:25.280 ","End":"03:30.530","Text":"This 2x is either equal to 4x"},{"Start":"03:30.530 ","End":"03:38.765","Text":"or this plus a whole number of 360 or 360K."},{"Start":"03:38.765 ","End":"03:50.195","Text":"The other possibility is to take this from 180 minus 4x also plus 360K."},{"Start":"03:50.195 ","End":"03:55.115","Text":"Let\u0027s see. Tell you what,"},{"Start":"03:55.115 ","End":"03:58.250","Text":"l continue with this 1 over here,"},{"Start":"03:58.250 ","End":"04:00.545","Text":"and then I\u0027ll be able to continue with this 1."},{"Start":"04:00.545 ","End":"04:02.090","Text":"From here I get,"},{"Start":"04:02.090 ","End":"04:06.420","Text":"if I bring this over to this side,"},{"Start":"04:06.420 ","End":"04:11.085","Text":"then I get that minus"},{"Start":"04:11.085 ","End":"04:18.630","Text":"2x is equal to 360K,"},{"Start":"04:18.630 ","End":"04:23.625","Text":"and then divide by minus 2 and get that"},{"Start":"04:23.625 ","End":"04:33.560","Text":"x equals minus 180K."},{"Start":"04:33.560 ","End":"04:35.285","Text":"I could leave it like this,"},{"Start":"04:35.285 ","End":"04:39.335","Text":"but I don\u0027t like the minus,"},{"Start":"04:39.335 ","End":"04:44.900","Text":"an actual fact minus K is a general whole number just as much as K. I would"},{"Start":"04:44.900 ","End":"04:50.780","Text":"prefer to write this as x equals 180K."},{"Start":"04:50.780 ","End":"04:57.230","Text":"It\u0027s not really the same K. This K is minus this K. But if you\u0027re not happy with this,"},{"Start":"04:57.230 ","End":"04:59.670","Text":"you could leave it like this."},{"Start":"05:01.120 ","End":"05:07.400","Text":"In this case, then if I bring this over to the other side,"},{"Start":"05:07.400 ","End":"05:16.065","Text":"I get 6x equals 180 plus 360K,"},{"Start":"05:16.065 ","End":"05:18.720","Text":"and then we need to divide by 6,"},{"Start":"05:18.720 ","End":"05:22.485","Text":"so x equals 30"},{"Start":"05:22.485 ","End":"05:30.150","Text":"plus 60K."},{"Start":"05:30.150 ","End":"05:34.730","Text":"We have this solution,"},{"Start":"05:34.730 ","End":"05:39.680","Text":"well, infinite number of solutions and we have this."},{"Start":"05:39.680 ","End":"05:45.350","Text":"Like I said, I just replaced minus K by K. Minus K is"},{"Start":"05:45.350 ","End":"05:48.200","Text":"a general whole number just as much as"},{"Start":"05:48.200 ","End":"05:54.660","Text":"K. This is the general solution of the part b and we are done here."}],"ID":5395},{"Watched":false,"Name":"Exercise 4","Duration":"10m 29s","ChapterTopicVideoID":5397,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5397.jpeg","UploadDate":"2016-03-10T21:11:12.6730000","DurationForVideoObject":"PT10M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.005","Text":"Here we have an exercise with two trigonometric equations."},{"Start":"00:07.005 ","End":"00:10.690","Text":"Let\u0027s start with part a."},{"Start":"00:11.960 ","End":"00:17.235","Text":"We have here sine of something plus sine of something."},{"Start":"00:17.235 ","End":"00:20.040","Text":"What we really would like is sine of something"},{"Start":"00:20.040 ","End":"00:23.535","Text":"equals sine of something and then we know how to solve that."},{"Start":"00:23.535 ","End":"00:30.045","Text":"We can make a start by writing sine 3x=minus sine x."},{"Start":"00:30.045 ","End":"00:34.580","Text":"This is still not my basic form where I want sine equals sine."},{"Start":"00:34.580 ","End":"00:35.990","Text":"I have this minus here."},{"Start":"00:35.990 ","End":"00:40.590","Text":"But I remembered there\u0027s a formula."},{"Start":"00:40.590 ","End":"00:47.870","Text":"Actually, it\u0027s a trigonometric identity that the sine of minus something, let\u0027s say,"},{"Start":"00:47.870 ","End":"00:51.995","Text":"let\u0027s use a nice Greek letter like Alpha is equal to"},{"Start":"00:51.995 ","End":"00:57.079","Text":"minus sine Alpha also useful in the other direction."},{"Start":"00:57.079 ","End":"00:59.780","Text":"Basically, what it means is that you can pull a minus in"},{"Start":"00:59.780 ","End":"01:02.735","Text":"an inside and out of a sine function."},{"Start":"01:02.735 ","End":"01:05.880","Text":"It doesn\u0027t work for the cosine by the way."},{"Start":"01:07.370 ","End":"01:13.495","Text":"Back here, left-hand side is okay at sine something,"},{"Start":"01:13.495 ","End":"01:15.550","Text":"right-hand side, I\u0027ll use this formula,"},{"Start":"01:15.550 ","End":"01:18.250","Text":"this minus here and put it in inside,"},{"Start":"01:18.250 ","End":"01:21.580","Text":"so it\u0027s sine of minus x,"},{"Start":"01:21.580 ","End":"01:24.385","Text":"but need a brackets here."},{"Start":"01:24.385 ","End":"01:31.450","Text":"Now we have our standard form where we have two infinite families of solutions."},{"Start":"01:31.450 ","End":"01:36.925","Text":"One of them is where we take 3x=minus x."},{"Start":"01:36.925 ","End":"01:39.370","Text":"This part is called the argument of the sine,"},{"Start":"01:39.370 ","End":"01:40.990","Text":"the thing that we take the sine up."},{"Start":"01:40.990 ","End":"01:43.297","Text":"We compare the arguments,"},{"Start":"01:43.297 ","End":"01:48.335","Text":"but in one of the arguments, we can add multiples of 360."},{"Start":"01:48.335 ","End":"01:55.820","Text":"We have to add multiples of 360 full circles and that\u0027s what the letter K is."},{"Start":"01:55.820 ","End":"01:59.610","Text":"It just means any whole number positive or negative."},{"Start":"02:00.080 ","End":"02:05.810","Text":"That\u0027s one, let\u0027s call it family of solutions because as K varies,"},{"Start":"02:05.810 ","End":"02:11.720","Text":"you get different solutions and the other family of solutions we get is similar to this."},{"Start":"02:11.720 ","End":"02:12.890","Text":"We take also 3."},{"Start":"02:12.890 ","End":"02:18.575","Text":"The 3x stays, the plus 360K is the same,"},{"Start":"02:18.575 ","End":"02:20.660","Text":"but instead of this angle,"},{"Start":"02:20.660 ","End":"02:23.510","Text":"we take 180 minus it,"},{"Start":"02:23.510 ","End":"02:27.830","Text":"like 180 minus whatever this is."},{"Start":"02:27.830 ","End":"02:29.600","Text":"If I figure out in my head,"},{"Start":"02:29.600 ","End":"02:32.210","Text":"what is 180 minus minus x?"},{"Start":"02:32.210 ","End":"02:36.509","Text":"It\u0027s 180 plus x."},{"Start":"02:38.350 ","End":"02:44.180","Text":"This is not the answer though because we need what x equals."},{"Start":"02:44.180 ","End":"02:50.240","Text":"Let\u0027s develop each of these."},{"Start":"02:50.240 ","End":"02:52.895","Text":"Let\u0027s say I start with the first one."},{"Start":"02:52.895 ","End":"02:55.205","Text":"I\u0027ll take it over here."},{"Start":"02:55.205 ","End":"02:58.445","Text":"What I get is if I move the x\u0027s to the left,"},{"Start":"02:58.445 ","End":"03:04.590","Text":"I\u0027ll get 4x=360K and"},{"Start":"03:04.590 ","End":"03:10.060","Text":"then divide by 4, I\u0027ve got x=90K."},{"Start":"03:10.060 ","End":"03:14.360","Text":"That\u0027s one infinite family of solutions."},{"Start":"03:14.360 ","End":"03:16.040","Text":"I\u0027ll highlight it."},{"Start":"03:16.040 ","End":"03:22.620","Text":"It means x=0 or 90 or 180 or 270,"},{"Start":"03:22.620 ","End":"03:27.455","Text":"whatever K you want to put or even minus 90, minus 180."},{"Start":"03:27.455 ","End":"03:28.910","Text":"All these are solutions,"},{"Start":"03:28.910 ","End":"03:34.015","Text":"that\u0027s one family of solutions and the other one will take from here."},{"Start":"03:34.015 ","End":"03:38.425","Text":"Let\u0027s see, the x is to the left, so that\u0027s 2x."},{"Start":"03:38.425 ","End":"03:40.180","Text":"What does that equal?"},{"Start":"03:40.180 ","End":"03:42.110","Text":"Just here. What was here?"},{"Start":"03:42.110 ","End":"03:47.178","Text":"180 plus 360K divide by 2,"},{"Start":"03:47.178 ","End":"03:53.210","Text":"x=90 plus 180K."},{"Start":"03:53.210 ","End":"03:58.200","Text":"That\u0027s our second family of solutions,"},{"Start":"03:58.200 ","End":"04:00.945","Text":"90 plus any multiple."},{"Start":"04:00.945 ","End":"04:03.840","Text":"90, 270,"},{"Start":"04:03.840 ","End":"04:05.870","Text":"450, and so on."},{"Start":"04:05.870 ","End":"04:09.089","Text":"I\u0027ll highlight this one also."},{"Start":"04:10.630 ","End":"04:13.740","Text":"Let\u0027s go on to part b."},{"Start":"04:15.800 ","End":"04:22.040","Text":"Part b is similar to part a in the sense that we don\u0027t quite have sine equals sine,"},{"Start":"04:22.040 ","End":"04:23.935","Text":"we have this minus."},{"Start":"04:23.935 ","End":"04:26.900","Text":"We already saw the trick."},{"Start":"04:26.900 ","End":"04:29.330","Text":"The trick is to put the minus inside,"},{"Start":"04:29.330 ","End":"04:35.590","Text":"which you can do in the case of sine by that formula that I\u0027ve previously wrote."},{"Start":"04:35.590 ","End":"04:40.980","Text":"We put the minus inside."},{"Start":"04:40.980 ","End":"04:44.840","Text":"We do minus of all of this,"},{"Start":"04:44.840 ","End":"04:53.630","Text":"we get minus x plus 30 and the right-hand side is good as it is, 60 plus x."},{"Start":"04:53.630 ","End":"04:58.850","Text":"Now I have the standard two families of solutions."},{"Start":"04:58.850 ","End":"05:07.175","Text":"Sometimes I put the curly brace that either minus x plus 30 equals this."},{"Start":"05:07.175 ","End":"05:09.980","Text":"These are called the argument of the sine."},{"Start":"05:09.980 ","End":"05:17.390","Text":"The argument of the sine here is this and 60 plus x is the argument of the sine here."},{"Start":"05:17.390 ","End":"05:18.935","Text":"It\u0027s what we take the sine of."},{"Start":"05:18.935 ","End":"05:22.387","Text":"Compare the arguments,"},{"Start":"05:22.387 ","End":"05:27.905","Text":"but also on one of them, we add whole circles, 360K."},{"Start":"05:27.905 ","End":"05:32.105","Text":"That\u0027s one family of possibilities."},{"Start":"05:32.105 ","End":"05:35.090","Text":"I say family because there\u0027s a lot of them because as K varies,"},{"Start":"05:35.090 ","End":"05:36.575","Text":"we get an infinite number of these."},{"Start":"05:36.575 ","End":"05:39.020","Text":"The other one is very similar,"},{"Start":"05:39.020 ","End":"05:42.830","Text":"it also has the same argument on the left,"},{"Start":"05:42.830 ","End":"05:46.280","Text":"it also adds full circles."},{"Start":"05:46.280 ","End":"05:48.725","Text":"But instead of this,"},{"Start":"05:48.725 ","End":"05:51.830","Text":"we take 180 minus that."},{"Start":"05:51.830 ","End":"05:53.615","Text":"Let me do at the other side,"},{"Start":"05:53.615 ","End":"05:58.400","Text":"180 minus (60 plus"},{"Start":"05:58.400 ","End":"06:08.895","Text":"x)=120 minus x and that\u0027s what I\u0027m putting here."},{"Start":"06:08.895 ","End":"06:12.195","Text":"Now, we have to solve these."},{"Start":"06:12.195 ","End":"06:17.989","Text":"With each of them, I\u0027m going to put the x\u0027s on the left and the rest on the right."},{"Start":"06:17.989 ","End":"06:20.270","Text":"Let\u0027s see, for the first one,"},{"Start":"06:20.270 ","End":"06:25.425","Text":"I will get that minus 2x,"},{"Start":"06:25.425 ","End":"06:27.197","Text":"it\u0027s breaking the x over,"},{"Start":"06:27.197 ","End":"06:29.100","Text":"the 30 over here,"},{"Start":"06:29.100 ","End":"06:37.794","Text":"that\u0027s 60 minus 30 is 30 plus 360K."},{"Start":"06:37.794 ","End":"06:39.680","Text":"Then we need to just get x,"},{"Start":"06:39.680 ","End":"06:41.660","Text":"so we divide by minus 2."},{"Start":"06:41.660 ","End":"06:48.200","Text":"So x is equal to this over minus 2 is minus 15,"},{"Start":"06:48.200 ","End":"06:54.915","Text":"and then minus 180K."},{"Start":"06:54.915 ","End":"06:58.250","Text":"This is one family of solutions."},{"Start":"06:58.250 ","End":"07:02.140","Text":"Now let\u0027s do the other one."},{"Start":"07:02.140 ","End":"07:04.560","Text":"From here, let\u0027s see what I get."},{"Start":"07:04.560 ","End":"07:09.710","Text":"Look, minus x and minus x, and they\u0027ll cancel."},{"Start":"07:09.710 ","End":"07:11.450","Text":"If I bring the x\u0027s to the left,"},{"Start":"07:11.450 ","End":"07:15.440","Text":"and there\u0027s nothing left of the x\u0027s it\u0027s just 0."},{"Start":"07:15.440 ","End":"07:17.930","Text":"The right-hand side, now less interesting,"},{"Start":"07:17.930 ","End":"07:19.399","Text":"but I\u0027ll do it anyway."},{"Start":"07:19.399 ","End":"07:23.450","Text":"We get 90 plus 360K,"},{"Start":"07:23.450 ","End":"07:25.670","Text":"but x has disappeared altogether."},{"Start":"07:25.670 ","End":"07:27.845","Text":"How do I solve for it?"},{"Start":"07:27.845 ","End":"07:30.215","Text":"I\u0027d like to remind you,"},{"Start":"07:30.215 ","End":"07:32.600","Text":"going back a bit to more basic algebra,"},{"Start":"07:32.600 ","End":"07:35.120","Text":"I\u0027ll give you an analogy."},{"Start":"07:35.120 ","End":"07:37.145","Text":"I\u0027ll write it at the side here."},{"Start":"07:37.145 ","End":"07:41.370","Text":"Suppose I had an equation like"},{"Start":"07:43.520 ","End":"07:54.030","Text":"2x=2(5 plus x) in"},{"Start":"07:54.030 ","End":"07:58.880","Text":"just plain algebra and this is just an example to illustrate a point,"},{"Start":"07:58.880 ","End":"08:02.240","Text":"and I want to solve for x."},{"Start":"08:02.240 ","End":"08:08.480","Text":"Then I would say, 2x=2 times 5 is 10 plus 2x,"},{"Start":"08:08.480 ","End":"08:11.540","Text":"then subtract 2x from both sides,"},{"Start":"08:11.540 ","End":"08:15.230","Text":"so bring the x\u0027s to the left and I\u0027d get 0=10,"},{"Start":"08:15.230 ","End":"08:18.230","Text":"which is simply a false statement."},{"Start":"08:18.230 ","End":"08:21.875","Text":"Now, when you come to an impossible thing,"},{"Start":"08:21.875 ","End":"08:25.775","Text":"that means that there is no solution."},{"Start":"08:25.775 ","End":"08:27.650","Text":"It\u0027s clear there\u0027s no solution."},{"Start":"08:27.650 ","End":"08:32.645","Text":"You can\u0027t have 2x=10 plus 2x, whatever it is."},{"Start":"08:32.645 ","End":"08:38.359","Text":"Here we would say, no solution, no such x."},{"Start":"08:38.359 ","End":"08:41.575","Text":"This is the same thing here."},{"Start":"08:41.575 ","End":"08:51.125","Text":"If you come to a situation where the x is missing and these two things are not equal,"},{"Start":"08:51.125 ","End":"08:55.320","Text":"then there is no solution."},{"Start":"08:55.480 ","End":"09:03.605","Text":"Wrong color. Now,"},{"Start":"09:03.605 ","End":"09:06.674","Text":"on the whole, we do have solutions,"},{"Start":"09:06.674 ","End":"09:09.890","Text":"there\u0027s just no solution for the second bit,"},{"Start":"09:09.890 ","End":"09:11.285","Text":"there were two rows,"},{"Start":"09:11.285 ","End":"09:13.025","Text":"this one has solutions."},{"Start":"09:13.025 ","End":"09:15.665","Text":"But instead of getting two families of solutions,"},{"Start":"09:15.665 ","End":"09:26.180","Text":"this is all that\u0027s left and this is the only family of solutions for the question."},{"Start":"09:26.180 ","End":"09:31.730","Text":"I just wanted to mention something just to be pedantic."},{"Start":"09:31.730 ","End":"09:34.640","Text":"If I had something else here,"},{"Start":"09:34.640 ","End":"09:37.340","Text":"suppose I got 0=0,"},{"Start":"09:37.340 ","End":"09:40.561","Text":"something which just happens to be true,"},{"Start":"09:40.561 ","End":"09:45.810","Text":"then something else happens that every x is a solution."},{"Start":"09:46.690 ","End":"09:50.390","Text":"But I just wanted to make the point that you only write"},{"Start":"09:50.390 ","End":"09:53.330","Text":"no solution in the case where this really is not equal."},{"Start":"09:53.330 ","End":"09:55.355","Text":"If I had 0=0 here,"},{"Start":"09:55.355 ","End":"09:58.300","Text":"that would be something else and I don\u0027t want to get deeply into that."},{"Start":"09:58.300 ","End":"10:02.090","Text":"But in this case, 0 is not equal to 90 plus 360K."},{"Start":"10:02.090 ","End":"10:04.340","Text":"In fact, even if you want to mess around with the K\u0027s,"},{"Start":"10:04.340 ","End":"10:05.765","Text":"wherever K you put,"},{"Start":"10:05.765 ","End":"10:09.515","Text":"there\u0027s no way you can add a multiple of 360 to 90 and get 0,"},{"Start":"10:09.515 ","End":"10:13.460","Text":"so there really is no solution in x."},{"Start":"10:13.460 ","End":"10:19.590","Text":"In fact, I can even emphasize no solution for x."},{"Start":"10:19.840 ","End":"10:25.175","Text":"This is something unusual that can happen and you\u0027ve seen it and just be prepared,"},{"Start":"10:25.175 ","End":"10:29.129","Text":"no big deal. We are done."}],"ID":5396},{"Watched":false,"Name":"Exercise 4c","Duration":"5m 16s","ChapterTopicVideoID":5398,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5398.jpeg","UploadDate":"2016-03-10T21:12:01.5370000","DurationForVideoObject":"PT5M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.620","Text":"For those who are interested,"},{"Start":"00:01.620 ","End":"00:05.730","Text":"I\u0027d like to give a totally optional bonus question."},{"Start":"00:05.730 ","End":"00:09.315","Text":"Let\u0027s make it a bonus part c, really optional,"},{"Start":"00:09.315 ","End":"00:13.573","Text":"but continuing this idea of what happens when x drops out."},{"Start":"00:13.573 ","End":"00:17.085","Text":"Let\u0027s take a similar example,"},{"Start":"00:17.085 ","End":"00:27.000","Text":"minus sin(x)=sin(180+x),"},{"Start":"00:27.000 ","End":"00:29.445","Text":"and see what we get."},{"Start":"00:29.445 ","End":"00:32.650","Text":"I need a bit more space here."},{"Start":"00:32.930 ","End":"00:35.250","Text":"Using the same, not trick,"},{"Start":"00:35.250 ","End":"00:37.610","Text":"the formula that we used above,"},{"Start":"00:37.610 ","End":"00:39.830","Text":"we put the minus inside and then we"},{"Start":"00:39.830 ","End":"00:47.900","Text":"have sin(-x)= sin(180+x),"},{"Start":"00:47.900 ","End":"00:51.555","Text":"a usual situation of sine something equals sine something,"},{"Start":"00:51.555 ","End":"00:55.655","Text":"and then we get 2 infinite families of solutions."},{"Start":"00:55.655 ","End":"00:59.165","Text":"We either have that the arguments are equal, that is,"},{"Start":"00:59.165 ","End":"01:03.350","Text":"minus x=180 plus x,"},{"Start":"01:03.350 ","End":"01:09.810","Text":"or it\u0027s equal plus a multiple of a circle,"},{"Start":"01:09.810 ","End":"01:11.630","Text":"in other words, 360 k,"},{"Start":"01:11.630 ","End":"01:13.235","Text":"where k is an integer."},{"Start":"01:13.235 ","End":"01:15.754","Text":"That\u0027s 1 family of solutions."},{"Start":"01:15.754 ","End":"01:21.870","Text":"The other is same format,"},{"Start":"01:21.870 ","End":"01:29.260","Text":"something plus 360k, but this time we take 180 minus this."},{"Start":"01:29.260 ","End":"01:31.800","Text":"I\u0027ll do that at the side."},{"Start":"01:33.970 ","End":"01:43.995","Text":"180 minus 180 plus x is just equal to minus x."},{"Start":"01:43.995 ","End":"01:48.960","Text":"Here I get minus x plus 360k."},{"Start":"01:48.960 ","End":"01:52.415","Text":"Now, the first part,"},{"Start":"01:52.415 ","End":"01:54.425","Text":"I\u0027ll take it over here,"},{"Start":"01:54.425 ","End":"01:56.660","Text":"will be, let\u0027s see,"},{"Start":"01:56.660 ","End":"02:04.970","Text":"minus 2x equals 180 plus 360k,"},{"Start":"02:04.970 ","End":"02:09.440","Text":"and then dividing by minus 2,"},{"Start":"02:09.440 ","End":"02:12.739","Text":"what I\u0027ll get is x equals"},{"Start":"02:12.739 ","End":"02:22.350","Text":"minus 90 minus 180k,"},{"Start":"02:22.350 ","End":"02:26.670","Text":"and that\u0027s perfectly fine and dandy."},{"Start":"02:26.670 ","End":"02:28.985","Text":"I\u0027ll highlight it."},{"Start":"02:28.985 ","End":"02:32.510","Text":"The peculiarity starts with the other 1."},{"Start":"02:32.510 ","End":"02:35.885","Text":"Because in this case, the minus x\u0027s cancel."},{"Start":"02:35.885 ","End":"02:38.900","Text":"If I bring the x\u0027s to 1 side I\u0027m left with 0=360k."},{"Start":"02:38.900 ","End":"02:44.370","Text":"Now, in this case,"},{"Start":"02:44.370 ","End":"02:51.050","Text":"it would not be okay to say no solution as we did before."},{"Start":"02:51.050 ","End":"02:56.040","Text":"Because it\u0027s certainly possible that 0=360k,"},{"Start":"02:56.040 ","End":"02:57.980","Text":"if k is 0."},{"Start":"02:57.980 ","End":"03:02.780","Text":"When this happens, when we don\u0027t get a false equality,"},{"Start":"03:02.780 ","End":"03:04.670","Text":"but something that could be true,"},{"Start":"03:04.670 ","End":"03:08.975","Text":"then it turns out that this thing is true for all x."},{"Start":"03:08.975 ","End":"03:15.275","Text":"Actually, this answer I would say either true for"},{"Start":"03:15.275 ","End":"03:22.575","Text":"all x or every x or something to that effect,"},{"Start":"03:22.575 ","End":"03:24.095","Text":"and actually that\u0027s true."},{"Start":"03:24.095 ","End":"03:26.780","Text":"It turns out that every x you substitute,"},{"Start":"03:26.780 ","End":"03:28.445","Text":"this will be true for."},{"Start":"03:28.445 ","End":"03:30.110","Text":"Let\u0027s even take an example."},{"Start":"03:30.110 ","End":"03:33.425","Text":"Suppose that x was,"},{"Start":"03:33.425 ","End":"03:36.545","Text":"let\u0027s say, minus 30."},{"Start":"03:36.545 ","End":"03:39.890","Text":"If I put x equals minus 30,"},{"Start":"03:39.890 ","End":"03:46.570","Text":"then what this says is that minus sin(-30) equals,"},{"Start":"03:46.570 ","End":"03:49.160","Text":"well, I\u0027m going to put a question mark here,"},{"Start":"03:49.160 ","End":"03:50.615","Text":"I\u0027m going to check that,"},{"Start":"03:50.615 ","End":"03:55.469","Text":"is equal to sin(180"},{"Start":"03:55.469 ","End":"04:01.735","Text":"minus minus 30) is 180 plus 30."},{"Start":"04:01.735 ","End":"04:05.295","Text":"The minus goes inside, so we\u0027re asking,"},{"Start":"04:05.295 ","End":"04:14.560","Text":"is sin(30) equal to sin(210)?"},{"Start":"04:16.010 ","End":"04:19.775","Text":"This meant to be minus 30."},{"Start":"04:19.775 ","End":"04:22.070","Text":"Of course, x is minus 30."},{"Start":"04:22.070 ","End":"04:25.910","Text":"That makes this 150."},{"Start":"04:26.020 ","End":"04:28.970","Text":"That\u0027s better. Now this thing is true."},{"Start":"04:28.970 ","End":"04:31.060","Text":"Sine(30) is sin(150)."},{"Start":"04:31.060 ","End":"04:37.930","Text":"They\u0027re supplementary angles or they complete each other to 180."},{"Start":"04:37.930 ","End":"04:42.095","Text":"In fact, it turns out that any x you put here,"},{"Start":"04:42.095 ","End":"04:43.595","Text":"this will be true."},{"Start":"04:43.595 ","End":"04:46.760","Text":"This is what we call a trigonometrical identity."},{"Start":"04:46.760 ","End":"04:50.180","Text":"Sometimes people write it like an equal with 3 lines,"},{"Start":"04:50.180 ","End":"04:52.710","Text":"meaning it\u0027s always true."},{"Start":"04:52.750 ","End":"04:58.490","Text":"This is really to illustrate the point that if x drops out,"},{"Start":"04:58.490 ","End":"05:05.155","Text":"you only can say no solution in case you get an impossible equality like this."},{"Start":"05:05.155 ","End":"05:08.430","Text":"So much for the bonus part."},{"Start":"05:08.430 ","End":"05:15.720","Text":"I hope it helped if you followed it. That\u0027s it."}],"ID":5397},{"Watched":false,"Name":"Exercise 5","Duration":"6m 26s","ChapterTopicVideoID":5399,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5399.jpeg","UploadDate":"2016-03-10T21:12:49.8270000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we have two equations,"},{"Start":"00:03.870 ","End":"00:08.820","Text":"both very similar and both on the same theme that"},{"Start":"00:08.820 ","End":"00:13.883","Text":"we have to find in one case all the angles which have a sine of 0,"},{"Start":"00:13.883 ","End":"00:15.608","Text":"and on the other case,"},{"Start":"00:15.608 ","End":"00:18.810","Text":"all angles which have a sine of 1."},{"Start":"00:18.810 ","End":"00:22.710","Text":"0 and 1 are very special values for the sine."},{"Start":"00:22.710 ","End":"00:24.480","Text":"Let\u0027s start with the first one,"},{"Start":"00:24.480 ","End":"00:29.100","Text":"with the case where sine is going to be 0."},{"Start":"00:29.100 ","End":"00:38.340","Text":"We can bring it to the form sine=sine if we write that sine x is equal to,"},{"Start":"00:38.340 ","End":"00:40.915","Text":"now 0 sine of 0."},{"Start":"00:40.915 ","End":"00:42.830","Text":"This is from the tables."},{"Start":"00:42.830 ","End":"00:47.180","Text":"Or you could use the calculator if you\u0027ve really forgotten everything."},{"Start":"00:47.180 ","End":"00:54.170","Text":"Then we use our usual technique of saying that x could be 0."},{"Start":"00:54.170 ","End":"00:56.840","Text":"The argument here could be the argument here,"},{"Start":"00:56.840 ","End":"01:03.815","Text":"but it also could be with an addition of a whole number of circles."},{"Start":"01:03.815 ","End":"01:05.510","Text":"In other words, 360K,"},{"Start":"01:05.510 ","End":"01:07.370","Text":"where K is a whole number."},{"Start":"01:07.370 ","End":"01:13.720","Text":"The other possibility is where we take 180 minus this angle."},{"Start":"01:13.720 ","End":"01:23.340","Text":"180 minus 0 is just 180 also plus 360K."},{"Start":"01:23.470 ","End":"01:28.940","Text":"Now, this could be left as is,"},{"Start":"01:28.940 ","End":"01:31.415","Text":"but there is a way of simplifying it."},{"Start":"01:31.415 ","End":"01:38.810","Text":"If you think about or if you actually write out some values with different values of K,"},{"Start":"01:38.810 ","End":"01:41.260","Text":"let\u0027s say K is 0."},{"Start":"01:41.260 ","End":"01:44.820","Text":"This would be 0. If k was 1,"},{"Start":"01:44.820 ","End":"01:47.160","Text":"I\u0027d get 360,"},{"Start":"01:47.160 ","End":"01:50.595","Text":"if K was 2, I\u0027d get 720,"},{"Start":"01:50.595 ","End":"01:52.830","Text":"if k is minus 1,"},{"Start":"01:52.830 ","End":"01:55.945","Text":"I\u0027d get minus 360,"},{"Start":"01:55.945 ","End":"02:00.275","Text":"and so on to infinity in both directions."},{"Start":"02:00.275 ","End":"02:04.070","Text":"If I look at the other one, this row,"},{"Start":"02:04.070 ","End":"02:10.460","Text":"if I let K equals 0, I have 180."},{"Start":"02:10.460 ","End":"02:13.024","Text":"I\u0027m deliberately writing it here."},{"Start":"02:13.024 ","End":"02:15.470","Text":"If I let K equals 1,"},{"Start":"02:15.470 ","End":"02:23.990","Text":"it\u0027s 180 plus 360 is 540, and so on."},{"Start":"02:23.990 ","End":"02:29.360","Text":"And if I let K equal minus 1,"},{"Start":"02:29.360 ","End":"02:34.050","Text":"then I\u0027ll get minus 180."},{"Start":"02:34.900 ","End":"02:38.600","Text":"Maybe do one more,"},{"Start":"02:38.600 ","End":"02:41.270","Text":"let\u0027s say K equals 2,"},{"Start":"02:41.270 ","End":"02:49.020","Text":"then it\u0027s 720 plus 180 is 900."},{"Start":"02:49.300 ","End":"02:55.970","Text":"If I look at all these values and if I combine them,"},{"Start":"02:55.970 ","End":"02:57.650","Text":"just by the way, I\u0027ve placed them,"},{"Start":"02:57.650 ","End":"03:00.560","Text":"what I get is minus 360,"},{"Start":"03:00.560 ","End":"03:04.793","Text":"minus 180, 0,"},{"Start":"03:04.793 ","End":"03:08.765","Text":"180, 360,"},{"Start":"03:08.765 ","End":"03:12.360","Text":"540, that\u0027s enough."},{"Start":"03:13.280 ","End":"03:18.200","Text":"Now, notice that the difference between each is 180."},{"Start":"03:18.200 ","End":"03:20.270","Text":"In fact, if you think about it,"},{"Start":"03:20.270 ","End":"03:22.685","Text":"these are all multiples of 180."},{"Start":"03:22.685 ","End":"03:25.886","Text":"This is 180 times 0,"},{"Start":"03:25.886 ","End":"03:27.584","Text":"180 times 1,"},{"Start":"03:27.584 ","End":"03:28.950","Text":"180 times 2,"},{"Start":"03:28.950 ","End":"03:30.840","Text":"180 times 3,"},{"Start":"03:30.840 ","End":"03:33.455","Text":"and so on with the minuses also."},{"Start":"03:33.455 ","End":"03:41.780","Text":"I could actually write this as x equals simply 180K, a different K,"},{"Start":"03:41.780 ","End":"03:43.490","Text":"not the same as this,"},{"Start":"03:43.490 ","End":"03:47.390","Text":"but it means all multiples of K. Like I said,"},{"Start":"03:47.390 ","End":"03:48.980","Text":"K equals 0, we get that,"},{"Start":"03:48.980 ","End":"03:51.905","Text":"K equals minus 1, we get that."},{"Start":"03:51.905 ","End":"03:56.090","Text":"Everything here is here and vice versa."},{"Start":"03:56.090 ","End":"03:59.330","Text":"This is much simpler than this, although this is not,"},{"Start":"03:59.330 ","End":"04:01.081","Text":"strictly speaking, wrong,"},{"Start":"04:01.081 ","End":"04:03.650","Text":"it\u0027s just this is much more aesthetic."},{"Start":"04:03.650 ","End":"04:11.490","Text":"This would be the general solution for the angle whose sine is 0."},{"Start":"04:11.590 ","End":"04:18.050","Text":"You might even want to check on the calculator some of these to try what is sine of 180?"},{"Start":"04:18.050 ","End":"04:20.525","Text":"What is sine of 540?"},{"Start":"04:20.525 ","End":"04:22.910","Text":"What is sine of minus 360,"},{"Start":"04:22.910 ","End":"04:27.847","Text":"and so on and verify that these are all indeed 0."},{"Start":"04:27.847 ","End":"04:30.280","Text":"Part B, same thing,"},{"Start":"04:30.280 ","End":"04:32.845","Text":"instead of 0, we have 1."},{"Start":"04:32.845 ","End":"04:38.980","Text":"Now, 1 happens to be the sine of 90 degrees."},{"Start":"04:38.980 ","End":"04:44.515","Text":"Sine of x is equal to sine of 90."},{"Start":"04:44.515 ","End":"04:47.395","Text":"We proceed as usual,"},{"Start":"04:47.395 ","End":"04:52.525","Text":"getting two sets of two families of solutions."},{"Start":"04:52.525 ","End":"04:56.560","Text":"One, by taking this argument equals this argument,"},{"Start":"04:56.560 ","End":"05:00.031","Text":"but adding multiples of 360,"},{"Start":"05:00.031 ","End":"05:08.125","Text":"the other possibility is to take 180 minus this,"},{"Start":"05:08.125 ","End":"05:15.760","Text":"which is also 90 plus 360K."},{"Start":"05:15.760 ","End":"05:18.970","Text":"This is the same as this."},{"Start":"05:18.970 ","End":"05:21.970","Text":"This is redundant. Not that it\u0027s wrong,"},{"Start":"05:21.970 ","End":"05:22.990","Text":"I\u0027m putting a line through it."},{"Start":"05:22.990 ","End":"05:24.730","Text":"It\u0027s just unnecessary."},{"Start":"05:24.730 ","End":"05:30.750","Text":"It\u0027s like if I solved an equation and I got x equals 2 or x equals 2,"},{"Start":"05:30.750 ","End":"05:33.665","Text":"then I would just say x is 2."},{"Start":"05:33.665 ","End":"05:38.515","Text":"If x is either 90 plus 360K or 90 plus 360K,"},{"Start":"05:38.515 ","End":"05:41.020","Text":"just throw out the extra one."},{"Start":"05:41.020 ","End":"05:44.154","Text":"I\u0027ll highlight this as the answer."},{"Start":"05:44.154 ","End":"05:48.625","Text":"If we really want to make things clear, concrete,"},{"Start":"05:48.625 ","End":"05:54.335","Text":"then we could say that x could be 90 or adding 360,"},{"Start":"05:54.335 ","End":"05:57.575","Text":"I could get 450,"},{"Start":"05:57.575 ","End":"06:01.328","Text":"or it could be, if K was 2,"},{"Start":"06:01.328 ","End":"06:06.050","Text":"720 plus 90 is 810, and so on,"},{"Start":"06:06.050 ","End":"06:09.470","Text":"and let\u0027s take a negative 1 also because minus 1,"},{"Start":"06:09.470 ","End":"06:14.960","Text":"90 minus 360, minus 270, and so on."},{"Start":"06:14.960 ","End":"06:18.380","Text":"The difference between each 2 is 360."},{"Start":"06:18.380 ","End":"06:25.890","Text":"These are the possibilities for x. We\u0027re done."}],"ID":5398},{"Watched":false,"Name":"Exercise 6","Duration":"4m 31s","ChapterTopicVideoID":5400,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5400.jpeg","UploadDate":"2016-03-10T21:13:24.6730000","DurationForVideoObject":"PT4M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this exercise, we have a couple of"},{"Start":"00:02.760 ","End":"00:07.120","Text":"trigonometric equations to solve involving the cosine."},{"Start":"00:07.700 ","End":"00:12.460","Text":"Let\u0027s get started with part a first."},{"Start":"00:13.850 ","End":"00:16.800","Text":"What we would like to have,"},{"Start":"00:16.800 ","End":"00:22.410","Text":"the ideal situation is where we have cosine of something equals cosine of something."},{"Start":"00:22.410 ","End":"00:24.420","Text":"Now this is not what we have here."},{"Start":"00:24.420 ","End":"00:26.204","Text":"We have the sine of something,"},{"Start":"00:26.204 ","End":"00:27.450","Text":"but this is just a number."},{"Start":"00:27.450 ","End":"00:28.575","Text":"There\u0027s no x\u0027s here."},{"Start":"00:28.575 ","End":"00:32.969","Text":"So we should be able to convert this to be the cosine of something."},{"Start":"00:32.969 ","End":"00:35.590","Text":"I\u0027ll do this at the side."},{"Start":"00:35.590 ","End":"00:38.740","Text":"Sine of 30,"},{"Start":"00:38.960 ","End":"00:42.410","Text":"if you look it up either on the calculator,"},{"Start":"00:42.410 ","End":"00:43.700","Text":"or on the special table,"},{"Start":"00:43.700 ","End":"00:44.840","Text":"I do it from memory,"},{"Start":"00:44.840 ","End":"00:47.794","Text":"I know that the sine of 30 years is a half."},{"Start":"00:47.794 ","End":"00:52.960","Text":"Then I also want to know cosine of what is the half."},{"Start":"00:52.960 ","End":"00:55.635","Text":"It\u0027s going to be cosine of something."},{"Start":"00:55.635 ","End":"01:02.525","Text":"Once again, either you use the table or from knowledge or the calculator,"},{"Start":"01:02.525 ","End":"01:09.515","Text":"you would do 0.5 and then shift cosine rather."},{"Start":"01:09.515 ","End":"01:16.710","Text":"However, your calculator supported inverse cosine and you would get 60 degrees."},{"Start":"01:18.370 ","End":"01:21.319","Text":"Cosine 60 is a half."},{"Start":"01:21.319 ","End":"01:23.540","Text":"You can also check it that way."},{"Start":"01:23.540 ","End":"01:32.630","Text":"Now we can rewrite this as cosine x equals cosine of 60."},{"Start":"01:32.630 ","End":"01:39.005","Text":"Now we\u0027re in the standard form and just like in the case of sine,"},{"Start":"01:39.005 ","End":"01:40.820","Text":"it has 2 families."},{"Start":"01:40.820 ","End":"01:44.809","Text":"Each of them is infinite of possibilities."},{"Start":"01:44.809 ","End":"01:50.825","Text":"In one of them, we take this argument to be equal to this argument."},{"Start":"01:50.825 ","End":"01:53.300","Text":"The x is the argument of the cosine."},{"Start":"01:53.300 ","End":"01:54.800","Text":"It\u0027s the thing we take the cosine of."},{"Start":"01:54.800 ","End":"01:58.190","Text":"So we compare the 2 arguments as 1 possibility,"},{"Start":"01:58.190 ","End":"02:04.130","Text":"but it could be generalized to adding multiples of a circle."},{"Start":"02:04.130 ","End":"02:09.170","Text":"So 360 times some whole number k. When we write k,"},{"Start":"02:09.170 ","End":"02:11.134","Text":"we mean it varies,"},{"Start":"02:11.134 ","End":"02:15.755","Text":"it\u0027s every or any whole number or integer,"},{"Start":"02:15.755 ","End":"02:19.430","Text":"negatives or positives, both good."},{"Start":"02:19.430 ","End":"02:27.920","Text":"The other possibility is that x is equal to minus the argument,"},{"Start":"02:27.920 ","End":"02:37.865","Text":"and again, plus 360 times k. So 60 or minus 60."},{"Start":"02:37.865 ","End":"02:40.580","Text":"This is because the cosine of an angle and"},{"Start":"02:40.580 ","End":"02:44.365","Text":"the cosine of minus an angle are the same thing."},{"Start":"02:44.365 ","End":"02:48.665","Text":"Those of you who know what functions off cosine is an even function."},{"Start":"02:48.665 ","End":"02:51.600","Text":"If you don\u0027t then forget what I just said."},{"Start":"02:52.120 ","End":"03:00.815","Text":"These are the solutions and we\u0027re done with part a."},{"Start":"03:00.815 ","End":"03:03.790","Text":"Let\u0027s move on to part b."},{"Start":"03:03.790 ","End":"03:07.010","Text":"In part b, once again,"},{"Start":"03:07.010 ","End":"03:13.320","Text":"I want to get it in the form cosine of x equals cosine of something."},{"Start":"03:17.030 ","End":"03:23.820","Text":"I need to know square root of 3 over 2 is cosine of what?"},{"Start":"03:23.820 ","End":"03:26.750","Text":"What I need is 1 particular answer here."},{"Start":"03:26.750 ","End":"03:28.535","Text":"You could use the calculator,"},{"Start":"03:28.535 ","End":"03:33.050","Text":"to take the square root of 3 divided by 2 and then shift cosine."},{"Start":"03:33.050 ","End":"03:39.125","Text":"Or hopefully you will have remembered the special angles and their cosines."},{"Start":"03:39.125 ","End":"03:44.340","Text":"This happens to be the cosine of 30 degrees."},{"Start":"03:44.690 ","End":"03:47.325","Text":"What I have here now,"},{"Start":"03:47.325 ","End":"03:49.725","Text":"let\u0027s scroll down,"},{"Start":"03:49.725 ","End":"03:59.320","Text":"is we have that cosine x equals cosine of 30."},{"Start":"03:59.320 ","End":"04:04.160","Text":"Then we get 2 sets of solutions."},{"Start":"04:04.160 ","End":"04:10.460","Text":"One where x is 30 plus whole circles,"},{"Start":"04:10.460 ","End":"04:17.510","Text":"which is what I call the 360k and the other possibility is the mirror image,"},{"Start":"04:17.510 ","End":"04:24.120","Text":"the minus angle, minus 30 also plus 360k."},{"Start":"04:26.930 ","End":"04:31.210","Text":"That\u0027s it for this exercise."}],"ID":5399},{"Watched":false,"Name":"Exercise 7","Duration":"4m 9s","ChapterTopicVideoID":5401,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5401.jpeg","UploadDate":"2016-03-10T21:13:57.2030000","DurationForVideoObject":"PT4M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:11.325","Text":"Here we have these couple of equations on cosine and we\u0027ll start with part a."},{"Start":"00:11.325 ","End":"00:14.910","Text":"The idea, as always is to get it"},{"Start":"00:14.910 ","End":"00:18.870","Text":"into the form cosine of something equals cosine of something."},{"Start":"00:18.870 ","End":"00:21.585","Text":"We\u0027re okay on the left-hand side,"},{"Start":"00:21.585 ","End":"00:26.115","Text":"we have cosine of something 3x to be precise."},{"Start":"00:26.115 ","End":"00:30.450","Text":"What we need is for this to be the cosine of something,"},{"Start":"00:30.450 ","End":"00:38.025","Text":"remember all these famous cosine of popular angles."},{"Start":"00:38.025 ","End":"00:40.860","Text":"I put it 45,60,"},{"Start":"00:40.860 ","End":"00:43.370","Text":"30,50, 0 and so on."},{"Start":"00:43.370 ","End":"00:49.055","Text":"I know that this is the cosine of 45 degrees."},{"Start":"00:49.055 ","End":"00:51.080","Text":"If you don\u0027t remember,"},{"Start":"00:51.080 ","End":"00:55.700","Text":"you can use your calculator to compute square root of 2 over 2"},{"Start":"00:55.700 ","End":"01:01.355","Text":"and then do shift sign or however it works on your calculator."},{"Start":"01:01.355 ","End":"01:05.075","Text":"It will come out to be 45 degrees."},{"Start":"01:05.075 ","End":"01:07.580","Text":"Once we have this, then it\u0027s standard."},{"Start":"01:07.580 ","End":"01:10.970","Text":"We have 2 infinite families of solutions."},{"Start":"01:10.970 ","End":"01:16.400","Text":"One we get by comparing this argument to this argument, 3x=45."},{"Start":"01:16.400 ","End":"01:21.405","Text":"But we also add whole circles,"},{"Start":"01:21.405 ","End":"01:24.180","Text":"in other words 360K."},{"Start":"01:24.180 ","End":"01:26.660","Text":"The other family of solutions is similar,"},{"Start":"01:26.660 ","End":"01:31.535","Text":"just that we take the negative of whatever this argument was,"},{"Start":"01:31.535 ","End":"01:36.990","Text":"minus 45 and again plus 360k."},{"Start":"01:37.190 ","End":"01:40.910","Text":"This makes sense with the minus because the cosine of"},{"Start":"01:40.910 ","End":"01:45.060","Text":"an angle and the cosine of minus the angle of the same."},{"Start":"01:45.140 ","End":"01:49.670","Text":"Cosine 45 and cosine minus 45 are the same and for any angle,"},{"Start":"01:49.670 ","End":"01:54.320","Text":"that\u0027s true that\u0027s why we have this plus and minus on"},{"Start":"01:54.320 ","End":"01:56.720","Text":"the whole circle also makes sense when you go around"},{"Start":"01:56.720 ","End":"02:00.420","Text":"the circle is the same point on the circle."},{"Start":"02:00.640 ","End":"02:05.720","Text":"We\u0027re not done yet because we have to find what x is and we have 3x."},{"Start":"02:05.720 ","End":"02:08.890","Text":"Just divide everything by 3."},{"Start":"02:08.890 ","End":"02:11.670","Text":"We get from the top one,"},{"Start":"02:11.670 ","End":"02:17.310","Text":"x =15 plus120K,"},{"Start":"02:17.310 ","End":"02:19.540","Text":"just dividing by 3."},{"Start":"02:19.540 ","End":"02:22.460","Text":"Mentally not hard."},{"Start":"02:22.460 ","End":"02:29.675","Text":"The second family becomes x = minus 15 also plus a 120K."},{"Start":"02:29.675 ","End":"02:31.590","Text":"That\u0027s all there is to it."},{"Start":"02:31.590 ","End":"02:34.810","Text":"Let\u0027s move on to part b."},{"Start":"02:35.860 ","End":"02:41.570","Text":"Now here also, I don\u0027t have quite cosine of something equals cosine something."},{"Start":"02:41.570 ","End":"02:43.220","Text":"In fact, I don\u0027t have an either side,"},{"Start":"02:43.220 ","End":"02:47.020","Text":"but the obvious thing to do is to divide everything by 2,"},{"Start":"02:47.020 ","End":"02:50.900","Text":"so cosine 2x =1/2."},{"Start":"02:50.900 ","End":"02:58.210","Text":"Now I have to find out a half is the cosine of what? I do this a lot."},{"Start":"02:58.210 ","End":"03:02.350","Text":"I know that the answer is 60 degrees,"},{"Start":"03:02.350 ","End":"03:05.140","Text":"but if you either remember it or you"},{"Start":"03:05.140 ","End":"03:08.125","Text":"have the table with you or you do it on the calculator,"},{"Start":"03:08.125 ","End":"03:11.710","Text":"take 0.5 and then shift cosine,"},{"Start":"03:11.710 ","End":"03:18.580","Text":"and you will get that the answer is 60 degrees."},{"Start":"03:18.580 ","End":"03:24.065","Text":"Cosine 2x= Cosine 60."},{"Start":"03:24.065 ","End":"03:29.240","Text":"Then we proceed to get the 2 infinite families of solutions."},{"Start":"03:29.240 ","End":"03:38.210","Text":"We have that 2x is 60 plus multiples of 360,"},{"Start":"03:38.210 ","End":"03:42.305","Text":"or we get the minus angle,"},{"Start":"03:42.305 ","End":"03:46.090","Text":"minus 60 also plus 360K."},{"Start":"03:46.090 ","End":"03:51.275","Text":"Finally, you just need to isolate x so we divide everything by 2."},{"Start":"03:51.275 ","End":"03:57.570","Text":"The first line gives us x=30.5 and half of this is 180K."},{"Start":"03:58.150 ","End":"04:05.255","Text":"The other one comes out to be minus 30 plus 180K."},{"Start":"04:05.255 ","End":"04:09.360","Text":"That\u0027s fairly straightforward. We\u0027re done."}],"ID":5400},{"Watched":false,"Name":"Exercise 8","Duration":"4m 36s","ChapterTopicVideoID":5402,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5402.jpeg","UploadDate":"2016-03-10T21:14:30.4670000","DurationForVideoObject":"PT4M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we have a pair of equations to solve."},{"Start":"00:04.500 ","End":"00:06.795","Text":"Each involving the cosine."},{"Start":"00:06.795 ","End":"00:09.975","Text":"The idea in both of them is to bring"},{"Start":"00:09.975 ","End":"00:14.325","Text":"the equation to the form cosine something equals cosine something."},{"Start":"00:14.325 ","End":"00:16.740","Text":"Now in part a,"},{"Start":"00:16.740 ","End":"00:18.360","Text":"it\u0027s already is in this form."},{"Start":"00:18.360 ","End":"00:20.430","Text":"Cosine something equals cosine something."},{"Start":"00:20.430 ","End":"00:22.815","Text":"We know how to solve this."},{"Start":"00:22.815 ","End":"00:25.275","Text":"First possibility is that,"},{"Start":"00:25.275 ","End":"00:27.420","Text":"this argument equals this argument."},{"Start":"00:27.420 ","End":"00:35.080","Text":"In other words, 2x+30=x."},{"Start":"00:35.390 ","End":"00:40.730","Text":"As usual, we add multiples of a whole circle."},{"Start":"00:40.730 ","End":"00:43.835","Text":"Which we write as 360K."},{"Start":"00:43.835 ","End":"00:48.980","Text":"Because the cosine doesn\u0027t change if you add a full circle."},{"Start":"00:48.980 ","End":"00:53.390","Text":"As usual, that\u0027s just one possibility out of two."},{"Start":"00:53.390 ","End":"00:55.820","Text":"The other possibility is that"},{"Start":"00:55.820 ","End":"01:00.700","Text":"this argument is not equal to this one but to it\u0027s negative."},{"Start":"01:00.700 ","End":"01:04.610","Text":"Because the cosine of a negative of something is the same."},{"Start":"01:04.610 ","End":"01:09.585","Text":"Again, we add whole circles 360K."},{"Start":"01:09.585 ","End":"01:13.265","Text":"It\u0027s two infinite families of solutions."},{"Start":"01:13.265 ","End":"01:19.085","Text":"But this is not the answer yet because we need to just get x on one side."},{"Start":"01:19.085 ","End":"01:21.895","Text":"We need to do a bit of algebra."},{"Start":"01:21.895 ","End":"01:27.580","Text":"Let\u0027s see. Let\u0027s take the first one and I\u0027ll do it over here."},{"Start":"01:27.580 ","End":"01:29.805","Text":"Move x to the left,"},{"Start":"01:29.805 ","End":"01:32.010","Text":"and the number to the right."},{"Start":"01:32.010 ","End":"01:34.785","Text":"We get that x equals,"},{"Start":"01:34.785 ","End":"01:41.860","Text":"this x is 2x-x-30+360K."},{"Start":"01:41.860 ","End":"01:45.545","Text":"That\u0027s one possibility and the other one I\u0027m just continuing here."},{"Start":"01:45.545 ","End":"01:47.240","Text":"Bring the x over,"},{"Start":"01:47.240 ","End":"01:50.810","Text":"I get 3x and then the 30 to"},{"Start":"01:50.810 ","End":"01:59.590","Text":"the right is equal to -30+360K."},{"Start":"02:00.140 ","End":"02:10.270","Text":"Then we divide by 3 we get x=-10+120K from dividing by 3."},{"Start":"02:12.200 ","End":"02:20.045","Text":"This is one family of solutions or say family of set because K can vary over any integer."},{"Start":"02:20.045 ","End":"02:23.065","Text":"The other family of solutions is this."},{"Start":"02:23.065 ","End":"02:26.530","Text":"Let\u0027s move on to part b."},{"Start":"02:26.990 ","End":"02:29.670","Text":"In Part b,"},{"Start":"02:29.670 ","End":"02:33.110","Text":"it\u0027s not quite in the form cosine equals cosine."},{"Start":"02:33.110 ","End":"02:37.740","Text":"But clearly if we bring this one to the right."},{"Start":"02:37.740 ","End":"02:39.825","Text":"Then it will be in that form."},{"Start":"02:39.825 ","End":"02:44.670","Text":"Cosine 2x=cosine 4x."},{"Start":"02:44.670 ","End":"02:47.025","Text":"Then we proceed as usual."},{"Start":"02:47.025 ","End":"02:56.115","Text":"That either the 2x=4x or we could add 360K."},{"Start":"02:56.115 ","End":"03:05.270","Text":"The other possibility is that it\u0027s minus 4x also plus whole circles."},{"Start":"03:05.270 ","End":"03:09.600","Text":"We have to extract x."},{"Start":"03:09.880 ","End":"03:15.590","Text":"Let\u0027s see, if I look at the first one."},{"Start":"03:15.590 ","End":"03:18.200","Text":"Let\u0027s see if we can do it in our heads."},{"Start":"03:18.200 ","End":"03:21.110","Text":"Maybe not. I\u0027ll do it. I\u0027ll write it out."},{"Start":"03:21.110 ","End":"03:31.110","Text":"We get 2x-4x is -2x=360K."},{"Start":"03:33.410 ","End":"03:36.390","Text":"Then we divide by minus 2."},{"Start":"03:36.390 ","End":"03:44.340","Text":"We get x=-180K. I\u0027ll highlight it."},{"Start":"03:44.340 ","End":"03:49.950","Text":"But actually, it\u0027s often customary"},{"Start":"03:49.950 ","End":"03:56.460","Text":"to forget about the minus because if K is a general whole number,"},{"Start":"03:56.460 ","End":"03:59.760","Text":"so is minus K. We don\u0027t need the minus replace K by"},{"Start":"03:59.760 ","End":"04:03.530","Text":"minus K. We can write it as just x=180K."},{"Start":"04:03.530 ","End":"04:05.060","Text":"But if that\u0027s confusing to you,"},{"Start":"04:05.060 ","End":"04:06.650","Text":"then leave it like that."},{"Start":"04:06.650 ","End":"04:12.600","Text":"That was from the first possibility."},{"Start":"04:12.600 ","End":"04:15.635","Text":"The second possibility I\u0027ll continue over here."},{"Start":"04:15.635 ","End":"04:21.410","Text":"This time we get 6x=360K."},{"Start":"04:21.410 ","End":"04:27.440","Text":"Dividing by 6 we get x=60K."},{"Start":"04:27.440 ","End":"04:28.610","Text":"No minuses here."},{"Start":"04:28.610 ","End":"04:30.800","Text":"No need for anything fancy."},{"Start":"04:30.800 ","End":"04:32.990","Text":"I\u0027ll just highlight it."},{"Start":"04:32.990 ","End":"04:36.600","Text":"That\u0027s Part B. We\u0027re done."}],"ID":5401},{"Watched":false,"Name":"Exercise 9","Duration":"6m 34s","ChapterTopicVideoID":5403,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5403.jpeg","UploadDate":"2016-03-10T21:15:24.3570000","DurationForVideoObject":"PT6M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.075","Text":"In this exercise, we have to solve a pair of trigonometric equations with cosine"},{"Start":"00:06.075 ","End":"00:09.690","Text":"and the strategy in both cases is to bring"},{"Start":"00:09.690 ","End":"00:14.565","Text":"the equation to the form cosine something equals cosine something."},{"Start":"00:14.565 ","End":"00:17.760","Text":"We\u0027re close to that in the first one."},{"Start":"00:17.760 ","End":"00:23.805","Text":"What I\u0027m going to do is move the cosine x to the other side,"},{"Start":"00:23.805 ","End":"00:29.670","Text":"so we get cosine 3x equals minus cosine x."},{"Start":"00:29.670 ","End":"00:32.250","Text":"Now, that\u0027s still not what we want because we want"},{"Start":"00:32.250 ","End":"00:35.205","Text":"just cosine of something without the minus."},{"Start":"00:35.205 ","End":"00:39.325","Text":"I\u0027ll use the formula that,"},{"Start":"00:39.325 ","End":"00:43.910","Text":"minus cosine of some angle could be x,"},{"Start":"00:43.910 ","End":"00:45.620","Text":"but let\u0027s use a Greek letter,"},{"Start":"00:45.620 ","End":"00:55.440","Text":"say Alpha is equal to the cosine of 180 minus Alpha in general."},{"Start":"00:55.440 ","End":"01:00.200","Text":"Now in our case, Alpha is just x and so this"},{"Start":"01:00.200 ","End":"01:05.570","Text":"becomes the cosine of 180 minus x."},{"Start":"01:05.570 ","End":"01:08.240","Text":"On the left-hand side is just the same."},{"Start":"01:08.240 ","End":"01:11.480","Text":"Now, we have cosine of one argument equals"},{"Start":"01:11.480 ","End":"01:14.900","Text":"cosine of another argument and we know how to solve that."},{"Start":"01:14.900 ","End":"01:18.990","Text":"The first possibility is that the arguments are equal,"},{"Start":"01:19.400 ","End":"01:21.630","Text":"but not just that."},{"Start":"01:21.630 ","End":"01:23.480","Text":"A whole family of solutions,"},{"Start":"01:23.480 ","End":"01:27.620","Text":"by adding 360k and a whole number of circles,"},{"Start":"01:27.620 ","End":"01:29.072","Text":"k is a whole number."},{"Start":"01:29.072 ","End":"01:31.850","Text":"The other possibility is similar to this,"},{"Start":"01:31.850 ","End":"01:37.295","Text":"except that we take minus of what was here."},{"Start":"01:37.295 ","End":"01:41.870","Text":"Here, I\u0027ll write minus 180 plus x,"},{"Start":"01:41.870 ","End":"01:43.790","Text":"that\u0027s just the minus of this thing,"},{"Start":"01:43.790 ","End":"01:46.170","Text":"and again, plus 360k."},{"Start":"01:46.330 ","End":"01:52.325","Text":"Of course, can\u0027t leave the answer like this because I need just x equals. Let\u0027s see."},{"Start":"01:52.325 ","End":"01:55.835","Text":"I\u0027ll start with the first one, continue over here,"},{"Start":"01:55.835 ","End":"02:00.900","Text":"bring the x to the left and then we get 4x equals"},{"Start":"02:00.900 ","End":"02:10.640","Text":"180 plus 360 times some whole number k. Then I divide by 4."},{"Start":"02:10.640 ","End":"02:20.505","Text":"I\u0027ve got x equals 180 over 4 is 45 and 360 over 4 is 90."},{"Start":"02:20.505 ","End":"02:25.470","Text":"This is the first family of solutions,"},{"Start":"02:25.470 ","End":"02:30.545","Text":"actually, it\u0027s an infinite number of solutions here, and highlight that."},{"Start":"02:30.545 ","End":"02:34.355","Text":"Then we also can continue with the second equation."},{"Start":"02:34.355 ","End":"02:36.140","Text":"Then if we bring the x over,"},{"Start":"02:36.140 ","End":"02:42.130","Text":"we get 2x equals minus 180 plus 360k"},{"Start":"02:42.130 ","End":"02:50.965","Text":"divide by 2 this time and we\u0027ve got x equals minus 90 plus 180k,"},{"Start":"02:50.965 ","End":"02:54.855","Text":"and that\u0027s our second family of solutions."},{"Start":"02:54.855 ","End":"02:59.900","Text":"On to part b. In part b, again,"},{"Start":"02:59.900 ","End":"03:02.855","Text":"we\u0027re close to cosine equals cosine,"},{"Start":"03:02.855 ","End":"03:05.975","Text":"but we have that minus again."},{"Start":"03:05.975 ","End":"03:07.850","Text":"I\u0027ll write that formula again,"},{"Start":"03:07.850 ","End":"03:09.515","Text":"in case you\u0027ve forgotten it already."},{"Start":"03:09.515 ","End":"03:16.050","Text":"Minus cosine Alpha is cosine of 180 minus Alpha."},{"Start":"03:16.050 ","End":"03:17.840","Text":"So using that here,"},{"Start":"03:17.840 ","End":"03:21.005","Text":"we get cosine of,"},{"Start":"03:21.005 ","End":"03:24.740","Text":"well, let\u0027s see, what is 180 minus this?"},{"Start":"03:24.740 ","End":"03:26.615","Text":"I\u0027ll do a computation here."},{"Start":"03:26.615 ","End":"03:34.175","Text":"180 minus x minus 30 is equal to,"},{"Start":"03:34.175 ","End":"03:37.055","Text":"let\u0027s see, that\u0027s going to be plus 30,"},{"Start":"03:37.055 ","End":"03:39.860","Text":"it\u0027s 210 minus x."},{"Start":"03:39.860 ","End":"03:41.465","Text":"That\u0027s what I\u0027m writing here,"},{"Start":"03:41.465 ","End":"03:44.160","Text":"210 minus x,"},{"Start":"03:44.160 ","End":"03:49.325","Text":"and the right-hand side already is just cosine of something which is 60 plus x."},{"Start":"03:49.325 ","End":"03:51.239","Text":"So we proceed, as usual,"},{"Start":"03:51.239 ","End":"03:54.935","Text":"we get 2 families from this,"},{"Start":"03:54.935 ","End":"04:04.320","Text":"either 210 minus x equals 60 plus x or plus a whole number of 360s."},{"Start":"04:04.320 ","End":"04:09.325","Text":"The other possibility is to take minus here,"},{"Start":"04:09.325 ","End":"04:14.050","Text":"so it\u0027s minus 60 minus x, again, plus 360k."},{"Start":"04:14.050 ","End":"04:19.860","Text":"Now, we just have to extract x from each side."},{"Start":"04:19.860 ","End":"04:23.110","Text":"If we go for the first one,"},{"Start":"04:23.110 ","End":"04:26.015","Text":"what we get is,"},{"Start":"04:26.015 ","End":"04:28.260","Text":"if I bring the x\u0027s to the left,"},{"Start":"04:28.260 ","End":"04:30.945","Text":"I have minus 2x,"},{"Start":"04:30.945 ","End":"04:33.315","Text":"and the numbers to the right,"},{"Start":"04:33.315 ","End":"04:39.990","Text":"I\u0027ve got 60 minus"},{"Start":"04:39.990 ","End":"04:45.610","Text":"210 minus 150 plus 360k."},{"Start":"04:46.850 ","End":"04:50.335","Text":"Then if I divide by minus 2,"},{"Start":"04:50.335 ","End":"04:58.430","Text":"I get that x equals this over minus 2 is 75,"},{"Start":"04:58.430 ","End":"05:03.360","Text":"and then minus 180k."},{"Start":"05:04.540 ","End":"05:08.180","Text":"This is one family of solutions."},{"Start":"05:08.180 ","End":"05:16.650","Text":"I\u0027ll just note that it\u0027s nicer to write it as 75 plus 180k."},{"Start":"05:16.650 ","End":"05:19.460","Text":"The reason we can do that is because k is"},{"Start":"05:19.460 ","End":"05:25.080","Text":"just any old integer and minus k is just any old integer also."},{"Start":"05:25.080 ","End":"05:28.305","Text":"You can replace k by minus k,"},{"Start":"05:28.305 ","End":"05:30.500","Text":"but if you\u0027re not sure about this,"},{"Start":"05:30.500 ","End":"05:32.255","Text":"then leave it like that."},{"Start":"05:32.255 ","End":"05:35.570","Text":"Now, let\u0027s look at the other one and what do we get here?"},{"Start":"05:35.570 ","End":"05:39.260","Text":"Let\u0027s see. If I bring the x\u0027s to the left,"},{"Start":"05:39.260 ","End":"05:43.100","Text":"it\u0027s going to cancel and I\u0027m going to get 0."},{"Start":"05:43.100 ","End":"05:45.695","Text":"What do we do with that? X has disappeared."},{"Start":"05:45.695 ","End":"05:48.905","Text":"Well, let\u0027s just write the right-hand side for the moment just to continue."},{"Start":"05:48.905 ","End":"05:56.820","Text":"We get minus 60 minus 210 is minus 270 plus 360k."},{"Start":"05:56.960 ","End":"06:05.090","Text":"X has disappeared and we have some impossible equality that just isn\u0027t true."},{"Start":"06:05.090 ","End":"06:11.675","Text":"Whatever k is, this thing is not going to be 0 and this is just a wrong statement."},{"Start":"06:11.675 ","End":"06:15.589","Text":"The way to interpret this is that we just don\u0027t get anything."},{"Start":"06:15.589 ","End":"06:21.987","Text":"There\u0027s no solution from this branch, from the second one."},{"Start":"06:21.987 ","End":"06:23.810","Text":"There still is this solution,"},{"Start":"06:23.810 ","End":"06:25.835","Text":"but we don\u0027t get anything extra,"},{"Start":"06:25.835 ","End":"06:30.780","Text":"so the answer is just this what I\u0027ve highlighted."},{"Start":"06:30.800 ","End":"06:34.960","Text":"Done with Part B, so that\u0027s it."}],"ID":5402},{"Watched":false,"Name":"Exercise 10","Duration":"5m 40s","ChapterTopicVideoID":5404,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5404.jpeg","UploadDate":"2016-03-10T21:16:04.8430000","DurationForVideoObject":"PT5M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.399","Text":"In this exercise, we have 2 equations to solve involving the cosine,"},{"Start":"00:05.399 ","End":"00:07.290","Text":"but they\u0027re very special."},{"Start":"00:07.290 ","End":"00:09.405","Text":"Then they\u0027re similar."},{"Start":"00:09.405 ","End":"00:13.995","Text":"In each case I have to find all the angles X,"},{"Start":"00:13.995 ","End":"00:18.945","Text":"which have either a cosine of 0 or 1 in this case."},{"Start":"00:18.945 ","End":"00:22.875","Text":"These are very special values for the cosine 0,"},{"Start":"00:22.875 ","End":"00:24.795","Text":"and 1 is the maximum."},{"Start":"00:24.795 ","End":"00:26.520","Text":"Let\u0027s see what we get."},{"Start":"00:26.520 ","End":"00:28.080","Text":"Start with part a,"},{"Start":"00:28.080 ","End":"00:33.565","Text":"which is to find the angles whose cosine is 0."},{"Start":"00:33.565 ","End":"00:40.099","Text":"We start off by trying to write it as cosine something equals cosine something,"},{"Start":"00:40.099 ","End":"00:47.750","Text":"I get cosine x equals and we just remember that cosine of 90 is 0,"},{"Start":"00:47.750 ","End":"00:51.690","Text":"or look it up on the calculator."},{"Start":"00:51.880 ","End":"00:55.880","Text":"We know what to do with cosine something equals cosine something."},{"Start":"00:55.880 ","End":"01:00.115","Text":"We get 2 families of solutions."},{"Start":"01:00.115 ","End":"01:04.030","Text":"1 is that x could be 90,"},{"Start":"01:04.030 ","End":"01:07.400","Text":"but also multiples of a whole circle,"},{"Start":"01:07.400 ","End":"01:09.680","Text":"that\u0027s 360 times k,"},{"Start":"01:09.680 ","End":"01:12.305","Text":"which is an integer and"},{"Start":"01:12.305 ","End":"01:16.325","Text":"the other possibilities where we take the minus of this thing here,"},{"Start":"01:16.325 ","End":"01:22.650","Text":"we also get minus 90 plus whole circles."},{"Start":"01:22.880 ","End":"01:27.690","Text":"Now if I write these out,"},{"Start":"01:27.690 ","End":"01:30.755","Text":"you\u0027ll see that there is a way of simplifying this."},{"Start":"01:30.755 ","End":"01:33.500","Text":"I mean, this is the correct answer and we could stop here,"},{"Start":"01:33.500 ","End":"01:37.520","Text":"but there\u0027s a way of simplifying it in mathematics."},{"Start":"01:37.520 ","End":"01:40.430","Text":"Mathematicians like things more simple."},{"Start":"01:40.430 ","End":"01:45.055","Text":"Let\u0027s write some of the values here."},{"Start":"01:45.055 ","End":"01:48.470","Text":"1 of them is if k is 0, we just get 90."},{"Start":"01:48.470 ","End":"01:52.130","Text":"Then if k is 1, we get 450."},{"Start":"01:52.130 ","End":"01:53.390","Text":"If k is 2,"},{"Start":"01:53.390 ","End":"01:58.730","Text":"we get 720 and 90 is 810, and so on."},{"Start":"01:58.730 ","End":"02:06.155","Text":"[inaudible] negative value k could be minus 1,"},{"Start":"02:06.155 ","End":"02:12.235","Text":"in which case we have minus 270 and that\u0027ll do."},{"Start":"02:12.235 ","End":"02:15.849","Text":"If I take these values,"},{"Start":"02:15.849 ","End":"02:18.400","Text":"let\u0027s say minus 90,"},{"Start":"02:18.400 ","End":"02:22.120","Text":"then I could write minus 90 here."},{"Start":"02:22.120 ","End":"02:28.570","Text":"If k is 0, if k is 1, I get 270."},{"Start":"02:28.570 ","End":"02:32.440","Text":"I\u0027m trying to write these in the spaces between these."},{"Start":"02:32.440 ","End":"02:41.710","Text":"720 minus 90 is 630, and so on."},{"Start":"02:41.710 ","End":"02:45.965","Text":"Now, if we combine these into 1 list,"},{"Start":"02:45.965 ","End":"02:52.550","Text":"then what I\u0027ll get is minus 270, minus 90,"},{"Start":"02:52.550 ","End":"02:59.130","Text":"90 then 270, 450,"},{"Start":"02:59.130 ","End":"03:03.950","Text":"630 a term more than enough,"},{"Start":"03:03.950 ","End":"03:06.625","Text":"infinite in both directions."},{"Start":"03:06.625 ","End":"03:11.560","Text":"Now if I look at the difference between any 2 of these,"},{"Start":"03:11.560 ","End":"03:15.325","Text":"I mean, for example, between this and this, it\u0027s 180."},{"Start":"03:15.325 ","End":"03:24.075","Text":"Between this and this it\u0027s a 180 and check each pair you\u0027ll see that we go in jumps of,"},{"Start":"03:24.075 ","End":"03:25.460","Text":"I\u0027ll just write it on of them."},{"Start":"03:25.460 ","End":"03:29.480","Text":"There\u0027s 180 gap between each."},{"Start":"03:29.480 ","End":"03:32.960","Text":"Another way of describing this is to pick any of these values."},{"Start":"03:32.960 ","End":"03:35.875","Text":"I\u0027ll pick this 1 and say a special 1,"},{"Start":"03:35.875 ","End":"03:40.520","Text":"and say that we take this value and add multiples of 180."},{"Start":"03:40.520 ","End":"03:42.845","Text":"Could be positive or negative multiples,"},{"Start":"03:42.845 ","End":"03:46.250","Text":"and we\u0027d get 90 plus 180k."},{"Start":"03:46.250 ","End":"03:49.010","Text":"We are not the same k as this, but k,"},{"Start":"03:49.010 ","End":"03:52.580","Text":"meaning any whole number, any integer."},{"Start":"03:52.580 ","End":"03:54.395","Text":"K is 0 here,"},{"Start":"03:54.395 ","End":"03:56.030","Text":"k is 1, we get this,"},{"Start":"03:56.030 ","End":"03:57.215","Text":"k is 2,"},{"Start":"03:57.215 ","End":"04:01.565","Text":"we get 360 plus 90 is this k is 4, k 5."},{"Start":"04:01.565 ","End":"04:04.170","Text":"Here, k is minus 1,"},{"Start":"04:04.170 ","End":"04:06.550","Text":"minus 2, and so on."},{"Start":"04:08.510 ","End":"04:10.970","Text":"I won\u0027t belabor the point anymore."},{"Start":"04:10.970 ","End":"04:12.655","Text":"This is the answer."},{"Start":"04:12.655 ","End":"04:14.600","Text":"I\u0027ll highlight it."},{"Start":"04:14.600 ","End":"04:18.320","Text":"Although technically you could have left the answer like this as I said,"},{"Start":"04:18.320 ","End":"04:19.655","Text":"this is much simpler."},{"Start":"04:19.655 ","End":"04:23.765","Text":"Let\u0027s get onto the other cosine and for the 0 we want the value"},{"Start":"04:23.765 ","End":"04:28.940","Text":"1 and again we proceed in the same way."},{"Start":"04:28.940 ","End":"04:31.505","Text":"We want the cosine equals cosine."},{"Start":"04:31.505 ","End":"04:37.520","Text":"Cosine x is, and we know that cosine of 0 is 1,"},{"Start":"04:37.520 ","End":"04:40.865","Text":"or you should know or look it up on your calculator."},{"Start":"04:40.865 ","End":"04:46.850","Text":"Then that means that if we just proceed mechanically,"},{"Start":"04:46.850 ","End":"04:53.150","Text":"we would say x=0 plus whole circles,"},{"Start":"04:53.150 ","End":"05:03.665","Text":"360k and the other family would be minus what\u0027s written here, plus 360k."},{"Start":"05:03.665 ","End":"05:07.730","Text":"Now, 0 and minus 0 are the same."},{"Start":"05:07.730 ","End":"05:10.925","Text":"So I haven\u0027t really added anything new here."},{"Start":"05:10.925 ","End":"05:13.885","Text":"I\u0027m just crossing it out not because it\u0027s wrong,"},{"Start":"05:13.885 ","End":"05:15.860","Text":"because it\u0027s redundant, it\u0027s unnecessary."},{"Start":"05:15.860 ","End":"05:17.990","Text":"We already have everything here."},{"Start":"05:17.990 ","End":"05:23.250","Text":"This would be the general solution only we wouldn\u0027t write the 0."},{"Start":"05:23.250 ","End":"05:31.950","Text":"We would just write X=360 times k multiples of 360 and that\u0027s the answer."},{"Start":"05:31.950 ","End":"05:34.060","Text":"I\u0027ll just highlight it."},{"Start":"05:35.090 ","End":"05:40.030","Text":"There we go. Okay. Done."}],"ID":5403},{"Watched":false,"Name":"Exercise 11","Duration":"6m 55s","ChapterTopicVideoID":5405,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5405.jpeg","UploadDate":"2016-03-10T21:16:57.1270000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.270","Text":"Here we have two trigonometric equations to solve."},{"Start":"00:05.270 ","End":"00:12.960","Text":"Up till now, we\u0027ve learned to solve equations of the type sine something"},{"Start":"00:12.960 ","End":"00:15.900","Text":"equals sine something or cosine of something equals cosine"},{"Start":"00:15.900 ","End":"00:20.370","Text":"something and here we have hybrid a mixture."},{"Start":"00:20.370 ","End":"00:25.890","Text":"Let\u0027s start with part a and I\u0027ll show you how we handle such a situation."},{"Start":"00:25.890 ","End":"00:33.185","Text":"What we want to do is either convert the sine to a cosine or the cosine to a sine,"},{"Start":"00:33.185 ","End":"00:38.450","Text":"either one will work and then rewrite the formula that we\u0027re going to use."},{"Start":"00:38.450 ","End":"00:41.780","Text":"The formula is that cosine of some angle,"},{"Start":"00:41.780 ","End":"00:43.490","Text":"let\u0027s call it Alpha,"},{"Start":"00:43.490 ","End":"00:47.870","Text":"is the same as the sine of what we call the complimentary angle,"},{"Start":"00:47.870 ","End":"00:53.280","Text":"or what it means is 90 minus Alpha."},{"Start":"00:53.480 ","End":"00:56.556","Text":"If Alpha is 30,"},{"Start":"00:56.556 ","End":"00:59.035","Text":"90 minus Alpha is 60."},{"Start":"00:59.035 ","End":"01:05.930","Text":"Sometimes this is written with three bars because it\u0027s not an equation, it\u0027s an identity,"},{"Start":"01:05.930 ","End":"01:08.610","Text":"it\u0027s always true and by the way,"},{"Start":"01:08.610 ","End":"01:12.500","Text":"there is also very similar formula."},{"Start":"01:12.500 ","End":"01:15.260","Text":"If I put sine instead of cosine, it also works,"},{"Start":"01:15.260 ","End":"01:16.610","Text":"could put the sine Alpha,"},{"Start":"01:16.610 ","End":"01:19.775","Text":"cosine 90 minus Alpha according to what I need."},{"Start":"01:19.775 ","End":"01:21.500","Text":"Now, like I said,"},{"Start":"01:21.500 ","End":"01:22.580","Text":"I could go both ways."},{"Start":"01:22.580 ","End":"01:25.085","Text":"I could decide to convert to sine or convert the cosine."},{"Start":"01:25.085 ","End":"01:28.280","Text":"I\u0027ll just choose to convert to sine."},{"Start":"01:28.280 ","End":"01:30.170","Text":"I guess the cosine here is simpler."},{"Start":"01:30.170 ","End":"01:34.135","Text":"It doesn\u0027t have the 3 and it will convert the cosine to a sine."},{"Start":"01:34.135 ","End":"01:43.475","Text":"What I get is sine 3x is using this sine of 90 minus x."},{"Start":"01:43.475 ","End":"01:45.470","Text":"Now we know how to solve this."},{"Start":"01:45.470 ","End":"01:48.544","Text":"We get two possibilities,"},{"Start":"01:48.544 ","End":"01:52.640","Text":"either 3x is 90 minus x,"},{"Start":"01:52.640 ","End":"01:57.080","Text":"I compare the arguments of the sine and we also must remember"},{"Start":"01:57.080 ","End":"02:04.040","Text":"to add multiples of 360 to get the complete solution,"},{"Start":"02:04.040 ","End":"02:06.200","Text":"at least for this family,"},{"Start":"02:06.200 ","End":"02:08.210","Text":"there\u0027s another family of solutions."},{"Start":"02:08.210 ","End":"02:11.780","Text":"I call it family because there\u0027s an infinite number of them and depending on k,"},{"Start":"02:11.780 ","End":"02:13.445","Text":"k could be any integer."},{"Start":"02:13.445 ","End":"02:18.125","Text":"The other one, I subtract this from 180,"},{"Start":"02:18.125 ","End":"02:19.490","Text":"and let\u0027s do that at the side."},{"Start":"02:19.490 ","End":"02:23.720","Text":"What is a 180 minus 90 minus x?"},{"Start":"02:23.720 ","End":"02:27.605","Text":"180 minus 90 is minus 90,"},{"Start":"02:27.605 ","End":"02:39.730","Text":"minus minus x is plus x. I also have the possibility that 3x is minus 90."},{"Start":"02:39.730 ","End":"02:42.135","Text":"Sorry, this is plus 90."},{"Start":"02:42.135 ","End":"02:46.830","Text":"No minus here, no minus here, sorry."},{"Start":"02:46.830 ","End":"02:50.960","Text":"Plus x plus 360k."},{"Start":"02:51.580 ","End":"02:54.607","Text":"Now we need just x."},{"Start":"02:54.607 ","End":"02:58.790","Text":"Let\u0027s start with the first one,"},{"Start":"02:58.790 ","End":"03:00.830","Text":"I\u0027ll continue it over here."},{"Start":"03:00.830 ","End":"03:07.350","Text":"Bring the x\u0027s to the left so what I have is 4x equals"},{"Start":"03:07.350 ","End":"03:14.655","Text":"90 plus 360k and then divide by 4."},{"Start":"03:14.655 ","End":"03:18.075","Text":"I\u0027ve got x equals 90 over 4."},{"Start":"03:18.075 ","End":"03:24.630","Text":"Let\u0027s see, that\u0027s 22.5 and 360 over 4 is 90."},{"Start":"03:24.630 ","End":"03:29.550","Text":"This is one family of solutions and I\u0027ll highlight it."},{"Start":"03:29.550 ","End":"03:31.440","Text":"Now back to the other one."},{"Start":"03:31.440 ","End":"03:33.090","Text":"Continuing this one,"},{"Start":"03:33.090 ","End":"03:34.200","Text":"I bring the x over,"},{"Start":"03:34.200 ","End":"03:40.920","Text":"I get 2x equals 90 plus 360k."},{"Start":"03:40.920 ","End":"03:42.180","Text":"I\u0027m just dividing by 2."},{"Start":"03:42.180 ","End":"03:48.370","Text":"I get x equals 45 plus 180k."},{"Start":"03:48.370 ","End":"03:56.700","Text":"This is my other set or family of solutions. On to part b."},{"Start":"03:56.990 ","End":"04:02.460","Text":"Again, we have one of these hybrids that has sine and cosine."},{"Start":"04:03.820 ","End":"04:07.610","Text":"I\u0027ll convert this cosine to"},{"Start":"04:07.610 ","End":"04:13.140","Text":"a sine and I\u0027ll use"},{"Start":"04:13.140 ","End":"04:20.075","Text":"the same trigonometric identity as above and write the cosine(x),"},{"Start":"04:20.075 ","End":"04:25.280","Text":"sine of 90 minus x."},{"Start":"04:25.280 ","End":"04:31.030","Text":"This is still not good enough for me because I still don\u0027t have sine equals sine."},{"Start":"04:31.030 ","End":"04:33.680","Text":"I have this minus here but if you"},{"Start":"04:33.680 ","End":"04:36.800","Text":"recall what we do if we have a minus in front of a sine is"},{"Start":"04:36.800 ","End":"04:43.880","Text":"we just put it inside and so we get the sine of minus of this,"},{"Start":"04:43.880 ","End":"04:48.309","Text":"which is minus 90 plus x."},{"Start":"04:48.309 ","End":"04:52.270","Text":"Now I have sine equals sine."},{"Start":"04:52.270 ","End":"04:55.220","Text":"Now we can proceed as usual,"},{"Start":"04:55.220 ","End":"05:01.390","Text":"so I won\u0027t copy this out just by ditto, same thing."},{"Start":"05:01.390 ","End":"05:05.120","Text":"Now we have either this argument equals this argument so"},{"Start":"05:05.120 ","End":"05:12.650","Text":"2x plus 60 could equal minus 90 plus x,"},{"Start":"05:12.650 ","End":"05:18.110","Text":"but also plus the usual 360k and the other"},{"Start":"05:18.110 ","End":"05:27.380","Text":"possibility 2x plus 60 in the case of sine is to take 180 minus this."},{"Start":"05:27.380 ","End":"05:29.464","Text":"I\u0027ll do that at the side."},{"Start":"05:29.464 ","End":"05:37.790","Text":"180 minus minus 90 plus x is equal to this."},{"Start":"05:37.790 ","End":"05:42.455","Text":"minus, minus is plus 270 and minus x"},{"Start":"05:42.455 ","End":"05:48.625","Text":"so here 270 minus x also plus 360k."},{"Start":"05:48.625 ","End":"05:52.070","Text":"Once again, I have two sets of possibilities,"},{"Start":"05:52.070 ","End":"05:54.050","Text":"but this is not x equals,"},{"Start":"05:54.050 ","End":"05:56.135","Text":"so I have to work a bit more."},{"Start":"05:56.135 ","End":"05:58.790","Text":"If I work on this one,"},{"Start":"05:58.790 ","End":"06:01.340","Text":"I will get, let\u0027s see,"},{"Start":"06:01.340 ","End":"06:07.460","Text":"x\u0027s on the left so that just gives me x right away and 60 on"},{"Start":"06:07.460 ","End":"06:16.080","Text":"the right means it\u0027s minus 150 plus 360k."},{"Start":"06:17.570 ","End":"06:24.650","Text":"That\u0027s one set of solutions which I will highlight and now back to the other."},{"Start":"06:24.650 ","End":"06:26.540","Text":"Here I bring the x over,"},{"Start":"06:26.540 ","End":"06:28.415","Text":"I get 3x,"},{"Start":"06:28.415 ","End":"06:30.680","Text":"and the 60 goes over to here."},{"Start":"06:30.680 ","End":"06:32.810","Text":"I subtract it from 270 and get"},{"Start":"06:32.810 ","End":"06:37.190","Text":"210 plus 360k and all I"},{"Start":"06:37.190 ","End":"06:42.530","Text":"need to do is divide by 3 and then I get x equals this over 3 is 70,"},{"Start":"06:42.530 ","End":"06:45.065","Text":"this over 3 is 120k."},{"Start":"06:45.065 ","End":"06:49.325","Text":"That\u0027s our second family of solutions."},{"Start":"06:49.325 ","End":"06:55.660","Text":"I will also highlight and so we\u0027re done with part b and the exercise."}],"ID":5404},{"Watched":false,"Name":"Exercise 12","Duration":"12m 33s","ChapterTopicVideoID":5406,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5406.jpeg","UploadDate":"2016-03-10T21:18:34.6530000","DurationForVideoObject":"PT12M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.115","Text":"In this exercise, we have a pair of trigonometric equations to solve."},{"Start":"00:05.115 ","End":"00:07.995","Text":"One involving sine, one involving cosine,"},{"Start":"00:07.995 ","End":"00:10.710","Text":"but not in the usual format we expect,"},{"Start":"00:10.710 ","End":"00:13.140","Text":"which is sine something equals something,"},{"Start":"00:13.140 ","End":"00:15.270","Text":"cosine something equals sine something."},{"Start":"00:15.270 ","End":"00:17.280","Text":"Here we have this square thing,"},{"Start":"00:17.280 ","End":"00:20.205","Text":"same with the cosine there\u0027s this square factor."},{"Start":"00:20.205 ","End":"00:25.300","Text":"Let\u0027s see how we go about doing this and we\u0027ll start with a, of course."},{"Start":"00:25.490 ","End":"00:32.790","Text":"We just have to use our algebraic techniques to somehow get it to a form that we know."},{"Start":"00:32.790 ","End":"00:39.320","Text":"Now, I just want to compare this to an example from algebra."},{"Start":"00:39.320 ","End":"00:44.430","Text":"Suppose I said that a^2 equals 9,"},{"Start":"00:44.430 ","End":"00:46.725","Text":"just to take a most basic case."},{"Start":"00:46.725 ","End":"00:52.330","Text":"The way I would solve it is I would say that a is equal to,"},{"Start":"00:52.330 ","End":"00:55.975","Text":"one possibility is the square root of 9,"},{"Start":"00:55.975 ","End":"00:58.440","Text":"which is 3, I do it on the side,"},{"Start":"00:58.440 ","End":"00:59.845","Text":"square root of 9,"},{"Start":"00:59.845 ","End":"01:02.570","Text":"I don\u0027t do it on the calculator because it\u0027s too simple,"},{"Start":"01:02.570 ","End":"01:04.729","Text":"but it could be done on the calculator."},{"Start":"01:04.729 ","End":"01:07.535","Text":"I get the 3, but that\u0027s not the complete story."},{"Start":"01:07.535 ","End":"01:12.140","Text":"Whenever we have a square and we want to solve it,"},{"Start":"01:12.140 ","End":"01:19.895","Text":"we actually have two solutions because minus 3 is also a solution."},{"Start":"01:19.895 ","End":"01:26.105","Text":"In fact, what we need is to take a to be plus or minus the square root of 9,"},{"Start":"01:26.105 ","End":"01:29.225","Text":"which means plus or minus 3,"},{"Start":"01:29.225 ","End":"01:32.670","Text":"and then we get both solutions."},{"Start":"01:34.490 ","End":"01:38.340","Text":"This was just an introduction to what I\u0027m going to do here."},{"Start":"01:38.340 ","End":"01:42.695","Text":"By analogy, we also have something squared equals something."},{"Start":"01:42.695 ","End":"01:49.040","Text":"Perhaps I should mention point on notation that when you write sine squared x,"},{"Start":"01:49.040 ","End":"01:53.495","Text":"this is the same as sine of x all squared."},{"Start":"01:53.495 ","End":"01:57.635","Text":"To save brackets, mathematicians agree that if we put the 2 here,"},{"Start":"01:57.635 ","End":"01:59.570","Text":"it means the whole thing squared."},{"Start":"01:59.570 ","End":"02:02.240","Text":"We have sine x squared is 1/4,"},{"Start":"02:02.240 ","End":"02:04.414","Text":"so following the same logic,"},{"Start":"02:04.414 ","End":"02:09.250","Text":"we get that sine x is not just the square root of 1/4,"},{"Start":"02:09.250 ","End":"02:14.210","Text":"but 2 possibilities plus or minus the square root of 1/4."},{"Start":"02:14.210 ","End":"02:17.240","Text":"Now, we have 2 possibilities."},{"Start":"02:17.240 ","End":"02:18.679","Text":"If we take the plus,"},{"Start":"02:18.679 ","End":"02:21.560","Text":"we get sine x equals,"},{"Start":"02:21.560 ","End":"02:25.190","Text":"and remember that the square root of a fraction,"},{"Start":"02:25.190 ","End":"02:29.180","Text":"I can take top and bottom separately assuming everything\u0027s positive."},{"Start":"02:29.180 ","End":"02:31.670","Text":"So square root of 1 is 1,"},{"Start":"02:31.670 ","End":"02:33.545","Text":"square root of 4 is 2,"},{"Start":"02:33.545 ","End":"02:34.955","Text":"that\u0027s for the plus,"},{"Start":"02:34.955 ","End":"02:40.745","Text":"but I also have the possibility that sine x is minus 1/2."},{"Start":"02:40.745 ","End":"02:47.090","Text":"So I have to solve each of these separately and then combine the solutions."},{"Start":"02:47.090 ","End":"02:48.410","Text":"Now for this one,"},{"Start":"02:48.410 ","End":"02:51.665","Text":"the usual recipe is to say yes,"},{"Start":"02:51.665 ","End":"02:54.350","Text":"sine x is sine of something."},{"Start":"02:54.350 ","End":"02:57.385","Text":"In this case sine of 30 degrees,"},{"Start":"02:57.385 ","End":"02:59.240","Text":"this should be memorized already,"},{"Start":"02:59.240 ","End":"03:01.325","Text":"shouldn\u0027t even need the calculator,"},{"Start":"03:01.325 ","End":"03:04.310","Text":"and then we have sine something equals sign something."},{"Start":"03:04.310 ","End":"03:07.880","Text":"The standard thing is to get two sets of solutions."},{"Start":"03:07.880 ","End":"03:13.400","Text":"We either have x being the same as the argument here"},{"Start":"03:13.400 ","End":"03:20.480","Text":"30 plus multiples of 360 or the supplement of the angle,"},{"Start":"03:20.480 ","End":"03:21.680","Text":"that\u0027s the old word."},{"Start":"03:21.680 ","End":"03:23.705","Text":"I mean 180 minus,"},{"Start":"03:23.705 ","End":"03:27.710","Text":"if I do 180 minus this it\u0027s 150,"},{"Start":"03:27.710 ","End":"03:32.860","Text":"and also plus 360k."},{"Start":"03:32.860 ","End":"03:37.070","Text":"That\u0027s 2 families of solutions,"},{"Start":"03:37.070 ","End":"03:41.360","Text":"but that\u0027s not all because we also have solutions from here."},{"Start":"03:41.360 ","End":"03:44.465","Text":"I\u0027m going to draw a separator here."},{"Start":"03:44.465 ","End":"03:47.555","Text":"We\u0027re going to work on this one separately."},{"Start":"03:47.555 ","End":"03:50.210","Text":"If you\u0027re using a calculator,"},{"Start":"03:50.210 ","End":"03:57.770","Text":"then you would find that the angle whose sine is minus 1/2 is minus 30 degrees,"},{"Start":"03:57.770 ","End":"04:00.830","Text":"but if you didn\u0027t have the calculator,"},{"Start":"04:00.830 ","End":"04:05.280","Text":"you might use the formula that"},{"Start":"04:05.430 ","End":"04:13.780","Text":"minus sine of whatever Alpha is equal to sine of minus Alpha."},{"Start":"04:13.780 ","End":"04:17.360","Text":"In other words, the minus can go in and out of the sine, this is by the way,"},{"Start":"04:17.360 ","End":"04:19.205","Text":"not true for the cosine,"},{"Start":"04:19.205 ","End":"04:21.834","Text":"and then we could say, okay,"},{"Start":"04:21.834 ","End":"04:29.280","Text":"minus sine x in this case is 1/2 by bringing"},{"Start":"04:29.280 ","End":"04:33.040","Text":"the minus over and then I could put the minus inside"},{"Start":"04:33.040 ","End":"04:37.420","Text":"and say that sine of minus x is 1/2,"},{"Start":"04:37.420 ","End":"04:43.740","Text":"and this I know is sine 30."},{"Start":"04:51.620 ","End":"05:00.200","Text":"Take 2 on that last bit and if you didn\u0027t have a calculator,"},{"Start":"05:00.200 ","End":"05:05.775","Text":"then you would write that minus 1/2 is"},{"Start":"05:05.775 ","End":"05:13.410","Text":"minus sine 30 because we know that 1/2 is sine 30,"},{"Start":"05:13.410 ","End":"05:14.730","Text":"that we do know,"},{"Start":"05:14.730 ","End":"05:19.540","Text":"and then we would put the minus inside here,"},{"Start":"05:19.540 ","End":"05:21.820","Text":"which is what we do with sine from here,"},{"Start":"05:21.820 ","End":"05:25.630","Text":"and then we would write it as sine of minus 30."},{"Start":"05:25.630 ","End":"05:33.510","Text":"Either by calculator or from knowledge that sine 30 is 1/2,"},{"Start":"05:33.510 ","End":"05:37.225","Text":"and this trigonometric identity."},{"Start":"05:37.225 ","End":"05:39.550","Text":"Now continuing as before,"},{"Start":"05:39.550 ","End":"05:42.250","Text":"we get 2 families of solutions."},{"Start":"05:42.250 ","End":"05:44.409","Text":"X is 1 family,"},{"Start":"05:44.409 ","End":"05:45.910","Text":"x is the other family."},{"Start":"05:45.910 ","End":"05:51.070","Text":"The first family of solutions is just to take the argument of this sine which is"},{"Start":"05:51.070 ","End":"05:58.010","Text":"minus 30 and add multiples of a whole circle, i.e, 360k."},{"Start":"05:58.040 ","End":"06:02.590","Text":"The other thing to do is to subtract this from 180."},{"Start":"06:02.590 ","End":"06:05.820","Text":"Now 180 minus this is"},{"Start":"06:05.820 ","End":"06:14.890","Text":"210 because 180 minus minus 30 is 180 plus 30 also plus 360k."},{"Start":"06:16.340 ","End":"06:24.300","Text":"Actually, we have really four families of solutions,"},{"Start":"06:24.300 ","End":"06:30.964","Text":"and x could be one of these or one of these,"},{"Start":"06:30.964 ","End":"06:32.554","Text":"or one of these,"},{"Start":"06:32.554 ","End":"06:38.925","Text":"or 1 of these four basic angles minus 30,"},{"Start":"06:38.925 ","End":"06:40.770","Text":"30, 150,"},{"Start":"06:40.770 ","End":"06:49.790","Text":"and 210 plus any one of those plus a multiple of 360 added to any one of those."},{"Start":"06:49.790 ","End":"06:56.130","Text":"Very well. On to part B."},{"Start":"06:56.920 ","End":"07:01.700","Text":"In part B, we\u0027re going to use a very similar trick."},{"Start":"07:01.700 ","End":"07:04.490","Text":"Again, we have something squared,"},{"Start":"07:04.490 ","End":"07:07.280","Text":"and when we have something squared,"},{"Start":"07:07.280 ","End":"07:11.330","Text":"then we just can solve it by saying that the cosine of"},{"Start":"07:11.330 ","End":"07:16.670","Text":"2x is equal to not just the square root of this,"},{"Start":"07:16.670 ","End":"07:21.875","Text":"but either plus or minus the square root of 3/4."},{"Start":"07:21.875 ","End":"07:25.010","Text":"Once again, I remind you of notation that when you write"},{"Start":"07:25.010 ","End":"07:28.940","Text":"cosine squared of something, say Alpha,"},{"Start":"07:28.940 ","End":"07:31.915","Text":"it\u0027s the same as cosine Alpha^2,"},{"Start":"07:31.915 ","End":"07:36.950","Text":"this is just a notation to save writing brackets."},{"Start":"07:36.950 ","End":"07:40.540","Text":"We split this up into 2 equations."},{"Start":"07:40.540 ","End":"07:43.775","Text":"We either have that cosine 2x,"},{"Start":"07:43.775 ","End":"07:45.690","Text":"perhaps I should put brackets here,"},{"Start":"07:45.690 ","End":"07:53.035","Text":"usually it\u0027s clear that the cosine ends here and the 2 begins here."},{"Start":"07:53.035 ","End":"07:55.855","Text":"Never hurts to hide extra brackets."},{"Start":"07:55.855 ","End":"08:00.100","Text":"This is either equal to the square root of 3/4,"},{"Start":"08:00.100 ","End":"08:02.330","Text":"but the square root of 3/4,"},{"Start":"08:02.330 ","End":"08:06.640","Text":"like before, we could take top and bottom square roots separately."},{"Start":"08:06.640 ","End":"08:11.545","Text":"It\u0027s square root of 3 over square root of 4, which is 2."},{"Start":"08:11.545 ","End":"08:16.440","Text":"The other possibility is that cosine, sorry,"},{"Start":"08:16.440 ","End":"08:24.625","Text":"that\u0027s cosine 2x is equal to minus the same thing,"},{"Start":"08:24.625 ","End":"08:28.030","Text":"square root of 3 over 2."},{"Start":"08:28.030 ","End":"08:30.760","Text":"Now, in this case,"},{"Start":"08:30.760 ","End":"08:36.925","Text":"square root of 3 over 2 is one of those familiar numbers and we\u0027re supposed to know,"},{"Start":"08:36.925 ","End":"08:39.100","Text":"even without a calculator,"},{"Start":"08:39.100 ","End":"08:43.140","Text":"that this is the cosine of 30 degrees,"},{"Start":"08:43.140 ","End":"08:46.095","Text":"but you could also do it on the calculator."},{"Start":"08:46.095 ","End":"08:48.360","Text":"We get cosine, actually,"},{"Start":"08:48.360 ","End":"08:50.565","Text":"I don\u0027t think we really do need the brackets,"},{"Start":"08:50.565 ","End":"08:53.340","Text":"I\u0027ll drop them, it equals this."},{"Start":"08:53.340 ","End":"08:57.175","Text":"Then we have our usual technique for cosine equals cosine,"},{"Start":"08:57.175 ","End":"09:03.270","Text":"which is to say that either 2x is 30 plus,"},{"Start":"09:03.270 ","End":"09:05.175","Text":"whole circles I call it,"},{"Start":"09:05.175 ","End":"09:10.970","Text":"360k or we have minus of"},{"Start":"09:10.970 ","End":"09:16.950","Text":"the angle minus 30 also plus 360k."},{"Start":"09:16.950 ","End":"09:18.770","Text":"Of course we want just x,"},{"Start":"09:18.770 ","End":"09:20.780","Text":"so we divide by 2."},{"Start":"09:20.780 ","End":"09:29.510","Text":"We get 1 family is 15 plus 180k,"},{"Start":"09:29.510 ","End":"09:36.000","Text":"and the other possibility is minus 15 plus 180k."},{"Start":"09:36.010 ","End":"09:42.410","Text":"I\u0027ll just mention that some people like to write this in a more shorthand way."},{"Start":"09:42.410 ","End":"09:50.795","Text":"You could write this as plus or minus 15 because the 180k is common,"},{"Start":"09:50.795 ","End":"09:54.619","Text":"and then here we have the 2 sets,"},{"Start":"09:54.619 ","End":"09:56.270","Text":"2 families of solutions."},{"Start":"09:56.270 ","End":"09:57.650","Text":"We have 2 choices to make,"},{"Start":"09:57.650 ","End":"09:58.910","Text":"you choose plus or minus,"},{"Start":"09:58.910 ","End":"10:04.210","Text":"and then you choose a value of k and any of those choices gives you a solution."},{"Start":"10:04.210 ","End":"10:10.785","Text":"Sometimes maybe I\u0027ll write it the shortcut way or we could leave it like that."},{"Start":"10:10.785 ","End":"10:15.825","Text":"I\u0027ll just highlight the old way of writing it, this and this,"},{"Start":"10:15.825 ","End":"10:18.190","Text":"but remember we\u0027re not done because we have"},{"Start":"10:18.190 ","End":"10:23.335","Text":"the other possibility that cosine 2x is minus square root of 3/2."},{"Start":"10:23.335 ","End":"10:28.960","Text":"In this case, square root of 3/2 is the cosine of 30,"},{"Start":"10:28.960 ","End":"10:31.465","Text":"but that minus has to stay."},{"Start":"10:31.465 ","End":"10:33.625","Text":"I don\u0027t want a minus,"},{"Start":"10:33.625 ","End":"10:37.050","Text":"and there is a way of putting the minus and not like with the sine,"},{"Start":"10:37.050 ","End":"10:38.925","Text":"there is a different formula."},{"Start":"10:38.925 ","End":"10:44.055","Text":"I\u0027ll just to remind you that the cosine of,"},{"Start":"10:44.055 ","End":"10:53.425","Text":"or rather the minus the cosine of Alpha is the same as cosine of 180 minus Alpha,"},{"Start":"10:53.425 ","End":"10:56.870","Text":"and we can read it from left to right or from right to left, doesn\u0027t matter."},{"Start":"10:56.870 ","End":"11:05.580","Text":"In this case, we get that this equals the cosine of 150 degrees."},{"Start":"11:06.410 ","End":"11:09.990","Text":"Same thing, ditto, ditto."},{"Start":"11:09.990 ","End":"11:13.590","Text":"Now we have cosine 2x equals cosine 150,"},{"Start":"11:13.590 ","End":"11:23.735","Text":"and we proceed as before that we say that either 2x is equal to 150,"},{"Start":"11:23.735 ","End":"11:25.775","Text":"that\u0027s this one here,"},{"Start":"11:25.775 ","End":"11:33.660","Text":"plus 360k or because it\u0027s a cosine we take minus of"},{"Start":"11:33.660 ","End":"11:42.490","Text":"this plus 360k and that gives us 2 more families."},{"Start":"11:43.090 ","End":"11:46.370","Text":"Sorry, we haven\u0027t got the x yet."},{"Start":"11:46.370 ","End":"11:48.470","Text":"Take 2 on the last bit."},{"Start":"11:48.470 ","End":"11:51.010","Text":"We still have to divide by 2,"},{"Start":"11:51.010 ","End":"11:53.625","Text":"and then we get that either x is"},{"Start":"11:53.625 ","End":"12:04.420","Text":"75 plus 180k or x is minus 75 plus 180k."},{"Start":"12:04.420 ","End":"12:06.410","Text":"In addition to these two,"},{"Start":"12:06.410 ","End":"12:10.505","Text":"I have a third and a fourth family of solutions,"},{"Start":"12:10.505 ","End":"12:13.250","Text":"and as before, if you like,"},{"Start":"12:13.250 ","End":"12:21.435","Text":"we could write this in more shorthand as plus or minus 75 plus 180k,"},{"Start":"12:21.435 ","End":"12:24.260","Text":"which means that we get a solution whether we choose plus"},{"Start":"12:24.260 ","End":"12:26.750","Text":"or minus and whatever integer k we choose,"},{"Start":"12:26.750 ","End":"12:28.770","Text":"we\u0027ll get a solution."},{"Start":"12:29.140 ","End":"12:33.390","Text":"That\u0027s part B done, so that\u0027s it."}],"ID":5405},{"Watched":false,"Name":"Exercise 13","Duration":"13m 37s","ChapterTopicVideoID":5407,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5407.jpeg","UploadDate":"2016-03-10T21:20:28.7230000","DurationForVideoObject":"PT13M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, there are two trigonometric equations to solve,"},{"Start":"00:04.260 ","End":"00:09.000","Text":"and they\u0027re starting to get a little bit more complicated."},{"Start":"00:09.000 ","End":"00:13.935","Text":"We\u0027re going to have to use some techniques to simplify these two forms we already know,"},{"Start":"00:13.935 ","End":"00:15.240","Text":"going to be two main things."},{"Start":"00:15.240 ","End":"00:18.270","Text":"One is just general algebra to simplify things,"},{"Start":"00:18.270 ","End":"00:21.315","Text":"and the other is trigonometric identities."},{"Start":"00:21.315 ","End":"00:25.350","Text":"The couple of trigonometric identities we\u0027ll need already for part"},{"Start":"00:25.350 ","End":"00:31.080","Text":"a is the double angle formulas."},{"Start":"00:31.080 ","End":"00:33.000","Text":"I\u0027m not exactly sure that\u0027s the name."},{"Start":"00:33.000 ","End":"00:40.400","Text":"But in any event, it says that the sine of"},{"Start":"00:40.400 ","End":"00:49.280","Text":"twice an angle Alpha is given by twice sine of Alpha, cosine Alpha."},{"Start":"00:49.280 ","End":"00:52.010","Text":"In other words, the sine of twice an angle is given in"},{"Start":"00:52.010 ","End":"00:55.265","Text":"terms of the sine and the cosine of the angle itself."},{"Start":"00:55.265 ","End":"00:59.120","Text":"There\u0027s corresponding 1 for cosine."},{"Start":"00:59.120 ","End":"01:09.440","Text":"Cosine of twice an angle is cosine^2 of Alpha minus sine^2 of Alpha."},{"Start":"01:09.440 ","End":"01:11.990","Text":"There are actually two other forms of this,"},{"Start":"01:11.990 ","End":"01:14.750","Text":"but this is the one I\u0027m going to use here."},{"Start":"01:14.750 ","End":"01:20.660","Text":"In general, I also want to remind you this will be useful,"},{"Start":"01:20.660 ","End":"01:28.935","Text":"part a and b is that there\u0027s a general identity that sine^2 of anything,"},{"Start":"01:28.935 ","End":"01:35.120","Text":"it could be Alpha plus cosine^2 of Alpha is equal to 1."},{"Start":"01:35.120 ","End":"01:38.780","Text":"These are actually all trigonometric identities and sometimes it\u0027s"},{"Start":"01:38.780 ","End":"01:43.770","Text":"written with an equals with 3 lines."},{"Start":"01:44.410 ","End":"01:49.115","Text":"I\u0027ll remove those, didn\u0027t want to confuse you, but you should know."},{"Start":"01:49.115 ","End":"01:55.860","Text":"Let\u0027s start with Part a and we\u0027ll scroll down a bit."},{"Start":"01:56.560 ","End":"02:03.410","Text":"I can already see in the left and in the right things that look like stuff from here."},{"Start":"02:03.410 ","End":"02:05.645","Text":"Let\u0027s see what we have here."},{"Start":"02:05.645 ","End":"02:08.900","Text":"The left-hand side is exactly this,"},{"Start":"02:08.900 ","End":"02:11.015","Text":"but with x instead of Alpha,"},{"Start":"02:11.015 ","End":"02:16.055","Text":"so a straightforward application of this rule from this identity"},{"Start":"02:16.055 ","End":"02:22.385","Text":"from right to left would give me that this is the sine of 2x."},{"Start":"02:22.385 ","End":"02:25.235","Text":"What about the right-hand side?"},{"Start":"02:25.235 ","End":"02:27.920","Text":"Looks very much like this,"},{"Start":"02:27.920 ","End":"02:31.460","Text":"except here is a minus and here is a plus."},{"Start":"02:31.990 ","End":"02:41.545","Text":"In one step I could say that this is minus of cosine 2 Alpha."},{"Start":"02:41.545 ","End":"02:44.075","Text":"Because if I took the minus outside,"},{"Start":"02:44.075 ","End":"02:45.440","Text":"I could do a middle step."},{"Start":"02:45.440 ","End":"02:55.180","Text":"It\u0027s the minus of cosine^2 Alpha minus sine^2 Alpha x rather in this case."},{"Start":"02:55.400 ","End":"02:59.325","Text":"I wrote Alpha instead of fixed."},{"Start":"02:59.325 ","End":"03:03.560","Text":"Now this is already more familiar,"},{"Start":"03:03.560 ","End":"03:12.245","Text":"but we still don\u0027t have the ideal situation of sine equals sine or cosine equals cosine,"},{"Start":"03:12.245 ","End":"03:16.470","Text":"and what I\u0027ll do is,"},{"Start":"03:16.470 ","End":"03:18.200","Text":"it doesn\u0027t really matter,"},{"Start":"03:18.200 ","End":"03:21.349","Text":"but let\u0027s say we go for sine equals sign."},{"Start":"03:21.349 ","End":"03:24.170","Text":"In that case, what I do is,"},{"Start":"03:24.170 ","End":"03:32.710","Text":"I write this in terms of the 180 minus the angle."},{"Start":"03:32.710 ","End":"03:34.745","Text":"I\u0027ll write it in a moment."},{"Start":"03:34.745 ","End":"03:36.155","Text":"If you have a minus,"},{"Start":"03:36.155 ","End":"03:37.745","Text":"when you have a cosine,"},{"Start":"03:37.745 ","End":"03:40.100","Text":"you can, instead of the argument,"},{"Start":"03:40.100 ","End":"03:43.369","Text":"put 180 minus the argument."},{"Start":"03:43.369 ","End":"03:45.485","Text":"In fact, you\u0027ve done this enough times."},{"Start":"03:45.485 ","End":"03:49.670","Text":"I\u0027ll just assume that you remember the formula minus cosine Alphas,"},{"Start":"03:49.670 ","End":"03:53.105","Text":"cosine 180 minus Alpha, I\u0027ve said it."},{"Start":"03:53.105 ","End":"03:58.640","Text":"But that gets rid of the minus."},{"Start":"03:58.640 ","End":"04:05.220","Text":"But that\u0027s still, we still have a mixture of sine and cosine."},{"Start":"04:05.390 ","End":"04:10.190","Text":"This is still not good enough."},{"Start":"04:10.190 ","End":"04:12.740","Text":"I want to convert the cosine to assign,"},{"Start":"04:12.740 ","End":"04:18.960","Text":"so I use yet another formula that the cosine,"},{"Start":"04:18.960 ","End":"04:21.275","Text":"perhaps I\u0027ll write it this time."},{"Start":"04:21.275 ","End":"04:24.745","Text":"I want if I\u0027m already writing the previous 1, 2."},{"Start":"04:24.745 ","End":"04:30.600","Text":"Minus cosine of Alpha is the cosine of 180 minus Alpha,"},{"Start":"04:30.600 ","End":"04:34.655","Text":"and then what I\u0027m going to use now is to convert a cosine to assign,"},{"Start":"04:34.655 ","End":"04:43.745","Text":"is to say that the cosine of an angle is the sine of 90 minus the angle."},{"Start":"04:43.745 ","End":"04:50.395","Text":"In this case, we will get sine of 90 minus this,"},{"Start":"04:50.395 ","End":"04:52.390","Text":"and I\u0027ll do that at the side."},{"Start":"04:52.390 ","End":"04:58.329","Text":"Let\u0027s see, 90 minus 180 minus"},{"Start":"04:58.329 ","End":"05:07.985","Text":"2x is equal to 90 minus a 180 is minus 90,"},{"Start":"05:07.985 ","End":"05:10.430","Text":"minus, minus is plus."},{"Start":"05:10.430 ","End":"05:15.390","Text":"Here we have this minus 90 plus 2x,"},{"Start":"05:15.390 ","End":"05:19.910","Text":"and finally, very familiar territory, sine equals sign."},{"Start":"05:19.910 ","End":"05:23.790","Text":"We use the usual technique,"},{"Start":"05:23.790 ","End":"05:33.740","Text":"and we say there are two possibilities that either 2x is equal to what\u0027s here,"},{"Start":"05:33.740 ","End":"05:38.980","Text":"minus 90 plus 2x plus"},{"Start":"05:38.980 ","End":"05:48.320","Text":"360 times k or x equals."},{"Start":"05:48.320 ","End":"05:55.805","Text":"Then we need in the case of sine 180 minus this, 180 minus,"},{"Start":"05:55.805 ","End":"06:01.845","Text":"minus 90 is 270,"},{"Start":"06:01.845 ","End":"06:09.970","Text":"and then minus 2x still plus the 360k."},{"Start":"06:11.090 ","End":"06:16.040","Text":"Something funny, get familiar happens in the first one."},{"Start":"06:16.040 ","End":"06:19.435","Text":"The 2x cancels,"},{"Start":"06:19.435 ","End":"06:27.590","Text":"and it leaves us with 0 equals minus 90 plus 360k,"},{"Start":"06:27.590 ","End":"06:32.620","Text":"which is impossible, not possible."},{"Start":"06:32.620 ","End":"06:38.465","Text":"What I can say is that this top row won\u0027t give me any solutions."},{"Start":"06:38.465 ","End":"06:42.830","Text":"You know what, I\u0027ll just put a red line through it because of this."},{"Start":"06:42.830 ","End":"06:46.670","Text":"But we still have solutions from the second possibility,"},{"Start":"06:46.670 ","End":"06:48.050","Text":"and that\u0027s the one I\u0027ll continue."},{"Start":"06:48.050 ","End":"06:53.270","Text":"Let\u0027s see, moving over here the 2x goes to the other side."},{"Start":"06:53.270 ","End":"07:03.120","Text":"I\u0027ve got 4x=270 plus 360 k and then dividing by 4,"},{"Start":"07:03.120 ","End":"07:05.300","Text":"and I get, let\u0027s see,"},{"Start":"07:05.300 ","End":"07:12.365","Text":"this divided by 4 would be 67.5."},{"Start":"07:12.365 ","End":"07:14.400","Text":"I believe."},{"Start":"07:15.080 ","End":"07:18.960","Text":"C divided by 2 is 135 divided by 2."},{"Start":"07:18.960 ","End":"07:20.970","Text":"Again here is 67.5."},{"Start":"07:20.970 ","End":"07:24.945","Text":"This divided by 4 is 90, so 90k,"},{"Start":"07:24.945 ","End":"07:30.030","Text":"so this is the general solution and I will highlight it,"},{"Start":"07:30.030 ","End":"07:32.190","Text":"so done with that."},{"Start":"07:32.190 ","End":"07:36.925","Text":"That was difficult but not too bad."},{"Start":"07:36.925 ","End":"07:39.350","Text":"Let\u0027s go on to part b."},{"Start":"07:42.510 ","End":"07:44.740","Text":"We don\u0027t have our formulas."},{"Start":"07:44.740 ","End":"07:46.480","Text":"Let me go and copy them."},{"Start":"07:46.480 ","End":"07:50.605","Text":"Here they are just copied the whole thing from part a."},{"Start":"07:50.605 ","End":"07:53.140","Text":"What do we do here?"},{"Start":"07:53.140 ","End":"07:58.210","Text":"Well, there\u0027s one more formula that I would like to introduce,"},{"Start":"07:58.210 ","End":"08:00.670","Text":"not from trigonometry, just from algebra."},{"Start":"08:00.670 ","End":"08:05.485","Text":"That in general says that a^2 minus b^2"},{"Start":"08:05.485 ","End":"08:12.040","Text":"is a plus b times a minus b or the other way round,"},{"Start":"08:12.040 ","End":"08:14.615","Text":"and it\u0027s called a difference of squares formula."},{"Start":"08:14.615 ","End":"08:16.415","Text":"Now how is this going to help me?"},{"Start":"08:16.415 ","End":"08:18.635","Text":"I want to use it on the left-hand side."},{"Start":"08:18.635 ","End":"08:22.700","Text":"It doesn\u0027t look like something squared minus something squared is the power of 4."},{"Start":"08:22.700 ","End":"08:25.865","Text":"But if you think the power of 4 is just the square root of a square,"},{"Start":"08:25.865 ","End":"08:31.490","Text":"I could write this as cosine squared x"},{"Start":"08:31.490 ","End":"08:37.895","Text":"squared minus sine squared x also squared."},{"Start":"08:37.895 ","End":"08:39.530","Text":"I broke up this power of 4,"},{"Start":"08:39.530 ","End":"08:42.095","Text":"the power of 2 to the power of 2."},{"Start":"08:42.095 ","End":"08:49.735","Text":"On the right, just as is 2 sine 2x, cosine 2x."},{"Start":"08:49.735 ","End":"08:52.115","Text":"The reason I did this is now I can use this."},{"Start":"08:52.115 ","End":"08:54.080","Text":"A can be cosine^2(x),"},{"Start":"08:54.080 ","End":"08:57.335","Text":"b can be sine squared x."},{"Start":"08:57.335 ","End":"09:01.655","Text":"On the left, if I use this difference of squares formula,"},{"Start":"09:01.655 ","End":"09:08.690","Text":"I\u0027ll get cosine^2(x) plus sine ^2(x),"},{"Start":"09:08.690 ","End":"09:10.345","Text":"that\u0027s my a plus b,"},{"Start":"09:10.345 ","End":"09:16.685","Text":"and the a minus b is cosine^2(x) minus sine^2(x)."},{"Start":"09:16.685 ","End":"09:18.919","Text":"All this is the left-hand side."},{"Start":"09:18.919 ","End":"09:20.930","Text":"Now on the right-hand side,"},{"Start":"09:20.930 ","End":"09:23.245","Text":"this looks like this,"},{"Start":"09:23.245 ","End":"09:27.035","Text":"and if I let Alpha be 2x,"},{"Start":"09:27.035 ","End":"09:30.480","Text":"this will be like my Alpha here and here,"},{"Start":"09:30.480 ","End":"09:39.020","Text":"then I can use this formula and say that this is equal to sine of 2-Alpha."},{"Start":"09:39.170 ","End":"09:41.625","Text":"What is 2-Alpha?"},{"Start":"09:41.625 ","End":"09:44.320","Text":"2-Alpha is 4x,"},{"Start":"09:45.290 ","End":"09:48.420","Text":"because Alpha is 2x,"},{"Start":"09:48.420 ","End":"09:54.650","Text":"so that\u0027s developing and see where we\u0027re going with this."},{"Start":"09:54.650 ","End":"09:58.955","Text":"The big thing, that\u0027s going to save us is this formula here,"},{"Start":"09:58.955 ","End":"10:02.945","Text":"that for any angle, the sine^2 plus cosine^2 is 1."},{"Start":"10:02.945 ","End":"10:07.400","Text":"This whole thing, this whole first factor disappears."},{"Start":"10:07.400 ","End":"10:10.050","Text":"Now it\u0027s a lot simpler."},{"Start":"10:10.630 ","End":"10:16.190","Text":"Look, here I can use where is it?"},{"Start":"10:16.190 ","End":"10:19.430","Text":"This here from right to left."},{"Start":"10:19.430 ","End":"10:24.185","Text":"Cosine^2 minus sine^2 is cosine of twice the angle,"},{"Start":"10:24.185 ","End":"10:26.510","Text":"so let\u0027s see where we stand now,"},{"Start":"10:26.510 ","End":"10:33.645","Text":"so this becomes cosine of 2x from this formula."},{"Start":"10:33.645 ","End":"10:35.345","Text":"On the right-hand side,"},{"Start":"10:35.345 ","End":"10:38.610","Text":"we have sine of 4x."},{"Start":"10:39.010 ","End":"10:43.325","Text":"Already a lot better than what we had originally."},{"Start":"10:43.325 ","End":"10:46.805","Text":"Now we\u0027ve learned how to do these mixed ones with cosine and sine,"},{"Start":"10:46.805 ","End":"10:49.070","Text":"you have to decide whether we\u0027re going with the cosine of"},{"Start":"10:49.070 ","End":"10:51.800","Text":"the science about the equal difficulty."},{"Start":"10:51.800 ","End":"10:55.640","Text":"I\u0027m just like to keep the left-hand side as it is,"},{"Start":"10:55.640 ","End":"10:57.905","Text":"so I\u0027m going to convert this to a cosine,"},{"Start":"10:57.905 ","End":"11:00.740","Text":"so this 1 stays as a cosine,"},{"Start":"11:00.740 ","End":"11:06.835","Text":"and here I\u0027m going to use the formula with 90 minus."},{"Start":"11:06.835 ","End":"11:11.410","Text":"It works the other way round too that,"},{"Start":"11:12.350 ","End":"11:16.460","Text":"if I have here the sine of Alpha,"},{"Start":"11:16.460 ","End":"11:23.435","Text":"then it\u0027s the cosine of 90 minus Alpha works either way."},{"Start":"11:23.435 ","End":"11:32.955","Text":"What we get is the cosine of 90 minus 4x."},{"Start":"11:32.955 ","End":"11:35.795","Text":"Now we have cosine equals cosine."},{"Start":"11:35.795 ","End":"11:38.410","Text":"Now we really know what to do."},{"Start":"11:38.410 ","End":"11:44.940","Text":"We either have the 2x equals 90 minus 4x,"},{"Start":"11:46.300 ","End":"11:51.790","Text":"and we have to add the 360k as always,"},{"Start":"11:51.790 ","End":"11:54.650","Text":"and the other possibility in the case of"},{"Start":"11:54.650 ","End":"11:59.210","Text":"cosine is to take the negative of what\u0027s written here,"},{"Start":"11:59.210 ","End":"12:03.580","Text":"so it\u0027ll be minus 90 plus 4x."},{"Start":"12:03.580 ","End":"12:06.610","Text":"Still with the plus 360k."},{"Start":"12:07.360 ","End":"12:10.430","Text":"We need to solve each one of these,"},{"Start":"12:10.430 ","End":"12:13.145","Text":"to start with the first one,"},{"Start":"12:13.145 ","End":"12:15.880","Text":"and I\u0027ll do it over here,"},{"Start":"12:15.880 ","End":"12:18.045","Text":"so what do we get?"},{"Start":"12:18.045 ","End":"12:19.350","Text":"We move the 4x over,"},{"Start":"12:19.350 ","End":"12:25.035","Text":"we get 6x equals 90 plus 360k,"},{"Start":"12:25.035 ","End":"12:29.920","Text":"and then of course we want to divide by 6,"},{"Start":"12:29.920 ","End":"12:32.825","Text":"so we get x is equal,"},{"Start":"12:32.825 ","End":"12:35.845","Text":"90 over 6 is 15,"},{"Start":"12:35.845 ","End":"12:39.685","Text":"360 over 6 is 60,"},{"Start":"12:39.685 ","End":"12:45.670","Text":"so we get that x equals 15 plus 60k."},{"Start":"12:45.670 ","End":"12:47.350","Text":"That\u0027s from the top row."},{"Start":"12:47.350 ","End":"12:50.230","Text":"Let\u0027s see what we get from the bottom row."},{"Start":"12:50.230 ","End":"12:57.220","Text":"Well, we get 2x minus 4x is minus 2x equals"},{"Start":"12:57.220 ","End":"13:04.635","Text":"minus 90 plus 360k divide by minus 2,"},{"Start":"13:04.635 ","End":"13:08.490","Text":"we\u0027ve got x equals plus 45,"},{"Start":"13:08.490 ","End":"13:11.860","Text":"and then minus 180k."},{"Start":"13:14.050 ","End":"13:17.615","Text":"This is the other set of solutions."},{"Start":"13:17.615 ","End":"13:20.270","Text":"There is an alternative way of writing it."},{"Start":"13:20.270 ","End":"13:25.295","Text":"I\u0027ve mentioned it before that you can actually instead of a minus,"},{"Start":"13:25.295 ","End":"13:31.370","Text":"you can actually write the plus because it\u0027s a general k. But if this doesn\u0027t suit,"},{"Start":"13:31.370 ","End":"13:33.260","Text":"you, just forget it and leave it like this."},{"Start":"13:33.260 ","End":"13:34.835","Text":"Whether I would leave it like this."},{"Start":"13:34.835 ","End":"13:37.440","Text":"Anyway, we are done."}],"ID":5406},{"Watched":false,"Name":"Exercise 14","Duration":"6m 58s","ChapterTopicVideoID":5408,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5408.jpeg","UploadDate":"2016-03-10T21:21:25.2170000","DurationForVideoObject":"PT6M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Here we have a pair of trigonometric equations,"},{"Start":"00:03.210 ","End":"00:05.295","Text":"one with sine, one with cosine."},{"Start":"00:05.295 ","End":"00:09.810","Text":"The difference from previous exercises is that this time we also have"},{"Start":"00:09.810 ","End":"00:14.865","Text":"a range for x that we have to limit ourselves to."},{"Start":"00:14.865 ","End":"00:18.280","Text":"Let\u0027s take the first one and we\u0027ll see."},{"Start":"00:21.680 ","End":"00:24.900","Text":"What we want to do here is first of all,"},{"Start":"00:24.900 ","End":"00:27.180","Text":"leave the range aside,"},{"Start":"00:27.180 ","End":"00:29.355","Text":"just concentrate on the equation."},{"Start":"00:29.355 ","End":"00:32.910","Text":"What we have is sine 3x,"},{"Start":"00:32.910 ","End":"00:34.080","Text":"on the right-hand side,"},{"Start":"00:34.080 ","End":"00:40.105","Text":"we also want to write a sine something and we know that 1/2 is sine 30,"},{"Start":"00:40.105 ","End":"00:45.500","Text":"and so we start solving as usual and say that we have two possibilities."},{"Start":"00:45.500 ","End":"00:48.365","Text":"Either 3x is 30,"},{"Start":"00:48.365 ","End":"00:51.040","Text":"plus multiples of 360,"},{"Start":"00:51.040 ","End":"00:56.569","Text":"or we take 180 minus this angle,"},{"Start":"00:56.569 ","End":"00:59.380","Text":"which gives us 150,"},{"Start":"00:59.380 ","End":"01:03.420","Text":"also plus 360k, and since it\u0027s 3x,"},{"Start":"01:03.420 ","End":"01:05.115","Text":"we need to divide by 3,"},{"Start":"01:05.115 ","End":"01:10.035","Text":"so we get that x is either 10,"},{"Start":"01:10.035 ","End":"01:13.000","Text":"plus and then 120k,"},{"Start":"01:13.340 ","End":"01:23.150","Text":"or we could also have another family of solutions, 50 plus 120k."},{"Start":"01:24.970 ","End":"01:28.570","Text":"If you don\u0027t want to mess with inequalities,"},{"Start":"01:28.570 ","End":"01:32.050","Text":"there is a very simple way you could do this and that is just to"},{"Start":"01:32.050 ","End":"01:36.160","Text":"start choosing different values of k. For example,"},{"Start":"01:36.160 ","End":"01:38.470","Text":"if k is 0, we have 10,"},{"Start":"01:38.470 ","End":"01:39.790","Text":"if k is 1,"},{"Start":"01:39.790 ","End":"01:43.998","Text":"we get 10 plus 120 is 130,"},{"Start":"01:43.998 ","End":"01:45.630","Text":"if k is 2,"},{"Start":"01:45.630 ","End":"01:49.785","Text":"we get 250, and so on."},{"Start":"01:49.785 ","End":"01:51.235","Text":"On the other side,"},{"Start":"01:51.235 ","End":"01:54.265","Text":"if k is minus 1,"},{"Start":"01:54.265 ","End":"01:57.140","Text":"we get minus 110,"},{"Start":"01:57.140 ","End":"02:00.360","Text":"and so on in both directions."},{"Start":"02:00.360 ","End":"02:05.525","Text":"Then we would look and see which of these is in the range,"},{"Start":"02:05.525 ","End":"02:10.795","Text":"and we\u0027d say from 0 to a 180, not inclusive."},{"Start":"02:10.795 ","End":"02:12.770","Text":"We only have two of them."},{"Start":"02:12.770 ","End":"02:16.830","Text":"That would be the 10 and the 130,"},{"Start":"02:16.830 ","End":"02:18.970","Text":"and then we can do the same thing here."},{"Start":"02:18.970 ","End":"02:21.800","Text":"We can start off with k=0,"},{"Start":"02:21.800 ","End":"02:27.540","Text":"so we would get 50, then k=1,"},{"Start":"02:27.540 ","End":"02:31.260","Text":"so we get 50 plus a 120 is a 170,"},{"Start":"02:31.260 ","End":"02:33.120","Text":"and then k=2,"},{"Start":"02:33.120 ","End":"02:36.065","Text":"we get 290,"},{"Start":"02:36.065 ","End":"02:39.125","Text":"and I\u0027m stopping because I see I\u0027m already over 180."},{"Start":"02:39.125 ","End":"02:41.710","Text":"On the other side, if k is minus 1,"},{"Start":"02:41.710 ","End":"02:45.465","Text":"I\u0027ve got 50 minus a 120 is minus 70,"},{"Start":"02:45.465 ","End":"02:47.300","Text":"and that\u0027s already too small."},{"Start":"02:47.300 ","End":"02:56.105","Text":"From here, I see that 50 is in the range of 0-180 and so is 170."},{"Start":"02:56.105 ","End":"03:02.855","Text":"Then we would write the answer as x could be any one of the following 4."},{"Start":"03:02.855 ","End":"03:04.745","Text":"We\u0027ll even arrange them in order,"},{"Start":"03:04.745 ","End":"03:08.810","Text":"10, 50,"},{"Start":"03:08.810 ","End":"03:13.480","Text":"130, 170,"},{"Start":"03:13.480 ","End":"03:16.985","Text":"any one of these, and they are in the range and that would be the solution."},{"Start":"03:16.985 ","End":"03:22.250","Text":"There is another way of messing with inequalities of saying that,"},{"Start":"03:22.250 ","End":"03:27.985","Text":"x is less than a 180,10 plus a 120k less than a 180 but,"},{"Start":"03:27.985 ","End":"03:29.540","Text":"if I don\u0027t have to,"},{"Start":"03:29.540 ","End":"03:34.585","Text":"this simple naive method usually is good enough."},{"Start":"03:34.585 ","End":"03:38.380","Text":"Let\u0027s move on to Part B."},{"Start":"03:39.290 ","End":"03:43.309","Text":"Here we also have a restricted range."},{"Start":"03:43.309 ","End":"03:46.865","Text":"X has to be between minus 90 and 90."},{"Start":"03:46.865 ","End":"03:51.740","Text":"What we want to do here is just solve the equation for the moment,"},{"Start":"03:51.740 ","End":"03:54.310","Text":"put aside the restriction."},{"Start":"03:54.310 ","End":"03:56.930","Text":"Using our usual techniques,"},{"Start":"03:56.930 ","End":"04:01.310","Text":"we write this as cosine 3x equals cosine of something,"},{"Start":"04:01.310 ","End":"04:06.574","Text":"and at this stage you\u0027re already supposed to remember your cosine and sine tables."},{"Start":"04:06.574 ","End":"04:12.510","Text":"This happens to be the cosine of 30 degrees."},{"Start":"04:12.960 ","End":"04:17.560","Text":"We solve it with the usual techniques."},{"Start":"04:17.560 ","End":"04:19.510","Text":"We put that,"},{"Start":"04:19.510 ","End":"04:22.360","Text":"there\u0027s two possibilities for 3x."},{"Start":"04:22.360 ","End":"04:26.154","Text":"One possibility is just equal to the 30,"},{"Start":"04:26.154 ","End":"04:30.850","Text":"but plus whole circles, which is the 360k,"},{"Start":"04:30.980 ","End":"04:38.650","Text":"and the other one for cosine is the negative minus 30 also plus 360k,"},{"Start":"04:39.080 ","End":"04:43.455","Text":"and then if we divide this by 3,"},{"Start":"04:43.455 ","End":"04:50.850","Text":"we get that x is either 10 plus 120k,"},{"Start":"04:50.850 ","End":"04:57.700","Text":"or it could be minus 10 plus 120k,"},{"Start":"04:58.700 ","End":"05:02.111","Text":"and k is an integer, 0,"},{"Start":"05:02.111 ","End":"05:04.900","Text":"1, 2, 3, 4, 5 or minus 1, minus 2."},{"Start":"05:04.900 ","End":"05:09.740","Text":"Let\u0027s take some examples of k\u0027s and see what x\u0027s we get."},{"Start":"05:09.740 ","End":"05:12.080","Text":"If k is 0,"},{"Start":"05:12.080 ","End":"05:16.550","Text":"it starts in the middle, then x=10."},{"Start":"05:16.550 ","End":"05:18.395","Text":"If k is 1,"},{"Start":"05:18.395 ","End":"05:20.855","Text":"then I get 130."},{"Start":"05:20.855 ","End":"05:24.050","Text":"If k is 2 each time I\u0027m adding 120,"},{"Start":"05:24.050 ","End":"05:27.970","Text":"so I\u0027d get 250, and so on."},{"Start":"05:27.970 ","End":"05:29.870","Text":"On the other side,"},{"Start":"05:29.870 ","End":"05:32.210","Text":"if k was minus 1,"},{"Start":"05:32.210 ","End":"05:35.375","Text":"I get 10 minus 120,"},{"Start":"05:35.375 ","End":"05:37.825","Text":"which is minus a 110,"},{"Start":"05:37.825 ","End":"05:41.700","Text":"and I\u0027m going to stop here because,"},{"Start":"05:41.700 ","End":"05:43.690","Text":"as I\u0027m writing this,"},{"Start":"05:43.690 ","End":"05:45.850","Text":"I\u0027m squinting and looking over here,"},{"Start":"05:45.850 ","End":"05:48.110","Text":"and seeing that I went from minus 90 to 90,"},{"Start":"05:48.110 ","End":"05:52.285","Text":"while here I\u0027ve already overshot the mark and already here overshot the mark."},{"Start":"05:52.285 ","End":"05:58.840","Text":"In fact, the only one that\u0027s in the range from minus 90 to 90,"},{"Start":"05:58.840 ","End":"06:03.955","Text":"is the x=10 value,"},{"Start":"06:03.955 ","End":"06:06.880","Text":"because 130 is already gone over,"},{"Start":"06:06.880 ","End":"06:10.690","Text":"and minus 110 is too low."},{"Start":"06:10.690 ","End":"06:14.260","Text":"That\u0027s for this series. Let\u0027s look at the second series."},{"Start":"06:14.260 ","End":"06:18.873","Text":"If k=0, we get minus 10,"},{"Start":"06:18.873 ","End":"06:26.525","Text":"if k=1, we get minus 10 plus 120 is 110."},{"Start":"06:26.525 ","End":"06:29.235","Text":"We\u0027ve already overshot the 90,"},{"Start":"06:29.235 ","End":"06:30.619","Text":"and then the other direction,"},{"Start":"06:30.619 ","End":"06:32.870","Text":"if k is minus 1,"},{"Start":"06:32.870 ","End":"06:35.360","Text":"I\u0027ve got minus a 130."},{"Start":"06:35.360 ","End":"06:37.190","Text":"That\u0027s already too low,"},{"Start":"06:37.190 ","End":"06:43.095","Text":"so only the minus 10 from this list."},{"Start":"06:43.095 ","End":"06:47.215","Text":"Altogether I collect the good ones that are in this range,"},{"Start":"06:47.215 ","End":"06:52.270","Text":"and x could be minus 10 or 10."},{"Start":"06:52.270 ","End":"06:54.785","Text":"These are the only two solutions,"},{"Start":"06:54.785 ","End":"06:58.350","Text":"and that\u0027s the end of Part B so we\u0027re done."}],"ID":5407},{"Watched":false,"Name":"Exercise 15","Duration":"9m 8s","ChapterTopicVideoID":5409,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5409.jpeg","UploadDate":"2016-03-10T21:22:34.1230000","DurationForVideoObject":"PT9M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"This exercise we have a couple of trigonometric equations and"},{"Start":"00:04.200 ","End":"00:08.865","Text":"each one restricts x to a certain range or interval."},{"Start":"00:08.865 ","End":"00:11.820","Text":"We\u0027ll start with the first one,"},{"Start":"00:11.820 ","End":"00:15.600","Text":"which has a mixture of sine and cosine."},{"Start":"00:15.600 ","End":"00:20.745","Text":"We\u0027re first going to solve the equation and then worry about the limit on x."},{"Start":"00:20.745 ","End":"00:23.520","Text":"We\u0027ve seen similar to this before,"},{"Start":"00:23.520 ","End":"00:25.890","Text":"what I\u0027m going to do is write it as sine x,"},{"Start":"00:25.890 ","End":"00:30.705","Text":"first of all, is equal to minus cosine x, separate them."},{"Start":"00:30.705 ","End":"00:34.260","Text":"Now I want to try and get this in terms of sine,"},{"Start":"00:34.260 ","End":"00:36.855","Text":"to get sine x equals sine something."},{"Start":"00:36.855 ","End":"00:39.300","Text":"Could\u0027ve done it the other way round and"},{"Start":"00:39.300 ","End":"00:42.810","Text":"brought the sine x over, it doesn\u0027t really matter."},{"Start":"00:42.810 ","End":"00:46.370","Text":"How am I going to get this to be sine of something?"},{"Start":"00:46.370 ","End":"00:51.290","Text":"First of all, I\u0027m going to write"},{"Start":"00:51.290 ","End":"00:57.790","Text":"the cosine x as the sine of 90 minus x,"},{"Start":"00:57.790 ","End":"01:01.520","Text":"that\u0027s one of the standard formulas and I\u0027m not going to repeat it."},{"Start":"01:01.520 ","End":"01:05.585","Text":"That the cosine of an angle is sine of 90 minus the angle."},{"Start":"01:05.585 ","End":"01:12.347","Text":"Now I\u0027m going to use the property of sine that a minus can go inside the sine,"},{"Start":"01:12.347 ","End":"01:18.560","Text":"so I\u0027ve got the sine of minus 90 plus x."},{"Start":"01:20.030 ","End":"01:24.440","Text":"Now I have the situation where sine of something equals sine of something."},{"Start":"01:24.440 ","End":"01:28.580","Text":"I\u0027ll just copy it here and here and so"},{"Start":"01:28.580 ","End":"01:35.090","Text":"we use our usual technique of saying that it could be that these two are equal."},{"Start":"01:35.090 ","End":"01:40.580","Text":"The arguments x equals minus 90 plus x and"},{"Start":"01:40.580 ","End":"01:47.265","Text":"then plus 360k that\u0027s for the full circles."},{"Start":"01:47.265 ","End":"01:52.265","Text":"The other possibility is that these are not equal,"},{"Start":"01:52.265 ","End":"01:58.355","Text":"but they are 180 degrees minus the other."},{"Start":"01:58.355 ","End":"02:01.925","Text":"So if I take this from 180 degrees,"},{"Start":"02:01.925 ","End":"02:11.500","Text":"let\u0027s do this mentally 180 minus this is a 180 plus 90 minus x plus 360k."},{"Start":"02:12.160 ","End":"02:15.215","Text":"Let\u0027s see what we get for each of these."},{"Start":"02:15.215 ","End":"02:17.615","Text":"If I simplify this one,"},{"Start":"02:17.615 ","End":"02:22.890","Text":"then look at that the x cancels."},{"Start":"02:22.890 ","End":"02:32.340","Text":"This goes with this and all we\u0027re left with is 0 equals minus 90 plus 360k,"},{"Start":"02:32.340 ","End":"02:38.720","Text":"the x is gone and this is a false statement because it doesn\u0027t even matter"},{"Start":"02:38.720 ","End":"02:46.175","Text":"what k you choose minus 90 plus full circles is never going to be 0 and this is just no."},{"Start":"02:46.175 ","End":"02:50.420","Text":"This doesn\u0027t give us anything, I\u0027m going to emphasize it."},{"Start":"02:50.420 ","End":"02:52.925","Text":"I\u0027ll just cross it out because it leads to a contradiction."},{"Start":"02:52.925 ","End":"02:55.910","Text":"But we still have the second row,"},{"Start":"02:55.910 ","End":"02:59.240","Text":"the second set of solutions and that will probably give us something,"},{"Start":"02:59.240 ","End":"03:03.260","Text":"so let\u0027s continue that one down here and here we get 2x,"},{"Start":"03:03.260 ","End":"03:12.125","Text":"I\u0027m just taking the x\u0027s to the left equals 270 plus 360k and if I divide by 2,"},{"Start":"03:12.125 ","End":"03:17.690","Text":"I get x equals 135 plus 180k."},{"Start":"03:17.690 ","End":"03:21.875","Text":"Now, this is the general solution to this trigonometric equation."},{"Start":"03:21.875 ","End":"03:29.690","Text":"Now I want to relate to the requirement that x should be between 0 and 360 inclusive."},{"Start":"03:29.690 ","End":"03:34.370","Text":"The easiest way to do it is not to mess with solving inequalities,"},{"Start":"03:34.370 ","End":"03:36.740","Text":"but just to write some values for k,"},{"Start":"03:36.740 ","End":"03:41.735","Text":"k is any integer 0, positive or negative."},{"Start":"03:41.735 ","End":"03:47.285","Text":"If we let k equal 0, we have 135."},{"Start":"03:47.285 ","End":"03:50.070","Text":"Maybe I\u0027ll even write that k equals 0."},{"Start":"03:50.070 ","End":"03:52.295","Text":"If I let k equals 1,"},{"Start":"03:52.295 ","End":"04:01.330","Text":"then I get 135 plus 180, which is 315."},{"Start":"04:02.140 ","End":"04:07.370","Text":"If I let k equals 2,"},{"Start":"04:07.370 ","End":"04:12.310","Text":"then I get 360 plus 135,"},{"Start":"04:12.310 ","End":"04:14.850","Text":"obviously too much but I\u0027ll write it anyway,"},{"Start":"04:14.850 ","End":"04:21.460","Text":"360 and 135 is 495 I believe."},{"Start":"04:21.830 ","End":"04:25.050","Text":"That\u0027s k equals 2, it\u0027s too much,"},{"Start":"04:25.050 ","End":"04:28.100","Text":"let\u0027s try k equals minus 1,"},{"Start":"04:28.100 ","End":"04:36.245","Text":"in which k is I got a 135 minus 180 and that\u0027s minus 45 degrees,"},{"Start":"04:36.245 ","End":"04:39.365","Text":"and so on, it\u0027s infinite in both directions."},{"Start":"04:39.365 ","End":"04:42.560","Text":"Now, strictly between 0 and 360,"},{"Start":"04:42.560 ","End":"04:43.970","Text":"I mean, including the endpoints,"},{"Start":"04:43.970 ","End":"04:46.795","Text":"I only have two values,"},{"Start":"04:46.795 ","End":"04:50.120","Text":"let me just erase it, I\u0027m used to writing degrees,"},{"Start":"04:50.120 ","End":"04:52.325","Text":"this one and this one."},{"Start":"04:52.325 ","End":"04:55.250","Text":"135 is in the range,"},{"Start":"04:55.250 ","End":"05:02.960","Text":"and so it\u0027s 315 and so I write my answer that x could be either a"},{"Start":"05:02.960 ","End":"05:12.440","Text":"135 or 315 and these are the two solutions and that\u0027s the end of part A."},{"Start":"05:12.440 ","End":"05:14.630","Text":"On to part B,"},{"Start":"05:14.630 ","End":"05:18.200","Text":"which is here somewhere there it is."},{"Start":"05:18.200 ","End":"05:22.625","Text":"Again, we have a trigonometric equation with a range."},{"Start":"05:22.625 ","End":"05:24.950","Text":"Forget about the range for the moment,"},{"Start":"05:24.950 ","End":"05:28.870","Text":"let\u0027s just solve the trigonometric equation."},{"Start":"05:28.870 ","End":"05:31.610","Text":"Remember that when you put a 2 here,"},{"Start":"05:31.610 ","End":"05:34.130","Text":"it\u0027s like the sine of 2x,"},{"Start":"05:34.130 ","End":"05:38.030","Text":"the whole thing squared is equal to a 1/4."},{"Start":"05:38.030 ","End":"05:40.205","Text":"Now we already know how to solve equation,"},{"Start":"05:40.205 ","End":"05:43.070","Text":"so something squared is some positive number,"},{"Start":"05:43.070 ","End":"05:46.430","Text":"then we get two possibilities that the thing itself, in this case,"},{"Start":"05:46.430 ","End":"05:51.380","Text":"sine 2x is plus or minus the square root of a 1/4."},{"Start":"05:51.380 ","End":"05:53.450","Text":"Now the square root of a 1/4,"},{"Start":"05:53.450 ","End":"05:56.330","Text":"I can do the square root of numerator and denominator separately,"},{"Start":"05:56.330 ","End":"05:58.190","Text":"so it gives me a 1/2,"},{"Start":"05:58.190 ","End":"06:01.775","Text":"so this is plus or minus a 1/2."},{"Start":"06:01.775 ","End":"06:04.620","Text":"This divides into two cases,"},{"Start":"06:04.620 ","End":"06:12.420","Text":"I either have that sine 2x equals a 1/2 or sine 2x equals minus a 1/2,"},{"Start":"06:12.420 ","End":"06:14.875","Text":"and let\u0027s pursue each of these."},{"Start":"06:14.875 ","End":"06:17.704","Text":"If it\u0027s equal to a 1/2,"},{"Start":"06:17.704 ","End":"06:22.670","Text":"a 1/2 is sine of 30 degrees,"},{"Start":"06:22.670 ","End":"06:24.485","Text":"you already know that by heart,"},{"Start":"06:24.485 ","End":"06:33.760","Text":"and so we solve it by saying that either 2x is 30 degrees."},{"Start":"06:34.180 ","End":"06:38.940","Text":"I apologize, I keep writing degrees, I\u0027m so used to it."},{"Start":"06:39.410 ","End":"06:49.150","Text":"It\u0027s either 2x equals 30 but plus whole circles, in other words, 360k."},{"Start":"06:50.000 ","End":"06:56.000","Text":"The other possibility with sine is that we take a 180 minus this thing,"},{"Start":"06:56.000 ","End":"06:58.535","Text":"I\u0027m just doing it in my head,180 minus 30,"},{"Start":"06:58.535 ","End":"07:02.620","Text":"150 also plus 360k,"},{"Start":"07:02.620 ","End":"07:07.200","Text":"two infinite sets of solutions."},{"Start":"07:07.820 ","End":"07:10.860","Text":"I wanted to just get x not 2x,"},{"Start":"07:10.860 ","End":"07:17.820","Text":"so divide by 2 so we have that x is equal to 15 plus 180k,"},{"Start":"07:18.310 ","End":"07:26.715","Text":"it\u0027s 1/2 of this and the other set of possibilities is 75,"},{"Start":"07:26.715 ","End":"07:31.390","Text":"1/2 of a 150 also plus a 180k."},{"Start":"07:31.790 ","End":"07:37.455","Text":"Look, we want results between 0 and 90,"},{"Start":"07:37.455 ","End":"07:39.975","Text":"not including 0 and 90,"},{"Start":"07:39.975 ","End":"07:43.910","Text":"let\u0027s just put values of k and the first and the second."},{"Start":"07:43.910 ","End":"07:49.340","Text":"In the first, if I put k equals 0, I get 15."},{"Start":"07:49.340 ","End":"07:55.010","Text":"If I put k equals 1, I get 195."},{"Start":"07:55.010 ","End":"07:56.540","Text":"I\u0027m already too much."},{"Start":"07:56.540 ","End":"07:59.450","Text":"If I put k equals minus 1,"},{"Start":"07:59.450 ","End":"08:06.655","Text":"then I get minus 165 too low and these only gets smaller and this only get bigger."},{"Start":"08:06.655 ","End":"08:12.255","Text":"In this row, only the 15 falls within the required range,"},{"Start":"08:12.255 ","End":"08:14.355","Text":"let\u0027s see what we get from this one."},{"Start":"08:14.355 ","End":"08:16.575","Text":"If k is 0,"},{"Start":"08:16.575 ","End":"08:19.725","Text":"I just get 75."},{"Start":"08:19.725 ","End":"08:21.835","Text":"If k is 1,"},{"Start":"08:21.835 ","End":"08:26.005","Text":"then that gives me 255,"},{"Start":"08:26.005 ","End":"08:32.400","Text":"which is way too much because my maximum is 90 less."},{"Start":"08:32.400 ","End":"08:34.240","Text":"On the other side,"},{"Start":"08:34.240 ","End":"08:35.530","Text":"if k is minus 1,"},{"Start":"08:35.530 ","End":"08:39.025","Text":"I got 75 minus 180,"},{"Start":"08:39.025 ","End":"08:45.449","Text":"minus 105 which is certainly less than 0, way too small."},{"Start":"08:45.449 ","End":"08:48.790","Text":"The only one that fits is the"},{"Start":"08:48.790 ","End":"08:56.325","Text":"75 and so there are only two answers to this,"},{"Start":"08:56.325 ","End":"09:03.540","Text":"x could equal 15 and it could equal 75."},{"Start":"09:03.540 ","End":"09:08.190","Text":"These are the two solutions and since this is part B, we are done."}],"ID":5408},{"Watched":false,"Name":"Exercise 16","Duration":"16m 9s","ChapterTopicVideoID":5410,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5410.jpeg","UploadDate":"2016-03-10T21:24:47.0530000","DurationForVideoObject":"PT16M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:10.455","Text":"Here we have another pair of trigonometric equations with restriction on the outcome x."},{"Start":"00:10.455 ","End":"00:13.515","Text":"The first one just involves the sine."},{"Start":"00:13.515 ","End":"00:16.215","Text":"Second one we have a mixture of sine and cosine."},{"Start":"00:16.215 ","End":"00:19.975","Text":"Let\u0027s start with the first and see what we can do here."},{"Start":"00:19.975 ","End":"00:23.700","Text":"At first, we ignore the limitation on x,"},{"Start":"00:23.700 ","End":"00:25.800","Text":"that is in the interval from 0 to 180,"},{"Start":"00:25.800 ","End":"00:27.540","Text":"we just go ahead and try and solve it."},{"Start":"00:27.540 ","End":"00:31.230","Text":"Now we want to get sine something equals sine something."},{"Start":"00:31.230 ","End":"00:37.157","Text":"Start out by moving the second term on to the right,"},{"Start":"00:37.157 ","End":"00:42.880","Text":"so we get sine x equals minus sine for x."},{"Start":"00:42.880 ","End":"00:46.985","Text":"Then we remember the way to deal with a minus."},{"Start":"00:46.985 ","End":"00:48.560","Text":"In the case of a sine function,"},{"Start":"00:48.560 ","End":"00:54.385","Text":"we just put the minus inside sine of minus 4x."},{"Start":"00:54.385 ","End":"00:58.805","Text":"Now we\u0027re all set for sine something equals sine something."},{"Start":"00:58.805 ","End":"01:03.485","Text":"We usually do it by saying x equals whatever this is."},{"Start":"01:03.485 ","End":"01:05.165","Text":"This equals this."},{"Start":"01:05.165 ","End":"01:08.035","Text":"But we add 360k,"},{"Start":"01:08.035 ","End":"01:13.535","Text":"whatever number of circles and the other one,"},{"Start":"01:13.535 ","End":"01:15.800","Text":"we take whatever was here,"},{"Start":"01:15.800 ","End":"01:18.215","Text":"but we subtract it from 180."},{"Start":"01:18.215 ","End":"01:20.750","Text":"If we subtract this from 180,"},{"Start":"01:20.750 ","End":"01:23.690","Text":"we get 180 minus,"},{"Start":"01:23.690 ","End":"01:29.189","Text":"minus 4x, so it\u0027s 180 plus 4x and again plus 360k."},{"Start":"01:29.189 ","End":"01:34.710","Text":"Of course you won\u0027t just x on the left."},{"Start":"01:34.710 ","End":"01:37.295","Text":"Let\u0027s see. Let\u0027s simplify them both."},{"Start":"01:37.295 ","End":"01:39.080","Text":"I\u0027ll start with the first one."},{"Start":"01:39.080 ","End":"01:42.440","Text":"I\u0027ll do it over here, 4x goes over,"},{"Start":"01:42.440 ","End":"01:47.185","Text":"we get 5x equals 360k,"},{"Start":"01:47.185 ","End":"01:53.065","Text":"and that makes x equal to 72k I divided by 5 of course."},{"Start":"01:53.065 ","End":"01:57.290","Text":"If I continue with this one on the left,"},{"Start":"01:57.290 ","End":"02:07.600","Text":"I get minus 3x and here just whatever was 180 plus 360k."},{"Start":"02:07.600 ","End":"02:10.450","Text":"In this case I want to divide by minus 3,"},{"Start":"02:10.450 ","End":"02:15.890","Text":"[inaudible] over minus 3 is minus 60 and here"},{"Start":"02:15.890 ","End":"02:22.080","Text":"I have minus 120k."},{"Start":"02:22.080 ","End":"02:24.170","Text":"These are the two sets of solutions."},{"Start":"02:24.170 ","End":"02:28.340","Text":"I think I mentioned once that if you have minus in front of something with the k,"},{"Start":"02:28.340 ","End":"02:30.490","Text":"you can make it a plus."},{"Start":"02:30.490 ","End":"02:34.310","Text":"Whether you understood me why or whether or"},{"Start":"02:34.310 ","End":"02:37.760","Text":"not we can just change it to a plus essentially,"},{"Start":"02:37.760 ","End":"02:41.060","Text":"k is just as much an integer as minus k is"},{"Start":"02:41.060 ","End":"02:45.660","Text":"an arbitrary integer and I leave it in this form here."},{"Start":"02:45.660 ","End":"02:49.655","Text":"If you really want to stick to this, That\u0027s also okay."},{"Start":"02:49.655 ","End":"02:51.875","Text":"You\u0027ll get the same answer."},{"Start":"02:51.875 ","End":"02:57.440","Text":"What we do now is with each of these sets of solutions as we start writing,"},{"Start":"02:57.440 ","End":"03:01.970","Text":"one way of doing it is to just write some samples values"},{"Start":"03:01.970 ","End":"03:06.470","Text":"for different values of k and just look and see which are in the range."},{"Start":"03:06.470 ","End":"03:10.715","Text":"I think this is easier this way than actually trying to solve inequalities."},{"Start":"03:10.715 ","End":"03:14.465","Text":"Although it\u0027s possible to do it completely by inequalities."},{"Start":"03:14.465 ","End":"03:20.495","Text":"Like you would write 72k less than 180 and solve what k is and k,"},{"Start":"03:20.495 ","End":"03:23.160","Text":"72 k bigger than 0."},{"Start":"03:23.870 ","End":"03:27.555","Text":"Let\u0027s just do it by writing values."},{"Start":"03:27.555 ","End":"03:29.790","Text":"If k is 0,"},{"Start":"03:29.790 ","End":"03:33.325","Text":"I get 72 times zero is zero."},{"Start":"03:33.325 ","End":"03:37.190","Text":"If k is 1, I get just 72."},{"Start":"03:37.190 ","End":"03:42.035","Text":"If k is 2, I get 144."},{"Start":"03:42.035 ","End":"03:43.730","Text":"If k is 3,"},{"Start":"03:43.730 ","End":"03:47.810","Text":"I get 216 and all the time I\u0027m looking up here and I see,"},{"Start":"03:47.810 ","End":"03:50.705","Text":"I\u0027ve gone over the limit, fine."},{"Start":"03:50.705 ","End":"03:57.800","Text":"No point going in the other direction with k equals minus 1."},{"Start":"03:57.800 ","End":"04:02.585","Text":"You could and then you\u0027d get minus 72."},{"Start":"04:02.585 ","End":"04:06.440","Text":"But certainly it\u0027s only getting away from where we want."},{"Start":"04:06.440 ","End":"04:09.110","Text":"Now, look at this inequality."},{"Start":"04:09.110 ","End":"04:10.400","Text":"It\u0027s a strict inequality,"},{"Start":"04:10.400 ","End":"04:13.730","Text":"0 is not good it is rejected,"},{"Start":"04:13.730 ","End":"04:21.035","Text":"so the only valid ones for this series, 72 and 144."},{"Start":"04:21.035 ","End":"04:23.465","Text":"For the next series."},{"Start":"04:23.465 ","End":"04:29.090","Text":"What we get if k is 0 minus 60,"},{"Start":"04:29.090 ","End":"04:30.950","Text":"which is already too low,"},{"Start":"04:30.950 ","End":"04:33.200","Text":"then if k is 1,"},{"Start":"04:33.200 ","End":"04:37.070","Text":"we get minus 60 plus 120 is 60."},{"Start":"04:37.070 ","End":"04:39.335","Text":"If k is 2,"},{"Start":"04:39.335 ","End":"04:49.270","Text":"we have 240 minus 60 is 180 and from then on, just gets bigger."},{"Start":"04:49.840 ","End":"04:55.730","Text":"I can see that a 180 is already too big because we have the strict inequality here."},{"Start":"04:55.730 ","End":"05:05.210","Text":"The only one from this series is the 60 that fits into the range and I highlighted it."},{"Start":"05:05.210 ","End":"05:09.050","Text":"Just by the way, if you did decide to stick with this,"},{"Start":"05:09.050 ","End":"05:14.809","Text":"also, you would put k equals 0, you would get 60."},{"Start":"05:14.809 ","End":"05:16.910","Text":"If you put k equals 1,"},{"Start":"05:16.910 ","End":"05:21.635","Text":"you\u0027d get minus 60."},{"Start":"05:21.635 ","End":"05:25.130","Text":"If you put k equals minus 1,"},{"Start":"05:25.130 ","End":"05:31.025","Text":"you\u0027d get the plus 60."},{"Start":"05:31.025 ","End":"05:35.630","Text":"Basically you get the same thing if you put k is minus 2,"},{"Start":"05:35.630 ","End":"05:38.135","Text":"you\u0027d get plus 240 minus 60."},{"Start":"05:38.135 ","End":"05:47.145","Text":"You get the same numbers just with different values of k with the opposite k. Anyway,"},{"Start":"05:47.145 ","End":"05:50.240","Text":"whatever it is, whichever one you use,"},{"Start":"05:50.240 ","End":"05:52.220","Text":"these are the numbers that occur,"},{"Start":"05:52.220 ","End":"05:54.695","Text":"and these are the 3 that are in our range."},{"Start":"05:54.695 ","End":"05:56.465","Text":"We write that x is,"},{"Start":"05:56.465 ","End":"05:58.940","Text":"I like to put them also in an increasing order,"},{"Start":"05:58.940 ","End":"06:06.170","Text":"could be 60, could be 72, could be 144."},{"Start":"06:06.170 ","End":"06:08.270","Text":"Those are the 3 answers to part a,"},{"Start":"06:08.270 ","End":"06:12.060","Text":"now let\u0027s look at part b."},{"Start":"06:14.980 ","End":"06:20.450","Text":"Again, we have a trigonometric equation"},{"Start":"06:20.450 ","End":"06:26.030","Text":"with a limitation that x has to be in a certain interval."},{"Start":"06:26.030 ","End":"06:28.730","Text":"Let\u0027s see This time we have a mixture."},{"Start":"06:28.730 ","End":"06:32.390","Text":"First thing to do is move this cosine to the other side."},{"Start":"06:32.390 ","End":"06:35.340","Text":"It\u0027s cosine 3x."},{"Start":"06:36.160 ","End":"06:43.455","Text":"I want to get everything in terms of sine,"},{"Start":"06:43.455 ","End":"06:47.505","Text":"so cosine of 3x is the same"},{"Start":"06:47.505 ","End":"06:53.935","Text":"as sine of 180 minus 3x."},{"Start":"06:53.935 ","End":"06:58.130","Text":"This technique of subtracting from 180,"},{"Start":"06:58.130 ","End":"07:00.856","Text":"sorry, I meant 90."},{"Start":"07:00.856 ","End":"07:02.845","Text":"Yes."},{"Start":"07:02.845 ","End":"07:05.770","Text":"This technique of subtracting 90 minus"},{"Start":"07:05.770 ","End":"07:09.130","Text":"an angle takes us from sine to cosine and vice versa."},{"Start":"07:09.130 ","End":"07:10.675","Text":"It\u0027s used both ways."},{"Start":"07:10.675 ","End":"07:13.510","Text":"In the old days, this was called the complement of the angle,"},{"Start":"07:13.510 ","End":"07:17.200","Text":"like sine of 20 is cosine 70 and so on."},{"Start":"07:17.200 ","End":"07:22.700","Text":"20 and 70 were complementary, they add up to 90."},{"Start":"07:22.700 ","End":"07:29.605","Text":"Now we have sine equals sine, and we know how to solve that."},{"Start":"07:29.605 ","End":"07:34.132","Text":"We say that either x is"},{"Start":"07:34.132 ","End":"07:41.245","Text":"equal to 90 minus 3x plus a whole number of 360s."},{"Start":"07:41.245 ","End":"07:46.870","Text":"The other possibility in the case of sine is to take 180 minus"},{"Start":"07:46.870 ","End":"07:54.655","Text":"this.180 minus this is 180 minus 90 plus 3x."},{"Start":"07:54.655 ","End":"07:58.340","Text":"We get, if I reverse the order,"},{"Start":"08:01.950 ","End":"08:08.545","Text":"3x and plus 90 again plus 360k."},{"Start":"08:08.545 ","End":"08:11.140","Text":"But we need just x,"},{"Start":"08:11.140 ","End":"08:15.175","Text":"and we have x on the right-hand side also, so simplify."},{"Start":"08:15.175 ","End":"08:17.965","Text":"The first one gives us that 4x,"},{"Start":"08:17.965 ","End":"08:19.674","Text":"that\u0027s by bringing the 3 over,"},{"Start":"08:19.674 ","End":"08:24.940","Text":"equals 90 plus 360k."},{"Start":"08:24.940 ","End":"08:27.160","Text":"Then x is equal to,"},{"Start":"08:27.160 ","End":"08:30.070","Text":"dividing by 4, so you half it is 45,"},{"Start":"08:30.070 ","End":"08:36.680","Text":"half it again 22 and 1/2 and quarter of this is 90."},{"Start":"08:37.170 ","End":"08:40.225","Text":"For this one, just continuing,"},{"Start":"08:40.225 ","End":"08:44.270","Text":"x minus 3 x is minus 2x."},{"Start":"08:45.150 ","End":"08:48.070","Text":"Then this equals the rest of it,"},{"Start":"08:48.070 ","End":"08:50.440","Text":"90 plus 360k,"},{"Start":"08:50.440 ","End":"08:52.720","Text":"and we divide by minus 2,"},{"Start":"08:52.720 ","End":"08:57.400","Text":"we get minus 45."},{"Start":"08:57.400 ","End":"09:03.010","Text":"Then just dividing this by minus 2,"},{"Start":"09:03.010 ","End":"09:06.175","Text":"we get minus 180k."},{"Start":"09:06.175 ","End":"09:10.720","Text":"Like I said, if you have minus k,"},{"Start":"09:10.720 ","End":"09:12.880","Text":"you can replace minus k by k,"},{"Start":"09:12.880 ","End":"09:14.860","Text":"it just looks nicer."},{"Start":"09:14.860 ","End":"09:21.350","Text":"I could write that x equals minus 45 plus 180k,"},{"Start":"09:21.960 ","End":"09:26.035","Text":"where k is any integer."},{"Start":"09:26.035 ","End":"09:32.035","Text":"Now we need to look at the limitation that x is between 0 and 270."},{"Start":"09:32.035 ","End":"09:35.440","Text":"I\u0027ll do it with my method of just writing values."},{"Start":"09:35.440 ","End":"09:42.685","Text":"If k=0 here, I start with 22 and 1/2 and that\u0027s good."},{"Start":"09:42.685 ","End":"09:48.910","Text":"Then I add 90 by setting k=1,"},{"Start":"09:48.910 ","End":"09:54.880","Text":"that gives me 112 and 1/2."},{"Start":"09:54.880 ","End":"10:02.485","Text":"If k=2, I get 180 plus 22 and 1/2."},{"Start":"10:02.485 ","End":"10:04.660","Text":"180 plus 20 is 200,"},{"Start":"10:04.660 ","End":"10:07.340","Text":"so it\u0027s 202 and 1/2."},{"Start":"10:11.580 ","End":"10:17.320","Text":"The next 1, where k is 3 is too big because we\u0027ve got 270, we\u0027re already over."},{"Start":"10:17.320 ","End":"10:20.139","Text":"But if I started right, 270 plus,"},{"Start":"10:20.139 ","End":"10:24.770","Text":"that would be 292 and 1/2."},{"Start":"10:25.440 ","End":"10:31.240","Text":"On the other side, if I took k equals minus 1,"},{"Start":"10:31.240 ","End":"10:37.225","Text":"then I\u0027d get 22 and 1/2 minus 90 and that would be minus 67 and 1/2."},{"Start":"10:37.225 ","End":"10:39.310","Text":"It\u0027s okay if you write too many."},{"Start":"10:39.310 ","End":"10:40.510","Text":"In the end, in any event,"},{"Start":"10:40.510 ","End":"10:43.390","Text":"you\u0027re only going to choose the good ones and the good ones"},{"Start":"10:43.390 ","End":"10:47.290","Text":"that are between 0 and 270 strictly,"},{"Start":"10:47.290 ","End":"10:50.480","Text":"this one is good, and this one is good, and this one is good."},{"Start":"10:56.640 ","End":"10:58.900","Text":"That\u0027s from this series."},{"Start":"10:58.900 ","End":"11:02.680","Text":"From this series, let\u0027s write a few values."},{"Start":"11:02.680 ","End":"11:06.220","Text":"If k is 0, we\u0027ve got minus 45, that\u0027s too small."},{"Start":"11:06.220 ","End":"11:09.510","Text":"Let\u0027s keep increasing k,"},{"Start":"11:09.510 ","End":"11:15.100","Text":"k=1, that gives me a 180 minus 45 is 135."},{"Start":"11:15.100 ","End":"11:17.350","Text":"If k is 2,"},{"Start":"11:17.350 ","End":"11:26.240","Text":"I\u0027ve got 360 minus 45 is 315 and that\u0027s too big already."},{"Start":"11:26.240 ","End":"11:32.820","Text":"The only one that is good for this range is the 135,"},{"Start":"11:32.820 ","End":"11:35.355","Text":"and I\u0027ll highlight that one."},{"Start":"11:35.355 ","End":"11:41.770","Text":"What I do at the end is to collect together the answers in this range."},{"Start":"11:41.770 ","End":"11:44.185","Text":"You could do them in any order."},{"Start":"11:44.185 ","End":"11:46.420","Text":"I like to do them in increasing order,"},{"Start":"11:46.420 ","End":"11:47.980","Text":"but that\u0027s just me,"},{"Start":"11:47.980 ","End":"11:52.695","Text":"22 and 1/2, 112 and 1/2."},{"Start":"11:52.695 ","End":"11:55.965","Text":"Next in line is the 135,"},{"Start":"11:55.965 ","End":"11:58.755","Text":"and the next is 202 and 1/2."},{"Start":"11:58.755 ","End":"12:02.300","Text":"All these 4 are solutions to this."},{"Start":"12:02.300 ","End":"12:06.805","Text":"I am done, but if you would like to see how to do"},{"Start":"12:06.805 ","End":"12:12.250","Text":"this end part without listing members and doing it with inequalities,"},{"Start":"12:12.250 ","End":"12:18.985","Text":"I\u0027ll show you, I\u0027ll take one of the examples."},{"Start":"12:18.985 ","End":"12:22.990","Text":"Let\u0027s say I\u0027ll take the easier one."},{"Start":"12:22.990 ","End":"12:26.875","Text":"Suppose I wanted to, say."},{"Start":"12:26.875 ","End":"12:30.250","Text":"I changed my mind, go for the more difficult one."},{"Start":"12:30.250 ","End":"12:32.860","Text":"We want to say that x,"},{"Start":"12:32.860 ","End":"12:35.300","Text":"which is 22 and 1/2,"},{"Start":"12:35.610 ","End":"12:40.870","Text":"plus 90k, this is my x,"},{"Start":"12:40.870 ","End":"12:44.990","Text":"I want it to be between 0 and 270."},{"Start":"12:45.060 ","End":"12:47.380","Text":"Now, I\u0027ve written it in one line,"},{"Start":"12:47.380 ","End":"12:49.270","Text":"but this is really 2 inequalities."},{"Start":"12:49.270 ","End":"12:50.350","Text":"We\u0027ll solve them separately."},{"Start":"12:50.350 ","End":"12:52.090","Text":"One inequality is this one."},{"Start":"12:52.090 ","End":"12:55.555","Text":"This is less than this, and the other inequality is this one."},{"Start":"12:55.555 ","End":"12:57.325","Text":"Let\u0027s take the first one."},{"Start":"12:57.325 ","End":"13:04.045","Text":"0 less than 22 and 1/2 plus 90k."},{"Start":"13:04.045 ","End":"13:09.055","Text":"Then, if I bring this to the left,"},{"Start":"13:09.055 ","End":"13:15.775","Text":"what I\u0027ll get is minus 22 and 1/2 is less than 90k."},{"Start":"13:15.775 ","End":"13:18.850","Text":"Then, if I divide by 90,"},{"Start":"13:18.850 ","End":"13:21.490","Text":"better continue over here where there\u0027s more room,"},{"Start":"13:21.490 ","End":"13:23.095","Text":"maybe I\u0027ll fit it in, let\u0027s see,"},{"Start":"13:23.095 ","End":"13:26.994","Text":"minus 22 and 1/2 over"},{"Start":"13:26.994 ","End":"13:33.950","Text":"90 is less than k. Now, I computed this."},{"Start":"13:33.950 ","End":"13:39.400","Text":"I know this is minus 1/4 because we got the 22 and 1/2 in"},{"Start":"13:39.400 ","End":"13:46.345","Text":"the first place by dividing 90 by 4."},{"Start":"13:46.345 ","End":"13:51.835","Text":"This comes out to be minus 1/4 is less"},{"Start":"13:51.835 ","End":"13:57.280","Text":"than k. That\u0027s for that one part,"},{"Start":"13:57.280 ","End":"13:58.900","Text":"and we\u0027ll see what to do with this in a moment."},{"Start":"13:58.900 ","End":"14:00.370","Text":"That\u0027s an intermediate result,"},{"Start":"14:00.370 ","End":"14:03.235","Text":"I\u0027ll just underline it."},{"Start":"14:03.235 ","End":"14:15.470","Text":"Now let\u0027s go for the other part and that\u0027s the 90k is less than 270."},{"Start":"14:18.030 ","End":"14:21.280","Text":"Sorry. This plus this is less than this,"},{"Start":"14:21.280 ","End":"14:28.345","Text":"so we get that 90k is less than,"},{"Start":"14:28.345 ","End":"14:40.370","Text":"and we can subtract this from the other side to 270 minus this is 247 and 1/2."},{"Start":"14:40.370 ","End":"14:42.725","Text":"Let me just check that."},{"Start":"14:42.725 ","End":"14:46.430","Text":"If I subtract 20, it\u0027s 250,"},{"Start":"14:46.430 ","End":"14:50.015","Text":"and I subtract another 2 and 1/2, that looks right."},{"Start":"14:50.015 ","End":"14:53.540","Text":"Now we divide by 90,"},{"Start":"14:53.540 ","End":"14:56.525","Text":"so we get k is less than."},{"Start":"14:56.525 ","End":"15:01.970","Text":"I computed it as 2 and 3/4 if I divide this by 90."},{"Start":"15:01.970 ","End":"15:05.555","Text":"Now, if I take the two bits that I\u0027ve underlined altogether,"},{"Start":"15:05.555 ","End":"15:10.970","Text":"I have that minus 1/4 is less than k,"},{"Start":"15:10.970 ","End":"15:14.815","Text":"is less than 2, and 3/4."},{"Start":"15:14.815 ","End":"15:19.735","Text":"But k is a whole number, an integer."},{"Start":"15:19.735 ","End":"15:21.620","Text":"What integers are,"},{"Start":"15:21.620 ","End":"15:25.235","Text":"they\u0027re whole numbers between minus 1/4 and 2 and 3/4?"},{"Start":"15:25.235 ","End":"15:29.320","Text":"There are only three possibilities: 0,"},{"Start":"15:29.320 ","End":"15:32.075","Text":"1, and 2, minus 1 is too low,"},{"Start":"15:32.075 ","End":"15:33.865","Text":"3 is too high."},{"Start":"15:33.865 ","End":"15:36.330","Text":"Then I take these 0, 1,"},{"Start":"15:36.330 ","End":"15:40.685","Text":"and 2 and substitute them in the formula."},{"Start":"15:40.685 ","End":"15:42.320","Text":"If I substitute 0,"},{"Start":"15:42.320 ","End":"15:44.120","Text":"I get precisely this."},{"Start":"15:44.120 ","End":"15:46.010","Text":"If I substitute 1,"},{"Start":"15:46.010 ","End":"15:49.880","Text":"I get this and if I substitute 2, I get this."},{"Start":"15:49.880 ","End":"15:51.995","Text":"I get the same 3 answers here."},{"Start":"15:51.995 ","End":"15:54.095","Text":"Similar technique for this."},{"Start":"15:54.095 ","End":"15:57.275","Text":"There\u0027s the more formal method,"},{"Start":"15:57.275 ","End":"16:00.410","Text":"but I like the method of just listing a few members of"},{"Start":"16:00.410 ","End":"16:09.260","Text":"the set family of solutions and seeing which ones fit in the range. That\u0027s it."}],"ID":5409},{"Watched":false,"Name":"Exercise 17","Duration":"2m 14s","ChapterTopicVideoID":5411,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5411.jpeg","UploadDate":"2016-03-10T21:25:04.7730000","DurationForVideoObject":"PT2M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this exercise, we have a pair of"},{"Start":"00:02.490 ","End":"00:06.029","Text":"trigonometric equations to solve each involving the tangent."},{"Start":"00:06.029 ","End":"00:10.125","Text":"Let\u0027s start with the first tangent x equals square root of 3."},{"Start":"00:10.125 ","End":"00:12.570","Text":"The idea is to bring it into the form where"},{"Start":"00:12.570 ","End":"00:15.825","Text":"tangent of something equals tangent of something."},{"Start":"00:15.825 ","End":"00:18.390","Text":"Well, left-hand side, we\u0027re okay,"},{"Start":"00:18.390 ","End":"00:21.945","Text":"right-hand side, I want to write as tangent of something."},{"Start":"00:21.945 ","End":"00:26.565","Text":"Square root of 3 is one of those famous tangent values,"},{"Start":"00:26.565 ","End":"00:28.790","Text":"and you should have memorized the table."},{"Start":"00:28.790 ","End":"00:31.805","Text":"It\u0027s actually the tangent of 60."},{"Start":"00:31.805 ","End":"00:34.250","Text":"Or you could look it up on the calculator square root of"},{"Start":"00:34.250 ","End":"00:37.580","Text":"3 and then shift tangent or inverse tangent."},{"Start":"00:37.580 ","End":"00:39.845","Text":"Once we have it in this form,"},{"Start":"00:39.845 ","End":"00:41.570","Text":"the solution is standard."},{"Start":"00:41.570 ","End":"00:43.730","Text":"If tangent something is tangent something,"},{"Start":"00:43.730 ","End":"00:46.759","Text":"then either those two somethings are equal,"},{"Start":"00:46.759 ","End":"00:52.325","Text":"or one is equal to the other plus a multiple of 180 degrees."},{"Start":"00:52.325 ","End":"00:56.390","Text":"I put a 180k where k is some whole number,"},{"Start":"00:56.390 ","End":"01:00.095","Text":"positive or negative, and that\u0027s the solution to a."},{"Start":"01:00.095 ","End":"01:02.960","Text":"Let\u0027s go on to Part b."},{"Start":"01:02.960 ","End":"01:09.090","Text":"In Part b, we have tangent of 3x equals 1."},{"Start":"01:09.090 ","End":"01:12.645","Text":"I want tangent of something equals tangent of something."},{"Start":"01:12.645 ","End":"01:19.190","Text":"Tangent 3x, 1 it happens to be the tangent of 45 degrees,"},{"Start":"01:19.190 ","End":"01:23.670","Text":"is something you should memorize."},{"Start":"01:23.740 ","End":"01:25.790","Text":"I\u0027m going to erase the degrees."},{"Start":"01:25.790 ","End":"01:29.750","Text":"We don\u0027t usually write the degrees."},{"Start":"01:29.750 ","End":"01:31.535","Text":"It\u0027s just a habit I have."},{"Start":"01:31.535 ","End":"01:34.730","Text":"Continuing the same method as in a."},{"Start":"01:34.730 ","End":"01:39.455","Text":"We say that 3x is equal to 45,"},{"Start":"01:39.455 ","End":"01:44.100","Text":"but possibly plus a multiple of 180 degrees."},{"Start":"01:44.100 ","End":"01:45.584","Text":"We said that in general,"},{"Start":"01:45.584 ","End":"01:47.005","Text":"it\u0027s a 180k,"},{"Start":"01:47.005 ","End":"01:50.780","Text":"where k is an integer positive or negative."},{"Start":"01:50.780 ","End":"01:54.830","Text":"Now, this is not good enough because I want x not 3x,"},{"Start":"01:54.830 ","End":"01:56.605","Text":"so divide by 3."},{"Start":"01:56.605 ","End":"02:01.410","Text":"Dividing by 3, we get x equals 45/3 is 15,"},{"Start":"02:01.410 ","End":"02:06.895","Text":"and indicate that I\u0027ve divided by 3 plus 60k."},{"Start":"02:06.895 ","End":"02:14.070","Text":"This is the general solution for this equation. We\u0027re done here."}],"ID":5410},{"Watched":false,"Name":"Exercise 18","Duration":"4m 59s","ChapterTopicVideoID":5412,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5412.jpeg","UploadDate":"2016-03-10T21:25:43.3700000","DurationForVideoObject":"PT4M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.558","Text":"Here we have a couple of equations to solve each of them involving the tangent,"},{"Start":"00:05.558 ","End":"00:08.250","Text":"and we\u0027ll start with the first one."},{"Start":"00:08.250 ","End":"00:14.639","Text":"The idea is to get it into the form of tangent of something equals tangent of something."},{"Start":"00:14.639 ","End":"00:17.280","Text":"Well, we\u0027re okay on the left-hand side,"},{"Start":"00:17.280 ","End":"00:19.290","Text":"we already have tangent of something."},{"Start":"00:19.290 ","End":"00:26.045","Text":"What we want to do is also on the right-hand side to have tangent of something."},{"Start":"00:26.045 ","End":"00:32.896","Text":"Now, I know that 1 over the square root of 3 is the tangent of 30 degrees,"},{"Start":"00:32.896 ","End":"00:34.700","Text":"because that\u0027s one of those famous ones."},{"Start":"00:34.700 ","End":"00:39.050","Text":"Here we have minus tangent of 30,"},{"Start":"00:39.050 ","End":"00:41.746","Text":"but the minus is not good for me."},{"Start":"00:41.746 ","End":"00:44.825","Text":"Then I remember there\u0027s a formula with the tangent,"},{"Start":"00:44.825 ","End":"00:47.230","Text":"but you can throw the minus inside,"},{"Start":"00:47.230 ","End":"00:49.130","Text":"or if I put it in more formal terms,"},{"Start":"00:49.130 ","End":"00:55.370","Text":"the tangent of minus Alpha is minus the tangent of Alpha."},{"Start":"00:55.370 ","End":"00:59.150","Text":"We\u0027re using it in this direction."},{"Start":"00:59.150 ","End":"01:02.090","Text":"I mean, equality works both ways."},{"Start":"01:02.090 ","End":"01:09.005","Text":"So tangent of x equals tangent of minus 30."},{"Start":"01:09.005 ","End":"01:10.760","Text":"Now, in this form,"},{"Start":"01:10.760 ","End":"01:12.250","Text":"we know how to solve it."},{"Start":"01:12.250 ","End":"01:15.143","Text":"The arguments of the tangent are either equal,"},{"Start":"01:15.143 ","End":"01:17.240","Text":"so that x could be minus 30,"},{"Start":"01:17.240 ","End":"01:20.455","Text":"but it could also be plus a multiple of 180,"},{"Start":"01:20.455 ","End":"01:22.535","Text":"so write that as 180k."},{"Start":"01:22.535 ","End":"01:26.300","Text":"Every 180 degrees the tangent is the same."},{"Start":"01:26.300 ","End":"01:29.375","Text":"This is the answer to Part a."},{"Start":"01:29.375 ","End":"01:32.395","Text":"Now onto Part b."},{"Start":"01:32.395 ","End":"01:35.380","Text":"Notice that in Part b,"},{"Start":"01:35.380 ","End":"01:36.890","Text":"we have a tangent equation,"},{"Start":"01:36.890 ","End":"01:40.040","Text":"but we also have a restriction on x."},{"Start":"01:40.040 ","End":"01:41.713","Text":"We first solve this,"},{"Start":"01:41.713 ","End":"01:45.260","Text":"and then we\u0027ll get the result in terms of k,"},{"Start":"01:45.260 ","End":"01:49.100","Text":"and this restriction will help us find what value or"},{"Start":"01:49.100 ","End":"01:53.529","Text":"values of k are okay to be within this range."},{"Start":"01:53.529 ","End":"01:59.420","Text":"Let\u0027s start by just getting a tangent on the left."},{"Start":"01:59.420 ","End":"02:02.525","Text":"I\u0027ll multiply both sides by square root of 3,"},{"Start":"02:02.525 ","End":"02:04.175","Text":"so tangent of 4x,"},{"Start":"02:04.175 ","End":"02:08.660","Text":"just bring the square root of 3 from a denominator here to here."},{"Start":"02:08.660 ","End":"02:13.935","Text":"Square root of 3 is a famous, I mean,"},{"Start":"02:13.935 ","End":"02:16.295","Text":"one of those values you should memorize,"},{"Start":"02:16.295 ","End":"02:19.985","Text":"and this is the tangent of 60 degrees."},{"Start":"02:19.985 ","End":"02:22.835","Text":"Now we have tangent equals tangent."},{"Start":"02:22.835 ","End":"02:25.970","Text":"We can say that 4x is 60,"},{"Start":"02:25.970 ","End":"02:28.160","Text":"but possibly plus whole number,"},{"Start":"02:28.160 ","End":"02:31.395","Text":"a whole number of semi-circles,"},{"Start":"02:31.395 ","End":"02:35.360","Text":"it\u0027s 180k, 180 is a 1/2 circle."},{"Start":"02:35.360 ","End":"02:38.180","Text":"Since I want x not 4x,"},{"Start":"02:38.180 ","End":"02:40.115","Text":"I divide by 4,"},{"Start":"02:40.115 ","End":"02:43.570","Text":"both sides I divide by 4."},{"Start":"02:44.780 ","End":"02:54.340","Text":"Fine. This is equal to 15 plus 180 over 4 is 45k."},{"Start":"02:55.090 ","End":"03:01.490","Text":"Now I want to relate to this restriction on x."},{"Start":"03:01.490 ","End":"03:04.310","Text":"Rather than solving inequalities,"},{"Start":"03:04.310 ","End":"03:08.390","Text":"I find it\u0027s easier to just write values of what x could"},{"Start":"03:08.390 ","End":"03:13.310","Text":"possibly be by taking different values of k. For example,"},{"Start":"03:13.310 ","End":"03:17.360","Text":"if I take k= 0,"},{"Start":"03:17.360 ","End":"03:19.695","Text":"I get 15 degrees,"},{"Start":"03:19.695 ","End":"03:21.320","Text":"that\u0027s for k= 0."},{"Start":"03:21.320 ","End":"03:22.970","Text":"If I take k= 1,"},{"Start":"03:22.970 ","End":"03:26.435","Text":"I get 15 plus 45 is 60."},{"Start":"03:26.435 ","End":"03:28.670","Text":"If I take k= 2,"},{"Start":"03:28.670 ","End":"03:38.205","Text":"I get 15 plus 90 is 105, and so on."},{"Start":"03:38.205 ","End":"03:41.150","Text":"While I\u0027m doing this, I\u0027m taking a look over here,"},{"Start":"03:41.150 ","End":"03:43.523","Text":"and I see, okay, I\u0027ve already gone over the top."},{"Start":"03:43.523 ","End":"03:45.500","Text":"Let\u0027s try the other direction,"},{"Start":"03:45.500 ","End":"03:48.105","Text":"k is minus 1,"},{"Start":"03:48.105 ","End":"03:53.220","Text":"then I get 15 minus 45 is minus 30."},{"Start":"03:53.220 ","End":"03:58.005","Text":"Got room for more, I got k= -2,"},{"Start":"03:58.005 ","End":"04:05.380","Text":"and then I get 15 minus 90 is minus 75."},{"Start":"04:07.040 ","End":"04:10.715","Text":"I guess I should have really written these over here."},{"Start":"04:10.715 ","End":"04:13.700","Text":"Yeah, that\u0027s better now got room for one more,"},{"Start":"04:13.700 ","End":"04:17.494","Text":"minus 3 although you can see it\u0027s going to go under."},{"Start":"04:17.494 ","End":"04:25.385","Text":"Minus 3 means we\u0027ve got minus 135 plus the 15 we\u0027re still at minus 120,"},{"Start":"04:25.385 ","End":"04:28.160","Text":"and that\u0027s already below the lower limit."},{"Start":"04:28.160 ","End":"04:31.480","Text":"If I want between 90 and 90,"},{"Start":"04:31.480 ","End":"04:37.535","Text":"the ones that are good are from here to here."},{"Start":"04:37.535 ","End":"04:41.360","Text":"Essentially that\u0027s the solution."},{"Start":"04:41.360 ","End":"04:47.405","Text":"I\u0027ve say that x can equal 1 of 4 things; minus 75,"},{"Start":"04:47.405 ","End":"04:52.400","Text":"minus 30, 15, or 60,"},{"Start":"04:52.400 ","End":"04:54.290","Text":"four solutions."},{"Start":"04:54.290 ","End":"04:58.380","Text":"We\u0027re done."}],"ID":5411},{"Watched":false,"Name":"Exercise 19","Duration":"4m 14s","ChapterTopicVideoID":5413,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5413.jpeg","UploadDate":"2016-03-10T21:26:18.4830000","DurationForVideoObject":"PT4M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"We have here a pair of equations to solve each of them involving the tangent function."},{"Start":"00:05.085 ","End":"00:12.135","Text":"We\u0027ll begin with the first tangent 5x=2 plus square root of 3."},{"Start":"00:12.135 ","End":"00:17.490","Text":"What we want is we want tangent of something equals tangent of something."},{"Start":"00:17.490 ","End":"00:21.885","Text":"Tangent 5x equals tangent of what?"},{"Start":"00:21.885 ","End":"00:27.355","Text":"Now this is not one of those famous tangents."},{"Start":"00:27.355 ","End":"00:29.865","Text":"I mean I have encountered it before."},{"Start":"00:29.865 ","End":"00:32.775","Text":"You\u0027ll need the calculator."},{"Start":"00:32.775 ","End":"00:34.380","Text":"If you compute this,"},{"Start":"00:34.380 ","End":"00:38.120","Text":"do plus 2 square root of 3 and then do something"},{"Start":"00:38.120 ","End":"00:42.545","Text":"like shift tangent or inverse tangent however your calculator works,"},{"Start":"00:42.545 ","End":"00:48.005","Text":"then we get that this is the tangent of 75 degrees."},{"Start":"00:48.005 ","End":"00:49.999","Text":"I don\u0027t usually write the degrees,"},{"Start":"00:49.999 ","End":"00:52.560","Text":"a force of habit there."},{"Start":"00:52.660 ","End":"00:57.200","Text":"Now we\u0027re in a familiar situation where we have tangent equals tangent,"},{"Start":"00:57.200 ","End":"01:01.850","Text":"so we compare the arguments, 5x=75,"},{"Start":"01:01.850 ","End":"01:12.005","Text":"but we always add a multiple of 180 because the tangent repeats itself every 180 degrees."},{"Start":"01:12.005 ","End":"01:16.700","Text":"Now all we have to do is divide both sides by 5,"},{"Start":"01:16.700 ","End":"01:19.675","Text":"and we\u0027ve got x=15."},{"Start":"01:19.675 ","End":"01:22.715","Text":"We have to divide this by 5 also,"},{"Start":"01:22.715 ","End":"01:28.400","Text":"180 over 5 is 36."},{"Start":"01:28.400 ","End":"01:30.290","Text":"It\u0027s 15 plus 36k."},{"Start":"01:30.290 ","End":"01:32.270","Text":"That\u0027s the answer."},{"Start":"01:32.270 ","End":"01:35.435","Text":"Now onto Part b."},{"Start":"01:35.435 ","End":"01:38.750","Text":"In Part b, we have tangent of something,"},{"Start":"01:38.750 ","End":"01:41.060","Text":"but here we don\u0027t have tangent of something."},{"Start":"01:41.060 ","End":"01:44.405","Text":"3 over square root of 3 doesn\u0027t look familiar."},{"Start":"01:44.405 ","End":"01:46.400","Text":"It doesn\u0027t seem to be one of the famous ones,"},{"Start":"01:46.400 ","End":"01:51.080","Text":"but there is a standard algebraic trick that"},{"Start":"01:51.080 ","End":"01:56.030","Text":"if I write 3 as square root of 3 times square root of 3,"},{"Start":"01:56.030 ","End":"01:57.875","Text":"just a tricky use often,"},{"Start":"01:57.875 ","End":"02:03.210","Text":"then 3 over the square root of 3 is square root of 3,"},{"Start":"02:03.210 ","End":"02:06.030","Text":"square root of 3 over square root of 3."},{"Start":"02:06.030 ","End":"02:07.530","Text":"One of these cancels,"},{"Start":"02:07.530 ","End":"02:10.275","Text":"so we\u0027re just left with square root of 3."},{"Start":"02:10.275 ","End":"02:28.890","Text":"Back here, we have the tangent(-3x)"},{"Start":"02:28.890 ","End":"02:31.820","Text":"is equal to minus. Let me say this. Let\u0027s just rewrite this"},{"Start":"02:31.820 ","End":"02:36.030","Text":"as the square root of 3 as we did here."},{"Start":"02:36.030 ","End":"02:38.950","Text":"This is minus the tangent(60)"},{"Start":"02:38.950 ","End":"02:43.550","Text":"degrees because the square root of 3 is a famous one that you have to remember."},{"Start":"02:43.550 ","End":"02:45.665","Text":"Of course you can still do it on the calculator."},{"Start":"02:45.665 ","End":"02:48.679","Text":"Anyway, it belongs to 60 degrees."},{"Start":"02:48.679 ","End":"02:50.750","Text":"We have this minus here."},{"Start":"02:50.750 ","End":"02:57.660","Text":"Let\u0027s put the minus inside."},{"Start":"02:57.660 ","End":"03:00.445","Text":"Let me quote the formula I\u0027m going to use."},{"Start":"03:00.445 ","End":"03:04.735","Text":"The tangent of minus Alpha is minus"},{"Start":"03:04.735 ","End":"03:07.030","Text":"the tangent of Alpha which basically means you"},{"Start":"03:07.030 ","End":"03:09.445","Text":"can take the minus in and out of the tangent."},{"Start":"03:09.445 ","End":"03:11.410","Text":"I actually have a choice of two things to do."},{"Start":"03:11.410 ","End":"03:15.620","Text":"We can either put the minus inside or we can take this minus outside."},{"Start":"03:15.620 ","End":"03:19.940","Text":"On second thought, it might be easier to take this minus outside."},{"Start":"03:20.280 ","End":"03:29.960","Text":"We get that minus tangent(3x)=minus tangent(60)."},{"Start":"03:30.980 ","End":"03:34.175","Text":"The minuses just cancel."},{"Start":"03:34.175 ","End":"03:40.285","Text":"Just put a line through them instead of writing an extra line of equation."},{"Start":"03:40.285 ","End":"03:42.335","Text":"Now I have the form I like;"},{"Start":"03:42.335 ","End":"03:45.425","Text":"tangent something equals tangent something."},{"Start":"03:45.425 ","End":"03:50.090","Text":"Those two somethings are equal or at least they might be equal,"},{"Start":"03:50.090 ","End":"03:54.064","Text":"but they could also differ by a multiple of 180 degrees."},{"Start":"03:54.064 ","End":"03:58.440","Text":"That\u0027s where we write the plus 180k here,"},{"Start":"03:58.440 ","End":"04:00.630","Text":"because I want just x not 3x,"},{"Start":"04:00.630 ","End":"04:06.480","Text":"so I\u0027ll divide by 3 and I get x=20 and"},{"Start":"04:06.480 ","End":"04:14.470","Text":"180 over 3 is 20 plus 60k and that\u0027s the answer for Part b. We\u0027re done."}],"ID":5412},{"Watched":false,"Name":"Exercise 20","Duration":"12m 40s","ChapterTopicVideoID":16679,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/16679.jpeg","UploadDate":"2019-01-31T18:41:54.2830000","DurationForVideoObject":"PT12M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:05.630 ","End":"00:11.745","Text":"We have here a pair of trigonometric equations to solve with the tangent."},{"Start":"00:11.745 ","End":"00:17.993","Text":"The first one has a restriction on the domain of x where x could be,"},{"Start":"00:17.993 ","End":"00:20.655","Text":"and the second one is unrestricted."},{"Start":"00:20.655 ","End":"00:22.875","Text":"We\u0027ll begin with the first one,"},{"Start":"00:22.875 ","End":"00:27.945","Text":"the tangent^2 of 4x=1/3."},{"Start":"00:27.945 ","End":"00:30.100","Text":"First of all, we focus on the equation pattern,"},{"Start":"00:30.100 ","End":"00:31.560","Text":"and later we\u0027ll see what to do about"},{"Start":"00:31.560 ","End":"00:36.150","Text":"the restriction that x is in the interval from 0-180."},{"Start":"00:36.150 ","End":"00:44.640","Text":"Now remember the tangent^2 is the same as tangent^2 when you put the 2 on the outside."},{"Start":"00:44.640 ","End":"00:46.280","Text":"This is just a shortcut way,"},{"Start":"00:46.280 ","End":"00:48.245","Text":"a shorthand way of writing this."},{"Start":"00:48.245 ","End":"00:51.400","Text":"We have that this=1/3."},{"Start":"00:51.400 ","End":"00:56.600","Text":"Now remember whenever we have something squared,"},{"Start":"00:56.600 ","End":"00:58.789","Text":"when we solve it, we get 2 solutions."},{"Start":"00:58.789 ","End":"01:03.935","Text":"For example, if I said that a^2 is equal to 9,"},{"Start":"01:03.935 ","End":"01:07.230","Text":"then I say that a is plus,"},{"Start":"01:07.230 ","End":"01:10.085","Text":"or minus the square root of 9."},{"Start":"01:10.085 ","End":"01:13.880","Text":"In other words, a is plus, or minus 3."},{"Start":"01:13.880 ","End":"01:16.240","Text":"You have to take both when you have a square."},{"Start":"01:16.240 ","End":"01:18.275","Text":"So same idea here,"},{"Start":"01:18.275 ","End":"01:22.130","Text":"we\u0027ve got tangent of 4x is plus,"},{"Start":"01:22.130 ","End":"01:25.555","Text":"or minus the square root of 1/3,"},{"Start":"01:25.555 ","End":"01:29.315","Text":"because the square root of something over something"},{"Start":"01:29.315 ","End":"01:33.500","Text":"like a over b is the square root of a over the square root of b."},{"Start":"01:33.500 ","End":"01:35.870","Text":"Didn\u0027t really need to remind you of that."},{"Start":"01:35.870 ","End":"01:38.779","Text":"We can write this as plus,"},{"Start":"01:38.779 ","End":"01:42.315","Text":"or minus 1 over the square root of 3."},{"Start":"01:42.315 ","End":"01:45.210","Text":"Square root of 1 is 1 and the square root of 3 is square root of 3."},{"Start":"01:45.210 ","End":"01:48.120","Text":"I really have 2 possibilities;"},{"Start":"01:48.120 ","End":"01:57.120","Text":"I could have tangent of 4x=1 over square root of 3."},{"Start":"01:57.120 ","End":"01:58.725","Text":"That\u0027s one possibility,"},{"Start":"01:58.725 ","End":"02:00.570","Text":"and also that here."},{"Start":"02:00.570 ","End":"02:08.335","Text":"The other possibility will be the tangent 4x is minus 1 over square root of 3."},{"Start":"02:08.335 ","End":"02:11.145","Text":"Let\u0027s go with this one first."},{"Start":"02:11.145 ","End":"02:15.995","Text":"Tangent of 4x, we want to be equal to tangent of something."},{"Start":"02:15.995 ","End":"02:19.220","Text":"Again, 1 over square root of 3 is one of those famous ones that"},{"Start":"02:19.220 ","End":"02:23.375","Text":"you really ought to memorize and if not, use the calculator."},{"Start":"02:23.375 ","End":"02:26.330","Text":"This is the tangent of 30 degrees."},{"Start":"02:26.330 ","End":"02:28.490","Text":"When we have tangent equals tangent,"},{"Start":"02:28.490 ","End":"02:30.400","Text":"we compare the 2 things,"},{"Start":"02:30.400 ","End":"02:37.826","Text":"but the equality is up to multiples of 180 so we add the 180k here."},{"Start":"02:37.826 ","End":"02:39.510","Text":"Because I want just x, not 4x,"},{"Start":"02:39.510 ","End":"02:42.990","Text":"so I\u0027m going to divide by 4 and get x equals,"},{"Start":"02:42.990 ","End":"02:47.310","Text":"let\u0027s see, 30 over 4 is 7-and-1/2,"},{"Start":"02:47.310 ","End":"02:51.584","Text":"so it\u0027s 7-and-1/2 plus"},{"Start":"02:51.584 ","End":"02:59.640","Text":"180 over 4 is 45k."},{"Start":"02:59.640 ","End":"03:01.425","Text":"That\u0027s the general solution."},{"Start":"03:01.425 ","End":"03:03.500","Text":"But coming back to this,"},{"Start":"03:03.500 ","End":"03:09.170","Text":"we do have this restriction that x has to be between 0 and 180."},{"Start":"03:09.170 ","End":"03:12.530","Text":"What we need to do here is choose the k"},{"Start":"03:12.530 ","End":"03:18.875","Text":"or values of k that make this thing fall inside this range."},{"Start":"03:18.875 ","End":"03:21.080","Text":"You can do it with solving inequalities,"},{"Start":"03:21.080 ","End":"03:26.990","Text":"but I find it\u0027s easier to just write down various values with different values of k,"},{"Start":"03:26.990 ","End":"03:28.370","Text":"and see what it could be."},{"Start":"03:28.370 ","End":"03:31.100","Text":"For example, if I take k=0,"},{"Start":"03:31.100 ","End":"03:34.960","Text":"I get x to be 7-and-1/2."},{"Start":"03:34.960 ","End":"03:38.610","Text":"If I take k=1,"},{"Start":"03:38.610 ","End":"03:45.034","Text":"I\u0027ve got 7.5 plus 45, that\u0027s 52-and-1/2."},{"Start":"03:45.034 ","End":"03:49.640","Text":"Now I\u0027m looking all the time here and I see I\u0027ve got room for more."},{"Start":"03:49.640 ","End":"03:52.130","Text":"Let\u0027s just go the other way first."},{"Start":"03:52.130 ","End":"03:55.270","Text":"If I took k=-1."},{"Start":"03:55.270 ","End":"03:57.890","Text":"I\u0027ll just write the values here to remind myself."},{"Start":"03:57.890 ","End":"03:59.150","Text":"This was k=0."},{"Start":"03:59.150 ","End":"04:00.920","Text":"This is k=1."},{"Start":"04:00.920 ","End":"04:03.245","Text":"Here we\u0027ll have k=-1."},{"Start":"04:03.245 ","End":"04:07.620","Text":"It would be 7.5 minus 45,"},{"Start":"04:07.620 ","End":"04:12.110","Text":"so that would be minus 37-and-1/2."},{"Start":"04:12.110 ","End":"04:14.975","Text":"We see we\u0027re already below the lower limit,"},{"Start":"04:14.975 ","End":"04:18.710","Text":"below 0, so no good continuing this direction."},{"Start":"04:18.710 ","End":"04:21.200","Text":"We continue in the other direction."},{"Start":"04:21.200 ","End":"04:25.160","Text":"Try k=2, that gives us"},{"Start":"04:25.160 ","End":"04:31.376","Text":"90 plus 7-and-1/2 is 97-and-1/2."},{"Start":"04:31.376 ","End":"04:32.390","Text":"Room for more,"},{"Start":"04:32.390 ","End":"04:42.630","Text":"k=3 gives us 135 plus 7-and-1/2 is 142-and-1/2."},{"Start":"04:45.010 ","End":"04:47.770","Text":"Then if I take k=4,"},{"Start":"04:47.770 ","End":"04:54.050","Text":"4 times 45 is 180,"},{"Start":"04:54.050 ","End":"04:57.990","Text":"I get 187-and-1/2,"},{"Start":"04:57.990 ","End":"05:00.720","Text":"so that\u0027s no good,"},{"Start":"05:00.720 ","End":"05:05.050","Text":"so really only from here to here,"},{"Start":"05:05.050 ","End":"05:09.260","Text":"the good values that fall in this range."},{"Start":"05:09.260 ","End":"05:12.035","Text":"They did a copy-paste job from here to here."},{"Start":"05:12.035 ","End":"05:17.660","Text":"But, that\u0027s not all because don\u0027t forget that we had another branch to solve."},{"Start":"05:17.660 ","End":"05:20.270","Text":"Let me just erase what I don\u0027t need."},{"Start":"05:20.270 ","End":"05:24.170","Text":"Just put a little dividing line here because we\u0027re"},{"Start":"05:24.170 ","End":"05:27.740","Text":"going to take this 1/2 now and this other possibility,"},{"Start":"05:27.740 ","End":"05:37.625","Text":"so tangent of 4x could be instead of tangent of 30 minus tangent of 30."},{"Start":"05:37.625 ","End":"05:42.435","Text":"Then we do our usual trick of putting the minus inside."},{"Start":"05:42.435 ","End":"05:46.780","Text":"One of the formulas of the tangent says that you can do that,"},{"Start":"05:46.780 ","End":"05:56.260","Text":"so we have the tangent of 4x is equal to tangent of minus 30."},{"Start":"05:56.260 ","End":"06:01.470","Text":"So 4x this time is minus 30,"},{"Start":"06:01.470 ","End":"06:05.105","Text":"instead of 30, but also plus 180k."},{"Start":"06:05.105 ","End":"06:16.120","Text":"This gives us that x equals minus 7-and-1/2 plus 45k."},{"Start":"06:16.940 ","End":"06:25.005","Text":"Continuing as before,"},{"Start":"06:25.005 ","End":"06:28.565","Text":"we just want the values that are within this range."},{"Start":"06:28.565 ","End":"06:32.090","Text":"If I say k=0,"},{"Start":"06:32.090 ","End":"06:34.770","Text":"obviously I\u0027ll get something negative that\u0027s already no good,"},{"Start":"06:34.770 ","End":"06:38.068","Text":"so I really have to start with k=1,"},{"Start":"06:38.068 ","End":"06:47.610","Text":"and then we get 45 minus 7-and-1/2 is 37-and-1/2."},{"Start":"06:47.610 ","End":"06:49.500","Text":"Then if k=2,"},{"Start":"06:49.500 ","End":"06:56.520","Text":"we get 90 minus 7-and-1/2, which is 82-and-1/2."},{"Start":"06:56.520 ","End":"07:04.275","Text":"If k=3, we\u0027ll get 135 minus 7-and-1/2,"},{"Start":"07:04.275 ","End":"07:10.410","Text":"which would be 127-and-1/2,"},{"Start":"07:10.410 ","End":"07:13.875","Text":"and then if k=4,"},{"Start":"07:13.875 ","End":"07:22.110","Text":"we get 180 minus 7-and-1/2, which is 172-and-1/2."},{"Start":"07:22.110 ","End":"07:25.020","Text":"Obviously, when k=5,"},{"Start":"07:25.020 ","End":"07:31.425","Text":"it\u0027s going to go over the 180 and these are the 4 values from here."},{"Start":"07:31.425 ","End":"07:35.100","Text":"From this one, we get these 4 values of x."},{"Start":"07:35.100 ","End":"07:42.305","Text":"Finally, we have to actually combine the 2 lists. Let me do that."},{"Start":"07:42.305 ","End":"07:44.270","Text":"You don\u0027t have to put them in order,"},{"Start":"07:44.270 ","End":"07:46.280","Text":"but I\u0027d like to actually put them in order,"},{"Start":"07:46.280 ","End":"07:51.035","Text":"and say what the values of x could be from 0-180."},{"Start":"07:51.035 ","End":"07:53.515","Text":"We could have 7-and-1/2,"},{"Start":"07:53.515 ","End":"07:56.700","Text":"then we could have 37-and-1/2,"},{"Start":"07:56.700 ","End":"08:02.160","Text":"we could have 52-and-1/2,"},{"Start":"08:02.160 ","End":"08:06.435","Text":"82-and-1/2, see one from here, one from here,"},{"Start":"08:06.435 ","End":"08:15.270","Text":"97-and-1/2, 127-and-1/2,"},{"Start":"08:15.270 ","End":"08:24.240","Text":"142-and-1/2, and finally 172-and-1/2."},{"Start":"08:24.240 ","End":"08:26.250","Text":"We had 1,2,3,4 solutions here,"},{"Start":"08:26.250 ","End":"08:29.520","Text":"4 solutions here, actually 8 possible solution."},{"Start":"08:29.520 ","End":"08:32.150","Text":"It\u0027s a lot of solutions,"},{"Start":"08:32.150 ","End":"08:33.440","Text":"and we didn\u0027t have to do this."},{"Start":"08:33.440 ","End":"08:35.240","Text":"You actually can write it in another way."},{"Start":"08:35.240 ","End":"08:39.890","Text":"You could say that either x is minus"},{"Start":"08:39.890 ","End":"08:45.706","Text":"7-and-1/2 plus 45k and then you write in brackets where k=1,"},{"Start":"08:45.706 ","End":"08:53.600","Text":"2, 3 or 4 or x=7-and-1/2 plus 45k,"},{"Start":"08:53.600 ","End":"08:57.770","Text":"where k equals, I didn\u0027t write it here."},{"Start":"08:57.770 ","End":"09:00.514","Text":"This was k=0, 1,"},{"Start":"09:00.514 ","End":"09:03.480","Text":"2, and 3."},{"Start":"09:03.480 ","End":"09:07.303","Text":"Then you say, x is this where k=0,"},{"Start":"09:07.303 ","End":"09:09.600","Text":"1, 2 or 3."},{"Start":"09:09.600 ","End":"09:12.885","Text":"Then I could highlight"},{"Start":"09:12.885 ","End":"09:19.445","Text":"this part and I could highlight this part and I could say x is either this or this."},{"Start":"09:19.445 ","End":"09:21.570","Text":"But if 8 is not so bad,"},{"Start":"09:21.570 ","End":"09:23.090","Text":"you could leave it like this."},{"Start":"09:23.090 ","End":"09:24.920","Text":"You don\u0027t have to also put them in order."},{"Start":"09:24.920 ","End":"09:26.565","Text":"I just like to do that."},{"Start":"09:26.565 ","End":"09:30.020","Text":"On to Part B now."},{"Start":"09:30.020 ","End":"09:34.950","Text":"Once again we have something squared."},{"Start":"09:35.350 ","End":"09:40.190","Text":"Once again we use the square root and we say that"},{"Start":"09:40.190 ","End":"09:46.890","Text":"tangent of 3x is either plus or minus the square root of 4,"},{"Start":"09:46.890 ","End":"09:49.525","Text":"and that splits up into 2 possibilities;"},{"Start":"09:49.525 ","End":"09:58.730","Text":"tangent 3x=2 or tangent 3x=-2 and we\u0027ll solve each one."},{"Start":"09:58.730 ","End":"10:02.910","Text":"Now tangent 3x=2,"},{"Start":"10:02.910 ","End":"10:09.360","Text":"I want to write it as tangent 3x equals tangent of something."},{"Start":"10:09.360 ","End":"10:13.540","Text":"2 is not some well-known or famous tangent,"},{"Start":"10:13.540 ","End":"10:15.895","Text":"so we just do it on the calculator."},{"Start":"10:15.895 ","End":"10:26.890","Text":"I make it 63.435 to round it to 3 decimal places so we know this is approximate."},{"Start":"10:27.320 ","End":"10:38.140","Text":"Then we take the tangent out and we say that 3x=63.435,"},{"Start":"10:38.140 ","End":"10:43.600","Text":"but you have to add multiples of 180, it\u0027s 180k."},{"Start":"10:43.600 ","End":"10:49.705","Text":"Finally, divide by 3 and we get that x is equal to,"},{"Start":"10:49.705 ","End":"10:52.908","Text":"let\u0027s see, divide this by 3,"},{"Start":"10:52.908 ","End":"11:00.322","Text":"21.145 plus, divide this by 3,"},{"Start":"11:00.322 ","End":"11:06.420","Text":"60k, where k is any integer."},{"Start":"11:06.420 ","End":"11:09.995","Text":"Now on this one, similar,"},{"Start":"11:09.995 ","End":"11:15.480","Text":"we get the tangent 3x,"},{"Start":"11:15.480 ","End":"11:21.830","Text":"minus 2 would be minus the tangent of 63.435."},{"Start":"11:21.830 ","End":"11:24.470","Text":"No need to use a calculator again."},{"Start":"11:24.470 ","End":"11:29.095","Text":"We use our usual trick with the tangent of putting the minus inside."},{"Start":"11:29.095 ","End":"11:39.875","Text":"This can be written as tangent of minus 63.435."},{"Start":"11:39.875 ","End":"11:45.433","Text":"Now once again, we throw out the tangent and say these things are equal,"},{"Start":"11:45.433 ","End":"11:49.820","Text":"but up to multiples of 180 so this time it\u0027s equal"},{"Start":"11:49.820 ","End":"11:58.445","Text":"to minus 63.435 again plus 180k."},{"Start":"11:58.445 ","End":"12:00.500","Text":"Once again, we divide by 3,"},{"Start":"12:00.500 ","End":"12:02.360","Text":"just like here, we divide it by 3,"},{"Start":"12:02.360 ","End":"12:11.685","Text":"here we\u0027ll divide by 3 and we get x equals minus 21.145 plus 60k."},{"Start":"12:11.685 ","End":"12:14.995","Text":"We say x is this or x equals this."},{"Start":"12:14.995 ","End":"12:17.735","Text":"Any one of these is a solution."},{"Start":"12:17.735 ","End":"12:28.295","Text":"You can, if you like, combine the answer and say x equals plus or minus 21.145 plus 60k."},{"Start":"12:28.295 ","End":"12:30.380","Text":"This is the general answer,"},{"Start":"12:30.380 ","End":"12:32.480","Text":"meaning you can do what you like."},{"Start":"12:32.480 ","End":"12:37.730","Text":"Take plus or minus take any value of k. That\u0027ll give you all the solutions."},{"Start":"12:37.730 ","End":"12:40.920","Text":"That\u0027s the end of this exercise."}],"ID":17438},{"Watched":false,"Name":"Exercise 21","Duration":"5m 5s","ChapterTopicVideoID":5415,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5415.jpeg","UploadDate":"2016-03-10T21:28:41.4200000","DurationForVideoObject":"PT5M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"We have here a pair of trigonometric equations to solve,"},{"Start":"00:04.050 ","End":"00:06.720","Text":"all involving the tangent."},{"Start":"00:06.720 ","End":"00:09.840","Text":"In 1 of them, we have a restriction on x,"},{"Start":"00:09.840 ","End":"00:13.820","Text":"that x should be between 0 and 90,"},{"Start":"00:13.820 ","End":"00:16.425","Text":"in the first 1 there is no restriction."},{"Start":"00:16.425 ","End":"00:19.200","Text":"So I\u0027ll start with the first 1."},{"Start":"00:19.200 ","End":"00:21.135","Text":"It\u0027s already in the form that we like"},{"Start":"00:21.135 ","End":"00:24.930","Text":"that tangent something equals tangent of something,"},{"Start":"00:24.930 ","End":"00:32.400","Text":"and so we say that the arguments are equal up to multiples of 180."},{"Start":"00:32.400 ","End":"00:34.030","Text":"In other words,"},{"Start":"00:35.690 ","End":"00:45.060","Text":"x-30=3x+180k."},{"Start":"00:45.060 ","End":"00:47.320","Text":"Let\u0027s see what we get."},{"Start":"00:49.010 ","End":"00:58.370","Text":"Let\u0027s bring the x\u0027s to the left and the numbers to the right,"},{"Start":"00:58.370 ","End":"01:05.000","Text":"we have -2x=30+180k,"},{"Start":"01:05.000 ","End":"01:10.940","Text":"and then"},{"Start":"01:10.940 ","End":"01:16.160","Text":"dividing by -2 both sides,"},{"Start":"01:16.160 ","End":"01:26.915","Text":"we get x= -15-90k."},{"Start":"01:26.915 ","End":"01:34.250","Text":"I mentioned before that if you have a minus k,"},{"Start":"01:34.250 ","End":"01:35.915","Text":"you can replace it with a plus k,"},{"Start":"01:35.915 ","End":"01:37.715","Text":"and that\u0027s what we often do."},{"Start":"01:37.715 ","End":"01:42.660","Text":"So I could write this as -15+90k,"},{"Start":"01:43.300 ","End":"01:52.145","Text":"but you could also just leave it with the minus if you\u0027re not sure about that."},{"Start":"01:52.145 ","End":"01:55.310","Text":"Because minus k is just a general whole number,"},{"Start":"01:55.310 ","End":"01:59.145","Text":"just as much as k either way."},{"Start":"01:59.145 ","End":"02:01.320","Text":"Okay, that\u0027s Part a."},{"Start":"02:01.320 ","End":"02:04.035","Text":"That was fairly straightforward."},{"Start":"02:04.035 ","End":"02:07.300","Text":"Let\u0027s go on to Part b."},{"Start":"02:10.390 ","End":"02:15.650","Text":"What we have here is this time we have a restriction on the x,"},{"Start":"02:15.650 ","End":"02:19.100","Text":"but we solve the general equation first and then worry about getting"},{"Start":"02:19.100 ","End":"02:23.210","Text":"x to be between 0 and 90, in this case."},{"Start":"02:23.210 ","End":"02:25.610","Text":"So tangent equals tangent."},{"Start":"02:25.610 ","End":"02:35.465","Text":"5x+15=95, but have to add multiples of 180."},{"Start":"02:35.465 ","End":"02:43.125","Text":"That\u0027s 180 times some general whole number. Let\u0027s see."},{"Start":"02:43.125 ","End":"02:47.810","Text":"Leave the x\u0027s on the left and numbers on the right."},{"Start":"02:47.810 ","End":"02:50.850","Text":"So that would be 95-15"},{"Start":"02:52.340 ","End":"02:58.140","Text":"is 80+180k."},{"Start":"02:58.140 ","End":"03:02.280","Text":"Then divide both sides by 5."},{"Start":"03:02.280 ","End":"03:07.470","Text":"So we have x equals 80 over 5 is 16,"},{"Start":"03:07.470 ","End":"03:11.535","Text":"180 over 5 is 36."},{"Start":"03:11.535 ","End":"03:16.470","Text":"A general thing would be 16 plus 36k."},{"Start":"03:16.470 ","End":"03:20.960","Text":"However, we are not general here because we have"},{"Start":"03:20.960 ","End":"03:25.160","Text":"this condition that we have to be between 0 and 90."},{"Start":"03:25.160 ","End":"03:29.400","Text":"The easiest thing to do is just choose values of"},{"Start":"03:29.400 ","End":"03:35.630","Text":"k. Try various ones and seeing which ones keep us inside the range."},{"Start":"03:35.630 ","End":"03:37.610","Text":"Let\u0027s start with k=0."},{"Start":"03:37.610 ","End":"03:39.875","Text":"Is the easiest to try."},{"Start":"03:39.875 ","End":"03:43.420","Text":"If k is 0, I get x=16."},{"Start":"03:43.420 ","End":"03:47.145","Text":"So that\u0027s 1 of my possible values for k=0."},{"Start":"03:47.145 ","End":"03:49.940","Text":"It happens to be inside the range."},{"Start":"03:49.940 ","End":"03:52.629","Text":"If I go to the left,"},{"Start":"03:52.629 ","End":"03:55.670","Text":"I can see it\u0027s going to be outside the range. Let\u0027s just try it."},{"Start":"03:55.670 ","End":"03:57.560","Text":"If I say k-1,"},{"Start":"03:57.560 ","End":"03:59.630","Text":"I\u0027ve got 16-36,"},{"Start":"03:59.630 ","End":"04:01.625","Text":"which is minus 20,"},{"Start":"04:01.625 ","End":"04:04.100","Text":"so that\u0027s now good."},{"Start":"04:04.100 ","End":"04:06.080","Text":"So let\u0027s go only to the right,"},{"Start":"04:06.080 ","End":"04:08.360","Text":"and we\u0027ll let k=1 next."},{"Start":"04:08.360 ","End":"04:13.490","Text":"Then 16+36 is, let\u0027s see,"},{"Start":"04:13.490 ","End":"04:16.500","Text":"that would be 52."},{"Start":"04:18.160 ","End":"04:22.610","Text":"If I take k=2,"},{"Start":"04:22.610 ","End":"04:32.140","Text":"then I have 72+16 is 88 but still good."},{"Start":"04:32.140 ","End":"04:33.570","Text":"It\u0027s still within the range."},{"Start":"04:33.570 ","End":"04:35.520","Text":"Obviously if I take k=3,"},{"Start":"04:35.520 ","End":"04:39.080","Text":"I\u0027m going to overflow just like I did with minus 1."},{"Start":"04:39.080 ","End":"04:45.090","Text":"So really, these are the only three solutions possible."},{"Start":"04:45.890 ","End":"04:51.150","Text":"In summary, that\u0027s what I say that there are three possible values of x,"},{"Start":"04:51.150 ","End":"04:56.700","Text":"16, 52 or 88."},{"Start":"04:56.700 ","End":"05:00.800","Text":"Everything else is either too high or too low."},{"Start":"05:00.800 ","End":"05:02.300","Text":"So that\u0027s it."},{"Start":"05:02.300 ","End":"05:04.770","Text":"We\u0027re done with Part B, so we\u0027re done."}],"ID":5414},{"Watched":false,"Name":"Exercise 22","Duration":"9m 2s","ChapterTopicVideoID":5416,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5416.jpeg","UploadDate":"2016-03-10T21:29:56.5470000","DurationForVideoObject":"PT9M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"Here we have 2 trigonometric equations to solve,"},{"Start":"00:03.690 ","End":"00:05.340","Text":"and in each case,"},{"Start":"00:05.340 ","End":"00:08.190","Text":"we\u0027re going to use the technique of substitution."},{"Start":"00:08.190 ","End":"00:10.215","Text":"Let\u0027s start with the first."},{"Start":"00:10.215 ","End":"00:15.120","Text":"If we look at it, the only way x appears is through cosine x."},{"Start":"00:15.120 ","End":"00:19.110","Text":"So the obvious thing to do is to substitute cosine x."},{"Start":"00:19.110 ","End":"00:24.450","Text":"Let\u0027s call it t. If I substitute,"},{"Start":"00:24.450 ","End":"00:26.670","Text":"what I get is,"},{"Start":"00:26.670 ","End":"00:33.630","Text":"this becomes 2t^2 because cosine(x)^2 means cosine(x)0^2."},{"Start":"00:33.630 ","End":"00:39.275","Text":"Here I have 3 times t - 2 = 0."},{"Start":"00:39.275 ","End":"00:45.485","Text":"This is a straightforward quadratic equation in t. Let\u0027s quickly solve that."},{"Start":"00:45.485 ","End":"00:47.345","Text":"We\u0027ve got the t is,"},{"Start":"00:47.345 ","End":"00:49.700","Text":"hope you remember your quadratics."},{"Start":"00:49.700 ","End":"00:59.780","Text":"-b + or - the √ b^2 - 4ac,"},{"Start":"00:59.780 ","End":"01:02.790","Text":"all this over 2a."},{"Start":"01:03.370 ","End":"01:12.780","Text":"This is = to c. -3 + or -."},{"Start":"01:12.780 ","End":"01:15.705","Text":"Now, what do we have here?"},{"Start":"01:15.705 ","End":"01:20.330","Text":"4.2.2 is 16 and there\u0027s a -,"},{"Start":"01:20.330 ","End":"01:22.235","Text":"-, so it\u0027s +."},{"Start":"01:22.235 ","End":"01:29.915","Text":"So it\u0027s the √25/2.2 is 4."},{"Start":"01:29.915 ","End":"01:33.455","Text":"The √25 is 5."},{"Start":"01:33.455 ","End":"01:43.510","Text":"So I\u0027ve got -3 + 5/4 and the other possibility is -3, -5/4."},{"Start":"01:43.510 ","End":"01:46.770","Text":"This one gives me 2/4,"},{"Start":"01:46.770 ","End":"01:48.165","Text":"which is a 1/2."},{"Start":"01:48.165 ","End":"01:54.330","Text":"This one gives me -8/4 which is -2."},{"Start":"01:54.330 ","End":"02:00.040","Text":"Now, these are the 1 values I get for t,"},{"Start":"02:00.110 ","End":"02:03.615","Text":"either a 1/2 or -2."},{"Start":"02:03.615 ","End":"02:07.775","Text":"But I don\u0027t want t, I want x."},{"Start":"02:07.775 ","End":"02:13.565","Text":"What I have is that cosine x"},{"Start":"02:13.565 ","End":"02:19.790","Text":"is = either 1.5 or -2."},{"Start":"02:19.790 ","End":"02:23.915","Text":"Now, I can immediately rule out this 1."},{"Start":"02:23.915 ","End":"02:29.645","Text":"The reason is that cosine of anything is always between 1 and -1."},{"Start":"02:29.645 ","End":"02:31.580","Text":"But, this is outside the range."},{"Start":"02:31.580 ","End":"02:36.660","Text":"There is no angle whose cosine is -2."},{"Start":"02:36.660 ","End":"02:40.465","Text":"We only have cosine x= 1/2."},{"Start":"02:40.465 ","End":"02:42.530","Text":"Solve that over here."},{"Start":"02:42.530 ","End":"02:45.560","Text":"So cosine x= 1/2."},{"Start":"02:45.560 ","End":"02:48.590","Text":"Remember the way we do this is we want cosine=cosine."},{"Start":"02:48.590 ","End":"02:55.910","Text":"1/2 is the cosine of 60."},{"Start":"02:55.910 ","End":"03:02.240","Text":"You could do this on the calculator if you don\u0027t remember the important angles."},{"Start":"03:02.240 ","End":"03:11.480","Text":"That means that the general solution is that x is, there\u0027s 2 possibilities."},{"Start":"03:11.480 ","End":"03:14.510","Text":"It\u0027s either equal to 60 as here,"},{"Start":"03:14.510 ","End":"03:17.690","Text":"plus multiples of a whole circle."},{"Start":"03:17.690 ","End":"03:20.605","Text":"Rewrite that as 360k,"},{"Start":"03:20.605 ","End":"03:24.560","Text":"where k is any integer positive, or negative."},{"Start":"03:24.560 ","End":"03:30.440","Text":"The other possibility, because cosine 60 is the same as cosine of -60,"},{"Start":"03:30.440 ","End":"03:36.035","Text":"is that x is -60 + 360k."},{"Start":"03:36.035 ","End":"03:41.105","Text":"So this is the complete solution for x,"},{"Start":"03:41.105 ","End":"03:44.520","Text":"and on to part b."},{"Start":"03:46.120 ","End":"03:52.130","Text":"In part b, we see that everything is written in terms of sine x,"},{"Start":"03:52.130 ","End":"03:57.300","Text":"and don\u0027t forget that sine(x)^2 means sine(x)0^2."},{"Start":"03:58.220 ","End":"04:04.770","Text":"We can substitute t = sine x."},{"Start":"04:04.770 ","End":"04:13.515","Text":"Then we get 2t^2 + 3t + 1= 0."},{"Start":"04:13.515 ","End":"04:17.680","Text":"Once again, the quadratic equation. Let\u0027s solve it."},{"Start":"04:17.680 ","End":"04:27.225","Text":"t= -3 + or - the √ b^2 - 4ac."},{"Start":"04:27.225 ","End":"04:30.879","Text":"9 -"},{"Start":"04:33.830 ","End":"04:39.470","Text":"4.2.1/2a."},{"Start":"04:39.470 ","End":"04:41.360","Text":"Let\u0027s see what this gives us."},{"Start":"04:41.360 ","End":"04:43.835","Text":"This gives us 9 - 8,"},{"Start":"04:43.835 ","End":"04:46.995","Text":"which is 1,"},{"Start":"04:46.995 ","End":"04:48.300","Text":"√ 1 is 1,"},{"Start":"04:48.300 ","End":"04:56.685","Text":"it\u0027s - 3 + or - 1/4 and there\u0027s 2 possibilities here."},{"Start":"04:56.685 ","End":"04:58.725","Text":"We take the +,"},{"Start":"04:58.725 ","End":"05:03.815","Text":"- 3 + 1 is -2 over 4 is -1/2."},{"Start":"05:03.815 ","End":"05:06.470","Text":"If I take the minus sign,"},{"Start":"05:06.470 ","End":"05:12.180","Text":"-3, -1 is -4 over 4 is -1."},{"Start":"05:12.610 ","End":"05:19.825","Text":"These are the 2 solutions to t. But,"},{"Start":"05:19.825 ","End":"05:22.850","Text":"these are not the solutions for x,"},{"Start":"05:22.850 ","End":"05:29.580","Text":"so we have to solve sine."},{"Start":"05:29.580 ","End":"05:32.750","Text":"Sine x, which is t,"},{"Start":"05:32.750 ","End":"05:38.665","Text":"has got to equal either -1/2 or -1."},{"Start":"05:38.665 ","End":"05:42.890","Text":"They\u0027re both in range of up to 1 inclusive,"},{"Start":"05:42.890 ","End":"05:44.750","Text":"and down to -1."},{"Start":"05:44.750 ","End":"05:47.520","Text":"We have to allow for both of them."},{"Start":"05:47.520 ","End":"05:49.095","Text":"Let\u0027s solve each 1 of them."},{"Start":"05:49.095 ","End":"05:56.535","Text":"Then, first of all, let\u0027s take sine x = -1/2."},{"Start":"05:56.535 ","End":"06:00.720","Text":"I\u0027m going to write it as sine x = sine of something."},{"Start":"06:00.720 ","End":"06:06.930","Text":"Well, -1/2 is -sine of 30."},{"Start":"06:06.930 ","End":"06:09.090","Text":"The sine of 30 is 1/2."},{"Start":"06:09.090 ","End":"06:10.600","Text":"Now with sine,"},{"Start":"06:10.600 ","End":"06:18.630","Text":"what we can do is put the minus inside because sine of -alpha is -sine alpha."},{"Start":"06:18.630 ","End":"06:23.415","Text":"This is the sine of -30."},{"Start":"06:23.415 ","End":"06:25.770","Text":"Then in the case of the sine,"},{"Start":"06:25.770 ","End":"06:28.935","Text":"we get 2 sets of solutions."},{"Start":"06:28.935 ","End":"06:37.155","Text":"1 is that x = -30 + multiples of 360."},{"Start":"06:37.155 ","End":"06:42.525","Text":"The other is we take 180 - this,"},{"Start":"06:42.525 ","End":"06:48.915","Text":"and if I do a 180 - -30, it\u0027s 210."},{"Start":"06:48.915 ","End":"06:51.120","Text":"Let\u0027s make a note of that."},{"Start":"06:51.120 ","End":"06:56.520","Text":"It\u0027s a 180 - -30 is 210,"},{"Start":"06:56.520 ","End":"06:58.620","Text":"that\u0027s where I get the 210 from,"},{"Start":"06:58.620 ","End":"07:02.200","Text":"and also + 360k."},{"Start":"07:02.200 ","End":"07:10.270","Text":"Now, that\u0027s not all because we also have the possibility that sine x is -1."},{"Start":"07:10.270 ","End":"07:14.665","Text":"If sine x is -1,"},{"Start":"07:14.665 ","End":"07:16.910","Text":"I\u0027ll just put a dividing line here."},{"Start":"07:16.910 ","End":"07:26.150","Text":"So sine of x is -1,"},{"Start":"07:26.150 ","End":"07:30.860","Text":"this is = -sine of 90."},{"Start":"07:30.860 ","End":"07:33.095","Text":"Cosine of 90 is 1."},{"Start":"07:33.095 ","End":"07:36.500","Text":"As before, we can put the minus inside."},{"Start":"07:36.500 ","End":"07:43.130","Text":"So sine x is sine of -90 and"},{"Start":"07:43.130 ","End":"07:51.790","Text":"so we get that either x is = -9t + 360k,"},{"Start":"07:53.030 ","End":"07:59.880","Text":"or x = 180 - this."},{"Start":"07:59.880 ","End":"08:06.585","Text":"Just say a 180 - -90 is 270,"},{"Start":"08:06.585 ","End":"08:12.280","Text":"or 270 + 360k."},{"Start":"08:12.280 ","End":"08:15.380","Text":"Now, in this case,"},{"Start":"08:15.380 ","End":"08:22.475","Text":"notice that the difference between these 2 is 360 degrees."},{"Start":"08:22.475 ","End":"08:26.150","Text":"So this solution is already included in this."},{"Start":"08:26.150 ","End":"08:30.290","Text":"For example, if I let k = 1 here I\u0027ve got 270."},{"Start":"08:30.290 ","End":"08:32.965","Text":"So this is redundant."},{"Start":"08:32.965 ","End":"08:35.175","Text":"It\u0027s not wrong,"},{"Start":"08:35.175 ","End":"08:38.060","Text":"it\u0027s perfectly correct but it\u0027s already included in here."},{"Start":"08:38.060 ","End":"08:39.770","Text":"That\u0027s what I mean by redundant."},{"Start":"08:39.770 ","End":"08:48.140","Text":"Really there are 3 possible families of solution for x. I\u0027ll just highlight them."},{"Start":"08:48.140 ","End":"08:52.445","Text":"We have this family of solutions for k being any integer."},{"Start":"08:52.445 ","End":"08:57.060","Text":"This possibility or this possibility,"},{"Start":"08:57.060 ","End":"09:02.320","Text":"and that gives the complete solution. We\u0027re done."}],"ID":5415},{"Watched":false,"Name":"Exercise 23","Duration":"13m 3s","ChapterTopicVideoID":5417,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5417.jpeg","UploadDate":"2016-03-10T21:31:42.0830000","DurationForVideoObject":"PT13M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.605","Text":"Here we have a couple of trigonometric equations to solve,"},{"Start":"00:04.605 ","End":"00:08.265","Text":"and in each one, we\u0027re going to use the technique of substitution."},{"Start":"00:08.265 ","End":"00:10.515","Text":"Let\u0027s look at the first one,"},{"Start":"00:10.515 ","End":"00:13.785","Text":"which is this one."},{"Start":"00:13.785 ","End":"00:19.695","Text":"We see that the only way that x appears is through cos(2x)."},{"Start":"00:19.695 ","End":"00:22.650","Text":"Don\u0027t forget that when you write the 2 here,"},{"Start":"00:22.650 ","End":"00:23.994","Text":"it\u0027s like the whole thing squared,"},{"Start":"00:23.994 ","End":"00:26.415","Text":"so this is cos(2x) all squared."},{"Start":"00:26.415 ","End":"00:31.620","Text":"The thing to substitute is the cos(2x),"},{"Start":"00:31.620 ","End":"00:36.539","Text":"and let\u0027s call it t. Now this equation becomes"},{"Start":"00:36.539 ","End":"00:45.367","Text":"4t^2 plus 3t equals 1."},{"Start":"00:45.367 ","End":"00:49.395","Text":"Let\u0027s already write it as minus 1 equals 0."},{"Start":"00:49.395 ","End":"00:52.425","Text":"Now we have a quadratic equation,"},{"Start":"00:52.425 ","End":"00:55.151","Text":"and that\u0027s fairly typical."},{"Start":"00:55.151 ","End":"01:00.215","Text":"Lot of exercises with substitutions reduce to a quadratic."},{"Start":"01:00.215 ","End":"01:06.680","Text":"Solve it using the formula: t is equal to minus b plus or"},{"Start":"01:06.680 ","End":"01:14.500","Text":"minus the square root of b^2 minus 4 times a times c,"},{"Start":"01:14.500 ","End":"01:18.280","Text":"and all this is over 2a."},{"Start":"01:18.680 ","End":"01:23.310","Text":"What we have is minus 3 plus or minus,"},{"Start":"01:23.310 ","End":"01:26.730","Text":"now let\u0027s see, 4 times 4 is 16,"},{"Start":"01:26.730 ","End":"01:29.063","Text":"and as a minus,"},{"Start":"01:29.063 ","End":"01:37.223","Text":"19 plus 16 is 25,"},{"Start":"01:37.223 ","End":"01:40.930","Text":"and 2 times 4 is 8."},{"Start":"01:40.970 ","End":"01:49.710","Text":"What we have is minus 3 plus 5 over 8,"},{"Start":"01:49.710 ","End":"01:54.600","Text":"or minus 3 minus 5 over 8."},{"Start":"01:54.600 ","End":"01:58.320","Text":"This one is 2 over 8,"},{"Start":"01:58.320 ","End":"02:01.545","Text":"which is 1/4,"},{"Start":"02:01.545 ","End":"02:03.300","Text":"and, minus 3 minus 5,"},{"Start":"02:03.300 ","End":"02:07.960","Text":"this is minus 1."},{"Start":"02:07.960 ","End":"02:10.640","Text":"Now these are not the solutions for x,"},{"Start":"02:10.640 ","End":"02:17.870","Text":"these are the solutions for t. We still have to go back to x."},{"Start":"02:17.870 ","End":"02:22.845","Text":"What we have is that cos(2x) is"},{"Start":"02:22.845 ","End":"02:28.575","Text":"equal to either 1/4, or minus 1."},{"Start":"02:28.575 ","End":"02:31.590","Text":"Let\u0027s solve each of these separately."},{"Start":"02:31.590 ","End":"02:34.350","Text":"Let\u0027s start with the 1/4."},{"Start":"02:34.350 ","End":"02:40.160","Text":"If cos(2x) equals 1/4,"},{"Start":"02:40.160 ","End":"02:44.915","Text":"what I have to do is write this 1/4 as cosine of something."},{"Start":"02:44.915 ","End":"02:49.860","Text":"This is not a famous value, well known,"},{"Start":"02:49.860 ","End":"02:52.552","Text":"we just have to use the calculator,"},{"Start":"02:52.552 ","End":"02:53.750","Text":"try, let\u0027s say,"},{"Start":"02:53.750 ","End":"02:59.165","Text":"0.25, and then inverse cosine or shift cosine."},{"Start":"02:59.165 ","End":"03:05.780","Text":"On my calculator, it gives me 75.5 something degrees."},{"Start":"03:05.780 ","End":"03:09.740","Text":"I\u0027ll just leave it as 75.5."},{"Start":"03:09.740 ","End":"03:15.760","Text":"That means that there are 2 families of solutions."},{"Start":"03:15.760 ","End":"03:18.530","Text":"First of all, we get what 2x equals."},{"Start":"03:18.530 ","End":"03:22.381","Text":"2x is either equal to 75.5,"},{"Start":"03:22.381 ","End":"03:25.956","Text":"actually it was 75.52 something;"},{"Start":"03:25.956 ","End":"03:27.400","Text":"because here I\u0027m going to divide by 2,"},{"Start":"03:27.400 ","End":"03:29.097","Text":"so I want to make it even,"},{"Start":"03:29.097 ","End":"03:32.905","Text":"plus 360k as always."},{"Start":"03:32.905 ","End":"03:41.980","Text":"The other thing with cosine is minus the same thing, 75.52 plus 360k."},{"Start":"03:41.980 ","End":"03:45.565","Text":"What we have to do now is divide by 2,"},{"Start":"03:45.565 ","End":"03:49.480","Text":"so we\u0027ve got that x is equal to."},{"Start":"03:49.480 ","End":"03:50.535","Text":"You know what?"},{"Start":"03:50.535 ","End":"03:52.426","Text":"I\u0027ll write it as plus or minus."},{"Start":"03:52.426 ","End":"03:54.280","Text":"Sometimes we combine them,"},{"Start":"03:54.280 ","End":"03:56.995","Text":"and saves a line. Plus or minus."},{"Start":"03:56.995 ","End":"04:04.660","Text":"Now this divided by 2 is 37.76,"},{"Start":"04:04.660 ","End":"04:08.800","Text":"and the 360k is just 180k."},{"Start":"04:08.800 ","End":"04:13.979","Text":"That is already 2 families of solutions,"},{"Start":"04:13.979 ","End":"04:18.575","Text":"37.76 and multiples of 180 or minus."},{"Start":"04:18.575 ","End":"04:25.010","Text":"But, let\u0027s not forget that we had another possibility that cos(2x) could be minus 1."},{"Start":"04:25.010 ","End":"04:27.320","Text":"See if I have some room here;"},{"Start":"04:27.320 ","End":"04:31.730","Text":"cos(2x) is minus 1."},{"Start":"04:31.730 ","End":"04:34.839","Text":"I\u0027ll just separate this."},{"Start":"04:34.839 ","End":"04:37.310","Text":"Now, I know that minus"},{"Start":"04:37.310 ","End":"04:39.590","Text":"1 is the"},{"Start":"04:39.590 ","End":"04:53.000","Text":"cos(180)."},{"Start":"04:53.000 ","End":"04:55.790","Text":"Remember, this is one of these famous angles,"},{"Start":"04:55.790 ","End":"04:59.610","Text":"but if not, your calculator should give it to you."},{"Start":"05:00.400 ","End":"05:05.891","Text":"Now, that\u0027s not the only possibility, we have to add,"},{"Start":"05:05.891 ","End":"05:07.295","Text":"I\u0027m running out of space,"},{"Start":"05:07.295 ","End":"05:12.623","Text":"so I\u0027ll write here, plus 360k."},{"Start":"05:12.623 ","End":"05:19.482","Text":"The other possibility,"},{"Start":"05:19.482 ","End":"05:22.992","Text":"take 2 on that last bit."},{"Start":"05:22.992 ","End":"05:27.690","Text":"What that gives us is that 2x is equal"},{"Start":"05:27.690 ","End":"05:36.890","Text":"to either 180 plus 360k,"},{"Start":"05:36.890 ","End":"05:43.340","Text":"or minus this, which is minus 180 plus 360k."},{"Start":"05:43.400 ","End":"05:47.191","Text":"The thing is that there\u0027s a redundancy here."},{"Start":"05:47.191 ","End":"05:49.630","Text":"If I look at the difference between these two,"},{"Start":"05:49.630 ","End":"05:54.445","Text":"these differ by 360 degrees."},{"Start":"05:54.445 ","End":"05:57.550","Text":"For example, if I take this,"},{"Start":"05:57.550 ","End":"05:59.995","Text":"and put k=1,"},{"Start":"05:59.995 ","End":"06:02.695","Text":"then I get 180,"},{"Start":"06:02.695 ","End":"06:03.710","Text":"which is already covered."},{"Start":"06:03.710 ","End":"06:05.705","Text":"If I put k=2,"},{"Start":"06:05.705 ","End":"06:09.290","Text":"then I get 180 plus 360."},{"Start":"06:09.290 ","End":"06:11.360","Text":"Basically, these are the same things,"},{"Start":"06:11.360 ","End":"06:13.025","Text":"just k is off by 1 here,"},{"Start":"06:13.025 ","End":"06:15.080","Text":"so this is included in this,"},{"Start":"06:15.080 ","End":"06:18.575","Text":"and therefore, I can just put a line through it."},{"Start":"06:18.575 ","End":"06:20.330","Text":"I can say, this is not wrong,"},{"Start":"06:20.330 ","End":"06:22.559","Text":"this is perfectly okay, just include it already."},{"Start":"06:22.559 ","End":"06:24.485","Text":"That\u0027s what I mean by redundant."},{"Start":"06:24.485 ","End":"06:28.460","Text":"Now all we have to do is divide by the 2,"},{"Start":"06:28.460 ","End":"06:35.880","Text":"and so x is equal to 90 plus 180k."},{"Start":"06:38.210 ","End":"06:43.346","Text":"Really we have 3 families of solutions,"},{"Start":"06:43.346 ","End":"06:44.524","Text":"these 2 of them,"},{"Start":"06:44.524 ","End":"06:48.795","Text":"we either have 37.76 plus 180k,"},{"Start":"06:48.795 ","End":"06:51.600","Text":"minus 37.76 plus 180k,"},{"Start":"06:51.600 ","End":"06:55.120","Text":"and 90 plus 180k."},{"Start":"06:55.840 ","End":"06:59.675","Text":"Let\u0027s move on to Part B now,"},{"Start":"06:59.675 ","End":"07:02.610","Text":"and here, something similar."},{"Start":"07:02.610 ","End":"07:06.440","Text":"Again, we have the 2x instead of x,"},{"Start":"07:06.440 ","End":"07:08.420","Text":"which we\u0027re more familiar with,"},{"Start":"07:08.420 ","End":"07:11.797","Text":"and it\u0027s sine instead of cosine,"},{"Start":"07:11.797 ","End":"07:17.633","Text":"so clearly what we have to do is to substitute the sin(2x),"},{"Start":"07:17.633 ","End":"07:19.557","Text":"and let\u0027s call it t,"},{"Start":"07:19.557 ","End":"07:23.798","Text":"favorite letter for substitution,"},{"Start":"07:23.798 ","End":"07:26.850","Text":"and what we get is 12t^2."},{"Start":"07:26.850 ","End":"07:30.922","Text":"Like I said, sin^2(2x) means sin(2x)^2,"},{"Start":"07:30.922 ","End":"07:33.174","Text":"just a shorthand notation."},{"Start":"07:33.174 ","End":"07:36.080","Text":"Minus 13 sin(2x) is t,"},{"Start":"07:36.080 ","End":"07:38.605","Text":"again, plus 3 equals 0."},{"Start":"07:38.605 ","End":"07:41.060","Text":"Once again, a quadratic equation,"},{"Start":"07:41.060 ","End":"07:42.950","Text":"solve it with the formula."},{"Start":"07:42.950 ","End":"07:50.270","Text":"Minus b is 13 plus or minus the square root of b^2, it\u0027s 169,"},{"Start":"07:50.270 ","End":"08:00.370","Text":"minus 4ac, minus 4 times 12 times 3, over 2a,24."},{"Start":"08:00.370 ","End":"08:05.020","Text":"Let\u0027s see, what do we have under the square root sign?"},{"Start":"08:05.020 ","End":"08:09.425","Text":"I just changed color there, sorry."},{"Start":"08:09.425 ","End":"08:11.230","Text":"Under the square root sign,"},{"Start":"08:11.230 ","End":"08:15.220","Text":"I have 4 times 12 is 48,"},{"Start":"08:15.220 ","End":"08:24.900","Text":"48 times 3 is 154,"},{"Start":"08:24.900 ","End":"08:27.975","Text":"no, make that 144."},{"Start":"08:27.975 ","End":"08:31.500","Text":"This minus this is 25,"},{"Start":"08:31.500 ","End":"08:37.294","Text":"so what I get is 13 plus or minus, let\u0027s see,"},{"Start":"08:37.294 ","End":"08:40.995","Text":"the square root of 25 is 5,"},{"Start":"08:40.995 ","End":"08:46.625","Text":"so 13 plus or minus 5 over 24."},{"Start":"08:46.625 ","End":"08:50.484","Text":"What do we get if we take the plus,"},{"Start":"08:50.484 ","End":"08:54.384","Text":"13 plus 5 over 24."},{"Start":"08:54.384 ","End":"08:58.288","Text":"This is 18/24,"},{"Start":"08:58.288 ","End":"09:01.567","Text":"and we reduce it to 3/4."},{"Start":"09:01.567 ","End":"09:06.035","Text":"If I take the minus,"},{"Start":"09:06.035 ","End":"09:09.700","Text":"then I have 13 minus 5 is 8,"},{"Start":"09:09.700 ","End":"09:15.807","Text":"8/24, and that is 1/3."},{"Start":"09:15.807 ","End":"09:19.260","Text":"These are the 2 possibilities for t,"},{"Start":"09:19.260 ","End":"09:22.170","Text":"but remember, t is sin(2x),"},{"Start":"09:22.170 ","End":"09:26.970","Text":"so what I get is that sin(2x) is equal"},{"Start":"09:26.970 ","End":"09:35.865","Text":"to either 3/4 or 1/3."},{"Start":"09:35.865 ","End":"09:39.503","Text":"I\u0027m going to solve each one separately."},{"Start":"09:39.503 ","End":"09:46.570","Text":"Let\u0027s start with sin(2x)=3/4."},{"Start":"09:46.570 ","End":"09:50.927","Text":"This is not a well known sine,"},{"Start":"09:50.927 ","End":"09:55.280","Text":"so we use the calculator and get"},{"Start":"09:55.280 ","End":"10:01.370","Text":"that this is equal to sine of, let\u0027s see."},{"Start":"10:01.370 ","End":"10:07.255","Text":"I\u0027ll punch in 0.75 and then shift sine,"},{"Start":"10:07.255 ","End":"10:10.910","Text":"and it\u0027s roughly 48.6."},{"Start":"10:10.910 ","End":"10:14.310","Text":"I\u0027ll just take it to one decimal place here."},{"Start":"10:14.680 ","End":"10:16.850","Text":"As usual with sine,"},{"Start":"10:16.850 ","End":"10:19.595","Text":"we get 2 families of solutions."},{"Start":"10:19.595 ","End":"10:25.940","Text":"We either get the 48.6 plus multiples of a whole circle,"},{"Start":"10:25.940 ","End":"10:30.400","Text":"that\u0027s 360k, or 180 minus this,"},{"Start":"10:30.400 ","End":"10:32.750","Text":"and on the calculator,"},{"Start":"10:32.750 ","End":"10:35.515","Text":"that gives me 131.4."},{"Start":"10:35.515 ","End":"10:42.450","Text":"Yeah, it looks right, this plus this is about 180, also 360k."},{"Start":"10:42.760 ","End":"10:51.275","Text":"Then we have to divide by 2 so we have x is equal to either,"},{"Start":"10:51.275 ","End":"10:53.060","Text":"let\u0027s see, I\u0027m going to just use my head,"},{"Start":"10:53.060 ","End":"10:57.785","Text":"24.3, and this will be, let\u0027s see,"},{"Start":"10:57.785 ","End":"11:03.555","Text":"13 divided by 2 is 6 carry 1,"},{"Start":"11:03.555 ","End":"11:08.520","Text":"11/2 is 5 carry 1.7."},{"Start":"11:08.520 ","End":"11:15.770","Text":"This one is plus 180k and so is this one because we divide the 360 by 2 also,"},{"Start":"11:15.770 ","End":"11:18.979","Text":"and that\u0027s 2 families of solutions."},{"Start":"11:18.979 ","End":"11:22.190","Text":"K is any integer, positive or negative."},{"Start":"11:22.190 ","End":"11:24.496","Text":"We\u0027re not done, we have more solutions."},{"Start":"11:24.496 ","End":"11:25.970","Text":"Because we\u0027ve done the 3/4 quarters,"},{"Start":"11:25.970 ","End":"11:28.800","Text":"we still have the 1/3."},{"Start":"11:29.120 ","End":"11:32.870","Text":"I\u0027ll use a different color then it will be easier to see."},{"Start":"11:32.870 ","End":"11:41.750","Text":"sin(2x)=1/3, and that is equal to sine of,"},{"Start":"11:41.750 ","End":"11:44.650","Text":"I\u0027ll use Mr. calculator,"},{"Start":"11:44.650 ","End":"11:50.160","Text":"comes out 19.47 something."},{"Start":"11:50.160 ","End":"11:54.900","Text":"Once again, we get 2 families of solutions."},{"Start":"11:54.900 ","End":"12:02.655","Text":"2x is either 19.47 plus 360k,"},{"Start":"12:02.655 ","End":"12:09.165","Text":"or subtract this from 180,"},{"Start":"12:09.165 ","End":"12:14.400","Text":"and I\u0027ve got 160.52 something,"},{"Start":"12:14.400 ","End":"12:18.270","Text":"but it\u0027s closer to .53."},{"Start":"12:18.270 ","End":"12:23.295","Text":"That\u0027s 360k, and when I divide by 2,"},{"Start":"12:23.295 ","End":"12:26.553","Text":"let me do it on the calculator,"},{"Start":"12:26.553 ","End":"12:30.390","Text":"here the calculator results to 2 decimal places."},{"Start":"12:30.390 ","End":"12:32.330","Text":"Dividing these by 2 also,"},{"Start":"12:32.330 ","End":"12:37.570","Text":"that\u0027s 180k and that\u0027s 180k,"},{"Start":"12:37.570 ","End":"12:46.605","Text":"so there are actually 4 families of solutions: this,"},{"Start":"12:46.605 ","End":"12:49.627","Text":"this, this, and this."},{"Start":"12:49.627 ","End":"12:53.412","Text":"In each case, k could be any integer positive or negative,"},{"Start":"12:53.412 ","End":"13:02.310","Text":"so we have 4 times infinity solutions and we\u0027re done with Part B, so that\u0027s it."}],"ID":5416},{"Watched":false,"Name":"Exercise 24","Duration":"12m 33s","ChapterTopicVideoID":5418,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5418.jpeg","UploadDate":"2016-03-10T21:33:20.6070000","DurationForVideoObject":"PT12M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.180","Text":"Here\u0027s another two-part exercise each one with a trigonometric substitution."},{"Start":"00:06.180 ","End":"00:10.470","Text":"You should be fairly familiar with this by now."},{"Start":"00:10.470 ","End":"00:12.435","Text":"I\u0027ll do it a little bit quicker."},{"Start":"00:12.435 ","End":"00:21.810","Text":"In part a, the thing to substitute would be the cos(4x) and we\u0027ll let that equal t,"},{"Start":"00:21.810 ","End":"00:24.870","Text":"our favorite letter, substitution,"},{"Start":"00:24.870 ","End":"00:29.890","Text":"and then we get the quadratic equation, 2t^2 plus 3t."},{"Start":"00:31.400 ","End":"00:38.680","Text":"I\u0027m going to bring the minus 1 over to this side and call it plus 1 equals 0."},{"Start":"00:39.200 ","End":"00:44.750","Text":"It\u0027s a very popular exercise that after substitution gives a quadratic,"},{"Start":"00:44.750 ","End":"00:46.730","Text":"but don\u0027t assume that\u0027s always the case."},{"Start":"00:46.730 ","End":"00:53.765","Text":"It\u0027s just a favorite thing with exercises and on exams also,"},{"Start":"00:53.765 ","End":"00:57.605","Text":"after a substitution, get a quadratic very frequent."},{"Start":"00:57.605 ","End":"01:00.635","Text":"Again, using the formula,"},{"Start":"01:00.635 ","End":"01:03.215","Text":"we get t equals,"},{"Start":"01:03.215 ","End":"01:05.480","Text":"this is the less important part"},{"Start":"01:05.480 ","End":"01:09.140","Text":"solving the quadratic equation and I\u0027ll just do it quickly,"},{"Start":"01:09.140 ","End":"01:18.065","Text":"plus or minus square root of b^2 minus 4ac over 2a is 4."},{"Start":"01:18.065 ","End":"01:24.020","Text":"Let\u0027s see, 9 minus 8 is 1."},{"Start":"01:24.020 ","End":"01:32.170","Text":"I\u0027ve got minus 3 plus or minus 1 over 4."},{"Start":"01:32.720 ","End":"01:36.090","Text":"Yeah, I just changed color there."},{"Start":"01:36.090 ","End":"01:43.185","Text":"This is equal to minus 3 plus 1 is minus 2 over 4 minus 1/2,"},{"Start":"01:43.185 ","End":"01:44.520","Text":"and minus 3, minus 1,"},{"Start":"01:44.520 ","End":"01:47.760","Text":"minus 4 over 4 minus 1."},{"Start":"01:47.760 ","End":"01:55.235","Text":"Now we substitute back from t to cos4x and get 2 sets of solutions."},{"Start":"01:55.235 ","End":"01:58.110","Text":"Let me just scroll a bit here."},{"Start":"01:58.960 ","End":"02:08.380","Text":"Since t is cos4x and that\u0027s minus 1/2 or minus 1,"},{"Start":"02:08.380 ","End":"02:09.770","Text":"I\u0027ll solve each one over here."},{"Start":"02:09.770 ","End":"02:10.985","Text":"Let\u0027s take the first one."},{"Start":"02:10.985 ","End":"02:15.245","Text":"Cos4x is minus 1/2."},{"Start":"02:15.245 ","End":"02:23.565","Text":"This time 1/2 is a well-known value for cosine."},{"Start":"02:23.565 ","End":"02:31.680","Text":"1/2 is cos(60) degrees."},{"Start":"02:31.990 ","End":"02:37.100","Text":"What we do with the minus is with the sign,"},{"Start":"02:37.100 ","End":"02:39.860","Text":"we could just throw the minus inside with the cosine,"},{"Start":"02:39.860 ","End":"02:41.660","Text":"to get rid of the minus,"},{"Start":"02:41.660 ","End":"02:44.645","Text":"we do 180 minus the angle."},{"Start":"02:44.645 ","End":"02:49.470","Text":"You\u0027ve seen this before, so I won\u0027t dwell on it."},{"Start":"02:49.470 ","End":"02:51.300","Text":"This is cos(120),"},{"Start":"02:51.300 ","End":"02:54.360","Text":"180 minus 60 is 120."},{"Start":"02:54.360 ","End":"03:00.105","Text":"We get that 4x is equal to,"},{"Start":"03:00.105 ","End":"03:07.590","Text":"with the cosine we get 2 families and we can combine them actually as plus or minus."},{"Start":"03:07.590 ","End":"03:10.410","Text":"It\u0027s plus or minus"},{"Start":"03:10.410 ","End":"03:17.750","Text":"120 because cosine of a negative is the same as cosine of the positive."},{"Start":"03:17.750 ","End":"03:23.180","Text":"We always have to add the 360K because of full circle,"},{"Start":"03:23.180 ","End":"03:25.670","Text":"it doesn\u0027t change the cosine either."},{"Start":"03:25.670 ","End":"03:28.880","Text":"Finally, we divide by 4."},{"Start":"03:28.880 ","End":"03:32.450","Text":"We\u0027ve got plus or minus 30."},{"Start":"03:32.450 ","End":"03:36.010","Text":"Then multiples of 90."},{"Start":"03:36.010 ","End":"03:39.260","Text":"Is any integer positive or negative?"},{"Start":"03:39.260 ","End":"03:44.149","Text":"That\u0027s 2 families of solutions,"},{"Start":"03:44.149 ","End":"03:47.000","Text":"30 and minus 30 plus 90K."},{"Start":"03:47.000 ","End":"03:48.335","Text":"Let\u0027s see what we get,"},{"Start":"03:48.335 ","End":"03:51.155","Text":"if we took the minus 1,"},{"Start":"03:51.155 ","End":"03:56.610","Text":"we\u0027ve got cos(4x)=minus 1."},{"Start":"04:01.220 ","End":"04:06.570","Text":"Since I remember doing this previously,"},{"Start":"04:06.860 ","End":"04:10.080","Text":"I\u0027ll just quote the solution"},{"Start":"04:10.080 ","End":"04:20.010","Text":"that 4x=180+360K."},{"Start":"04:20.010 ","End":"04:21.420","Text":"Just to briefly remind you,"},{"Start":"04:21.420 ","End":"04:31.275","Text":"we don\u0027t need the minus 180 because minus 180 is just 360 degrees away from 180."},{"Start":"04:31.275 ","End":"04:34.110","Text":"It\u0027s the same family of solutions."},{"Start":"04:34.110 ","End":"04:36.120","Text":"We don\u0027t get the minus 180."},{"Start":"04:36.120 ","End":"04:38.574","Text":"We don\u0027t need it. It\u0027s redundant."},{"Start":"04:38.574 ","End":"04:42.695","Text":"Now we have 3 possibilities."},{"Start":"04:42.695 ","End":"04:46.490","Text":"I forgot to divide by 4, excuse me."},{"Start":"04:46.490 ","End":"04:57.555","Text":"x is equal to this over 4 is 45 plus 90K."},{"Start":"04:57.555 ","End":"04:59.900","Text":"Now we have 3 possibilities,"},{"Start":"04:59.900 ","End":"05:01.460","Text":"all with 90K,"},{"Start":"05:01.460 ","End":"05:08.225","Text":"we have this, which is really 2 sets plus 30 and minus 30."},{"Start":"05:08.225 ","End":"05:12.100","Text":"Here we have 45 and in all cases plus 90K."},{"Start":"05:12.100 ","End":"05:14.090","Text":"That ends part a,"},{"Start":"05:14.090 ","End":"05:17.510","Text":"and now we can move on to part b."},{"Start":"05:17.510 ","End":"05:21.500","Text":"In part b this is also a substitution."},{"Start":"05:21.500 ","End":"05:26.030","Text":"We see that essentially everything is in terms of sine 3x."},{"Start":"05:26.030 ","End":"05:27.995","Text":"That\u0027s what we\u0027ll substitute."},{"Start":"05:27.995 ","End":"05:37.235","Text":"Sine 3x we\u0027ll let that equal t. This is just a constant, 2 root 3."},{"Start":"05:37.235 ","End":"05:38.450","Text":"This is t^2."},{"Start":"05:38.450 ","End":"05:41.225","Text":"Remember sine squared means sine of the thing,"},{"Start":"05:41.225 ","End":"05:42.470","Text":"all squared."},{"Start":"05:42.470 ","End":"05:44.510","Text":"This is t^2."},{"Start":"05:44.510 ","End":"05:47.640","Text":"This is just t."},{"Start":"05:48.920 ","End":"05:56.450","Text":"You\u0027ll allow me to move this to the other side in one go equals 0."},{"Start":"05:56.450 ","End":"06:02.461","Text":"I don\u0027t want to waste the whole line just to transfer the constant."},{"Start":"06:02.461 ","End":"06:08.060","Text":"It\u0027s got some funny coefficients with square roots but it\u0027s just a quadratic equation."},{"Start":"06:08.060 ","End":"06:12.185","Text":"We will solve it same as any other quadratic with the formula"},{"Start":"06:12.185 ","End":"06:17.630","Text":"minus b plus or minus the square root of b^2,"},{"Start":"06:17.630 ","End":"06:21.920","Text":"which is 1 minus 4 times a,"},{"Start":"06:21.920 ","End":"06:28.340","Text":"which is 2 root 3 times c,"},{"Start":"06:28.340 ","End":"06:34.695","Text":"which is minus 2 root 3,"},{"Start":"06:34.695 ","End":"06:38.385","Text":"and all this over 2a,"},{"Start":"06:38.385 ","End":"06:44.100","Text":"which will make it 4 root 3 if I double this."},{"Start":"06:44.100 ","End":"06:47.690","Text":"Let\u0027s see what\u0027s under the square root sign."},{"Start":"06:47.690 ","End":"06:53.195","Text":"Let me just do the side exercise here."},{"Start":"06:53.195 ","End":"06:55.320","Text":"I\u0027ll do it up here."},{"Start":"06:55.320 ","End":"07:02.305","Text":"Let\u0027s see,1 minus 4 times 2 root 3 times minus 2 root 3."},{"Start":"07:02.305 ","End":"07:04.340","Text":"What this equals, first of all,"},{"Start":"07:04.340 ","End":"07:06.860","Text":"the minus and minus makes it plus,"},{"Start":"07:06.860 ","End":"07:09.140","Text":"so it\u0027s 1 plus."},{"Start":"07:09.140 ","End":"07:11.210","Text":"Now let\u0027s take the whole numbers first,"},{"Start":"07:11.210 ","End":"07:16.085","Text":"I have a 4 and a 2 and a 2, that\u0027s 16."},{"Start":"07:16.085 ","End":"07:22.120","Text":"Then root 3 with root 3 gives me 3."},{"Start":"07:22.120 ","End":"07:25.250","Text":"These two combine to give me 3."},{"Start":"07:25.250 ","End":"07:27.860","Text":"16 times 3 is 48."},{"Start":"07:27.860 ","End":"07:30.575","Text":"What I get is 49."},{"Start":"07:30.575 ","End":"07:35.370","Text":"I know that the root of 49 is 7."},{"Start":"07:36.680 ","End":"07:48.000","Text":"Back to here, what I have is minus 1 plus or minus 7 over 4 root 3."},{"Start":"07:48.320 ","End":"07:53.040","Text":"If I take the plus,"},{"Start":"07:53.040 ","End":"07:59.985","Text":"I\u0027ve got 6/4 root 3."},{"Start":"07:59.985 ","End":"08:03.590","Text":"If I take the minus,"},{"Start":"08:03.590 ","End":"08:05.330","Text":"I have minus 1,"},{"Start":"08:05.330 ","End":"08:14.350","Text":"minus 7 is minus 8/4 root 3."},{"Start":"08:16.330 ","End":"08:22.220","Text":"Let\u0027s multiply top and bottom by root"},{"Start":"08:22.220 ","End":"08:28.200","Text":"3 and get rid of the irrationality in the denominator."},{"Start":"08:29.720 ","End":"08:33.635","Text":"I\u0027ll work on both of them."},{"Start":"08:33.635 ","End":"08:36.635","Text":"If I multiply top and bottom by root 3,"},{"Start":"08:36.635 ","End":"08:41.940","Text":"I\u0027ve got 6 root 3 over 4."},{"Start":"08:41.940 ","End":"08:45.645","Text":"Root 3 root 3 is 3."},{"Start":"08:45.645 ","End":"08:47.830","Text":"I see if I can simplify this."},{"Start":"08:47.830 ","End":"08:50.890","Text":"This is 6 over 12."},{"Start":"08:50.890 ","End":"08:54.925","Text":"Notice that 6 over 12 is 1/2,"},{"Start":"08:54.925 ","End":"08:58.970","Text":"so we get root 3 over 2."},{"Start":"08:59.530 ","End":"09:06.085","Text":"Here, if I multiply by root 3,"},{"Start":"09:06.085 ","End":"09:09.960","Text":"I get 8 root 3,"},{"Start":"09:09.960 ","End":"09:11.955","Text":"there\u0027s a minus,"},{"Start":"09:11.955 ","End":"09:17.530","Text":"over 4 root 3 root 3."},{"Start":"09:17.840 ","End":"09:20.820","Text":"Root.3 root 3 is just 3."},{"Start":"09:20.820 ","End":"09:24.720","Text":"I\u0027ll erase and write 3."},{"Start":"09:24.720 ","End":"09:31.500","Text":"Let\u0027s see, here we have 8 minus 8."},{"Start":"09:31.500 ","End":"09:36.150","Text":"Well, why don\u0027t I just cancel first of all,"},{"Start":"09:36.150 ","End":"09:38.805","Text":"4 into 8 goes twice."},{"Start":"09:38.805 ","End":"09:47.845","Text":"I have minus 2/3 root 3 minus 2 root 3 over 3."},{"Start":"09:47.845 ","End":"09:56.845","Text":"I\u0027ve the feeling this comes out to be less than minus 1 because root 3 is roughly 1.7,"},{"Start":"09:56.845 ","End":"10:01.885","Text":"and 2 times 1.7 is 3.4."},{"Start":"10:01.885 ","End":"10:06.155","Text":"I can give you a more precise answer on the calculator."},{"Start":"10:06.155 ","End":"10:11.090","Text":"This one comes out to minus 1.15 something."},{"Start":"10:11.090 ","End":"10:13.350","Text":"It\u0027s out of range."},{"Start":"10:15.980 ","End":"10:18.350","Text":"Well, you know what? I\u0027ll leave it like"},{"Start":"10:18.350 ","End":"10:20.555","Text":"that for now then I will rule it out in a moment."},{"Start":"10:20.555 ","End":"10:29.655","Text":"What I want to say is that t is sin(3x)."},{"Start":"10:29.655 ","End":"10:40.174","Text":"We have that sin(3x) is equal to either one of these."},{"Start":"10:40.174 ","End":"10:43.070","Text":"In other words, root 3 over 2,"},{"Start":"10:43.070 ","End":"10:47.855","Text":"or this number here, minus 1.15."},{"Start":"10:47.855 ","End":"10:51.700","Text":"Like I said, now we see it\u0027s the sine of something and"},{"Start":"10:51.700 ","End":"10:55.340","Text":"the sine of something can never be less than minus 1."},{"Start":"10:55.340 ","End":"11:00.710","Text":"Sine is always between minus 1 and 1."},{"Start":"11:00.710 ","End":"11:04.465","Text":"This is not possible,"},{"Start":"11:04.465 ","End":"11:06.060","Text":"we rule it out."},{"Start":"11:06.060 ","End":"11:13.890","Text":"The only solution of these two is root 3 over 2."},{"Start":"11:13.890 ","End":"11:21.190","Text":"Now let\u0027s continue over here."},{"Start":"11:21.440 ","End":"11:29.900","Text":"We only have sin(3x) is equal to root 3 over 2."},{"Start":"11:29.900 ","End":"11:33.725","Text":"The other possibility was eliminated."},{"Start":"11:33.725 ","End":"11:37.990","Text":"We have to say that this is the sine of something,"},{"Start":"11:37.990 ","End":"11:43.250","Text":"and this happens to be the sine of 60 degrees."},{"Start":"11:43.250 ","End":"11:45.380","Text":"It\u0027s one of those famous ones."},{"Start":"11:45.380 ","End":"11:47.810","Text":"But if you didn\u0027t remember that,"},{"Start":"11:47.810 ","End":"11:50.420","Text":"then there\u0027s always the calculator."},{"Start":"11:50.420 ","End":"11:54.560","Text":"We now get 2 families of solutions of possibilities."},{"Start":"11:54.560 ","End":"11:57.870","Text":"3x is either 60"},{"Start":"11:58.430 ","End":"12:05.555","Text":"plus 360K or subtract this from 180,"},{"Start":"12:05.555 ","End":"12:11.250","Text":"and that gives us a 120 also plus 360K."},{"Start":"12:11.250 ","End":"12:15.450","Text":"Finally, we divide by 3 because we need just x."},{"Start":"12:15.450 ","End":"12:17.160","Text":"This comes out 20,"},{"Start":"12:17.160 ","End":"12:19.350","Text":"and this comes out 40."},{"Start":"12:19.350 ","End":"12:22.300","Text":"Each of them is 120K."},{"Start":"12:24.470 ","End":"12:28.990","Text":"These are the two sets of solutions."},{"Start":"12:29.120 ","End":"12:31.740","Text":"That\u0027s it. Done with part b."},{"Start":"12:31.740 ","End":"12:33.850","Text":"That\u0027s the end."}],"ID":5417},{"Watched":false,"Name":"Exercise 25","Duration":"13m 27s","ChapterTopicVideoID":5419,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5419.jpeg","UploadDate":"2016-03-10T21:35:11.4530000","DurationForVideoObject":"PT13M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.215","Text":"In this exercise, we have 2 parts and each of them is a trigonometric equation."},{"Start":"00:07.215 ","End":"00:11.160","Text":"Now, this is in the section on substitutions,"},{"Start":"00:11.160 ","End":"00:14.190","Text":"but it\u0027s not immediately clear in"},{"Start":"00:14.190 ","End":"00:18.570","Text":"either of them because we have a mixture both of cosine, and sine."},{"Start":"00:18.570 ","End":"00:25.800","Text":"The solution to this in both cases is that we have a trigonometrical identity,"},{"Start":"00:25.800 ","End":"00:28.900","Text":"which has the sine squared of anything, say,"},{"Start":"00:28.900 ","End":"00:34.285","Text":"Alpha plus cosine squared of the same thing is equal to 1."},{"Start":"00:34.285 ","End":"00:38.480","Text":"We use this if we bring the sine squared over cosine squared to"},{"Start":"00:38.480 ","End":"00:43.085","Text":"the other side to get cosine squared in terms of sine squared and vice versa."},{"Start":"00:43.085 ","End":"00:45.320","Text":"Notice that here we have sine squared,"},{"Start":"00:45.320 ","End":"00:46.939","Text":"and here we have cosine squared,"},{"Start":"00:46.939 ","End":"00:50.230","Text":"so we\u0027ll be able to do a conversion."},{"Start":"00:50.230 ","End":"00:53.550","Text":"Let\u0027s just begin, and we\u0027ll see."},{"Start":"00:53.550 ","End":"00:58.370","Text":"For part A, I\u0027ll convert the sine"},{"Start":"00:58.370 ","End":"01:02.420","Text":"to cosine because sine squared is 1 minus cosine squared,"},{"Start":"01:02.420 ","End":"01:04.295","Text":"bring this to the other side."},{"Start":"01:04.295 ","End":"01:13.100","Text":"I\u0027ve got 1 minus cosine squared x plus cosine x equals 0."},{"Start":"01:13.100 ","End":"01:19.490","Text":"Now I can substitute t equals cosine x,"},{"Start":"01:19.490 ","End":"01:21.400","Text":"or cosine x equals t,"},{"Start":"01:21.400 ","End":"01:25.895","Text":"and then we get square root of 2,"},{"Start":"01:25.895 ","End":"01:32.535","Text":"1 minus t squared plus t equals 0."},{"Start":"01:32.535 ","End":"01:36.210","Text":"Let\u0027s tidy this up a bit."},{"Start":"01:36.210 ","End":"01:43.365","Text":"We get minus the square root of 2t squared plus"},{"Start":"01:43.365 ","End":"01:51.970","Text":"t and plus square root of 2 equals 0."},{"Start":"01:53.090 ","End":"01:57.140","Text":"I think there\u0027s too many square roots here."},{"Start":"01:57.140 ","End":"01:58.700","Text":"We could work with it as is."},{"Start":"01:58.700 ","End":"02:05.110","Text":"But, I think if we multiply both sides by square root of 2,"},{"Start":"02:05.110 ","End":"02:07.910","Text":"we might be able to get rid of some of the roots."},{"Start":"02:07.910 ","End":"02:14.250","Text":"In fact, you know what, let\u0027s make it even minus square root of 2,"},{"Start":"02:14.250 ","End":"02:17.285","Text":"and then we\u0027ll have a positive a, which is what I like."},{"Start":"02:17.285 ","End":"02:20.690","Text":"If I multiply by minus root 2 here we have minus,"},{"Start":"02:20.690 ","End":"02:28.005","Text":"minus is plus root 2 times root 2 is 2, t squared."},{"Start":"02:28.005 ","End":"02:34.440","Text":"Then because we multiplied by minus root 2t,"},{"Start":"02:34.440 ","End":"02:39.340","Text":"and then we have minus root 2 times root 2 is 2 equals 0."},{"Start":"02:39.340 ","End":"02:41.450","Text":"Maybe this is a bit better than this."},{"Start":"02:41.450 ","End":"02:43.470","Text":"I\u0027m not sure."},{"Start":"02:43.480 ","End":"02:48.495","Text":"Let\u0027s see. Scroll a bit,"},{"Start":"02:48.495 ","End":"02:50.770","Text":"get more space here."},{"Start":"02:51.230 ","End":"03:02.180","Text":"We use the formula t equals minus b plus or minus the square root of b squared is"},{"Start":"03:02.180 ","End":"03:10.940","Text":"2 minus 4ac over"},{"Start":"03:10.940 ","End":"03:13.480","Text":"2a is 4."},{"Start":"03:13.480 ","End":"03:16.710","Text":"Let\u0027s see. Under the square root sign,"},{"Start":"03:16.710 ","End":"03:23.730","Text":"we have 4 times 2 times 2 is 16,"},{"Start":"03:23.730 ","End":"03:28.780","Text":"it\u0027s plus 16 plus 2 is 18."},{"Start":"03:28.960 ","End":"03:31.190","Text":"Let me do this at the side."},{"Start":"03:31.190 ","End":"03:32.720","Text":"What we got here was,"},{"Start":"03:32.720 ","End":"03:35.450","Text":"as I said, square root of 18."},{"Start":"03:35.450 ","End":"03:37.310","Text":"It doesn\u0027t come out of whole number,"},{"Start":"03:37.310 ","End":"03:46.340","Text":"but we can write this as the square root of 9 times 2 because 9 is a square,"},{"Start":"03:46.340 ","End":"03:48.710","Text":"and this becomes root 9 root 2,"},{"Start":"03:48.710 ","End":"03:51.640","Text":"which is 3 root 2."},{"Start":"03:51.640 ","End":"03:53.870","Text":"That looks a lot better."},{"Start":"03:53.870 ","End":"03:56.075","Text":"Now, put this back here,"},{"Start":"03:56.075 ","End":"04:04.725","Text":"and we get root 2 plus 3 root 2 over 4,"},{"Start":"04:04.725 ","End":"04:07.090","Text":"sorry, plus or minus."},{"Start":"04:07.520 ","End":"04:12.180","Text":"What we get then is two possibilities."},{"Start":"04:12.180 ","End":"04:15.370","Text":"If we take the plus,"},{"Start":"04:16.010 ","End":"04:19.740","Text":"this is like 1 root 2 plus 3 root 2,"},{"Start":"04:19.740 ","End":"04:22.360","Text":"we get 4 root 2/4."},{"Start":"04:24.230 ","End":"04:26.250","Text":"If we take the minus,"},{"Start":"04:26.250 ","End":"04:32.080","Text":"we get 1 minus 3 is minus 2 root 2/4."},{"Start":"04:32.750 ","End":"04:35.940","Text":"Now, 4/4 cancels,"},{"Start":"04:35.940 ","End":"04:39.315","Text":"and this is equal to root 2."},{"Start":"04:39.315 ","End":"04:43.980","Text":"Here minus 2/4 is minus a 1/2,"},{"Start":"04:43.980 ","End":"04:51.320","Text":"so we get minus root 2/2."},{"Start":"04:51.320 ","End":"04:55.970","Text":"Now, remember, t is cosine x."},{"Start":"04:55.970 ","End":"05:07.530","Text":"What we get is that cosine x is equal to either root 2 or minus root 2/2."},{"Start":"05:08.180 ","End":"05:13.150","Text":"Now, this first one can be ruled out."},{"Start":"05:13.150 ","End":"05:16.865","Text":"I claim this is impossible because"},{"Start":"05:16.865 ","End":"05:21.430","Text":"the cosine x is always between minus 1 and 1 inclusive."},{"Start":"05:21.430 ","End":"05:23.910","Text":"Root 2 is bigger than 1,"},{"Start":"05:23.910 ","End":"05:27.995","Text":"it\u0027s 1 point something, it\u0027s 1.414, blah-blah-blah."},{"Start":"05:27.995 ","End":"05:30.050","Text":"But it\u0027s bigger than 1,"},{"Start":"05:30.050 ","End":"05:32.720","Text":"which is what I care about, so I can rule it out."},{"Start":"05:32.720 ","End":"05:34.535","Text":"We\u0027re just left with this."},{"Start":"05:34.535 ","End":"05:38.720","Text":"Now we get an ordinary trigonometric equation."},{"Start":"05:38.720 ","End":"05:41.640","Text":"I\u0027ll continue that over here."},{"Start":"05:42.190 ","End":"05:49.190","Text":"Cosine x equals minus root 2/2,"},{"Start":"05:49.190 ","End":"05:58.745","Text":"and I know that root 2/2 is the cosine of 45 degrees."},{"Start":"05:58.745 ","End":"06:01.650","Text":"It\u0027s a well-known one."},{"Start":"06:02.900 ","End":"06:10.625","Text":"There\u0027s a familiar a way to get the minus inside with a sign,"},{"Start":"06:10.625 ","End":"06:12.860","Text":"we just throw it inside with the cosine."},{"Start":"06:12.860 ","End":"06:14.060","Text":"To get rid of the minus,"},{"Start":"06:14.060 ","End":"06:16.975","Text":"we subtract this from 180."},{"Start":"06:16.975 ","End":"06:20.925","Text":"This is cosine of 135."},{"Start":"06:20.925 ","End":"06:27.350","Text":"Like I said, I got the 135 by doing 180 minus 45."},{"Start":"06:27.350 ","End":"06:31.180","Text":"Now I have cosine something equals cosine something."},{"Start":"06:31.180 ","End":"06:33.720","Text":"We get that x equals"},{"Start":"06:33.720 ","End":"06:41.820","Text":"135 plus multiples of"},{"Start":"06:43.040 ","End":"06:47.370","Text":"360k or minus this,"},{"Start":"06:47.370 ","End":"06:53.485","Text":"and tell you what, I\u0027ll just write it as plus or minus that of writing another line."},{"Start":"06:53.485 ","End":"06:56.360","Text":"This is the general answer."},{"Start":"06:56.360 ","End":"06:57.820","Text":"Basically, you have two choices."},{"Start":"06:57.820 ","End":"06:59.380","Text":"You can choose plus or minus,"},{"Start":"06:59.380 ","End":"07:01.180","Text":"and you can choose whichever k you want,"},{"Start":"07:01.180 ","End":"07:02.380","Text":"k is an integer."},{"Start":"07:02.380 ","End":"07:03.550","Text":"I don\u0027t write that every time,"},{"Start":"07:03.550 ","End":"07:04.855","Text":"but k is an integer,"},{"Start":"07:04.855 ","End":"07:06.220","Text":"0, 1, 2, 3, 4,"},{"Start":"07:06.220 ","End":"07:08.050","Text":"5, or minus 1, minus 2."},{"Start":"07:08.050 ","End":"07:12.820","Text":"This gives us the complete impossibilities for solutions."},{"Start":"07:13.070 ","End":"07:15.150","Text":"That\u0027s part A."},{"Start":"07:15.150 ","End":"07:22.820","Text":"Let\u0027s move on to part B where I said we have a similar situation."},{"Start":"07:22.820 ","End":"07:28.010","Text":"I\u0027m not going to repeat the formula."},{"Start":"07:28.010 ","End":"07:32.280","Text":"It was the cosine squared plus sine squared is 1,"},{"Start":"07:32.290 ","End":"07:38.495","Text":"which means that the cosine squared is 1 minus the sine squared."},{"Start":"07:38.495 ","End":"07:39.920","Text":"It works both ways."},{"Start":"07:39.920 ","End":"07:41.900","Text":"Sine squared is 1 minus cosine squared,"},{"Start":"07:41.900 ","End":"07:46.130","Text":"and cosine squared is 1 minus sine squared, and so on."},{"Start":"07:46.130 ","End":"07:47.810","Text":"If I do that,"},{"Start":"07:47.810 ","End":"07:53.750","Text":"then I see that I only have sine x in my equation."},{"Start":"07:53.750 ","End":"07:57.200","Text":"Naturally we substitute sine x."},{"Start":"07:57.200 ","End":"08:04.485","Text":"Let\u0027s call it t. Then what we get is minus 4,"},{"Start":"08:04.485 ","End":"08:12.240","Text":"1 minus t squared plus 3t plus 3 equals 0."},{"Start":"08:12.240 ","End":"08:14.210","Text":"This is going to be a quadratic."},{"Start":"08:14.210 ","End":"08:16.400","Text":"Let\u0027s simplify it. Let\u0027s see."},{"Start":"08:16.400 ","End":"08:17.960","Text":"Where do we get t squared from?"},{"Start":"08:17.960 ","End":"08:21.650","Text":"Minus 4 times minus 1 is 4t squared."},{"Start":"08:21.650 ","End":"08:23.600","Text":"Where can we get just t?"},{"Start":"08:23.600 ","End":"08:27.035","Text":"Only from here, 3t are numbers."},{"Start":"08:27.035 ","End":"08:29.030","Text":"We have a minus 4 from here,"},{"Start":"08:29.030 ","End":"08:30.635","Text":"a plus 3 from here,"},{"Start":"08:30.635 ","End":"08:33.215","Text":"that\u0027s minus 1 equals 0."},{"Start":"08:33.215 ","End":"08:35.975","Text":"Now applying the quadratic formula,"},{"Start":"08:35.975 ","End":"08:39.800","Text":"we get minus b plus,"},{"Start":"08:39.800 ","End":"08:47.740","Text":"or minus the square root of b squared minus 4 times a times c,"},{"Start":"08:47.740 ","End":"08:51.870","Text":"and all over 2, which is 8."},{"Start":"08:51.870 ","End":"08:55.040","Text":"Let\u0027s see. 4 times 4 is 16,"},{"Start":"08:55.040 ","End":"08:57.184","Text":"minus times minus is plus."},{"Start":"08:57.184 ","End":"09:05.700","Text":"What we have under here is the square root of 25, which is 5."},{"Start":"09:05.700 ","End":"09:12.870","Text":"I have minus 3 plus or minus 5/8."},{"Start":"09:12.870 ","End":"09:19.690","Text":"That gives us first taking the plus and then take the minus."},{"Start":"09:19.690 ","End":"09:24.552","Text":"For the plus I get minus 3 plus 5 is 2,"},{"Start":"09:24.552 ","End":"09:26.145","Text":"2/8 is a 1/4."},{"Start":"09:26.145 ","End":"09:33.160","Text":"Minus 3 minus 5 is minus 8/8 is minus 1."},{"Start":"09:33.230 ","End":"09:43.635","Text":"Now, remember, t is equal to sine x. There it is."},{"Start":"09:43.635 ","End":"09:50.920","Text":"Sine x is going to be equal to either 1/4 or minus 1."},{"Start":"09:50.920 ","End":"09:52.705","Text":"They are both in range."},{"Start":"09:52.705 ","End":"09:56.689","Text":"This one just made it because it\u0027s from minus 1 to 1 inclusive,"},{"Start":"09:56.689 ","End":"09:58.126","Text":"that\u0027s what sine can be,"},{"Start":"09:58.126 ","End":"10:02.645","Text":"so we\u0027ll have to solve two possibilities."},{"Start":"10:02.645 ","End":"10:07.350","Text":"Let\u0027s start with the top one,"},{"Start":"10:07.350 ","End":"10:09.690","Text":"sine x equals 1/4."},{"Start":"10:09.690 ","End":"10:14.590","Text":"If sine x equals 1/4,"},{"Start":"10:14.590 ","End":"10:18.980","Text":"I have to find out what this is as a sine."},{"Start":"10:18.980 ","End":"10:20.795","Text":"This is sine of what?"},{"Start":"10:20.795 ","End":"10:23.870","Text":"This is not a well-known one of the standard one."},{"Start":"10:23.870 ","End":"10:27.155","Text":"So use the calculator."},{"Start":"10:27.155 ","End":"10:31.520","Text":"This comes out to 14.477 something."},{"Start":"10:31.520 ","End":"10:35.030","Text":"I\u0027ll round it off to one decimal place, 14.5,"},{"Start":"10:35.030 ","End":"10:38.060","Text":"I should really say approximately equal to,"},{"Start":"10:38.060 ","End":"10:42.240","Text":"but we always round off."},{"Start":"10:42.590 ","End":"10:48.844","Text":"Now we have a standard format of sine something equals sine something."},{"Start":"10:48.844 ","End":"10:57.620","Text":"So we get two possibilities that x is equal to either 14.5 plus whole circles,"},{"Start":"10:57.620 ","End":"10:59.730","Text":"I call it 360k."},{"Start":"11:00.040 ","End":"11:06.665","Text":"The other possibility is by subtracting this from 180."},{"Start":"11:06.665 ","End":"11:14.600","Text":"This gives 165.5 subtract"},{"Start":"11:14.600 ","End":"11:18.595","Text":"from 180 also plus 360k."},{"Start":"11:18.595 ","End":"11:26.405","Text":"These are 2 sets of families of solution where k could be any integer."},{"Start":"11:26.405 ","End":"11:28.865","Text":"But we\u0027re not done yet."},{"Start":"11:28.865 ","End":"11:36.745","Text":"We also have the other possibility that sine x is minus 1."},{"Start":"11:36.745 ","End":"11:40.160","Text":"I know that we\u0027ve done this before,"},{"Start":"11:40.160 ","End":"11:42.470","Text":"but we\u0027ll do it again."},{"Start":"11:42.470 ","End":"11:47.945","Text":"This is equal to minus."},{"Start":"11:47.945 ","End":"11:51.400","Text":"Now, 1 is the sine of 90."},{"Start":"11:51.400 ","End":"11:55.490","Text":"With the sine we can put the minus inside."},{"Start":"11:55.490 ","End":"12:01.300","Text":"This is the sine of minus 90,"},{"Start":"12:01.300 ","End":"12:08.120","Text":"and that gives us that x could be"},{"Start":"12:09.200 ","End":"12:18.430","Text":"minus 9t plus whole circle is 360k."},{"Start":"12:18.430 ","End":"12:21.850","Text":"Now we\u0027ve seen this before that the other family,"},{"Start":"12:21.850 ","End":"12:23.110","Text":"I remember is redundant."},{"Start":"12:23.110 ","End":"12:24.520","Text":"But you know what? Let\u0027s do it again."},{"Start":"12:24.520 ","End":"12:31.095","Text":"The other possibility is I take this from 180, 180 minus,"},{"Start":"12:31.095 ","End":"12:33.930","Text":"minus 90 is 270,"},{"Start":"12:33.930 ","End":"12:39.160","Text":"and we\u0027ve got 270 plus whole circles."},{"Start":"12:39.160 ","End":"12:48.650","Text":"But if I let k equals 1, I get 270."},{"Start":"12:49.260 ","End":"12:54.550","Text":"Basically, these solutions are included in these may be with a different k,"},{"Start":"12:54.550 ","End":"12:55.735","Text":"it\u0027s off by 1,"},{"Start":"12:55.735 ","End":"12:59.090","Text":"but this is redundant."},{"Start":"12:59.690 ","End":"13:04.810","Text":"Not illegal or wrong perfectly okay,"},{"Start":"13:04.810 ","End":"13:07.330","Text":"but redundant because it included in this."},{"Start":"13:07.330 ","End":"13:11.480","Text":"We really get 3 families of solutions."},{"Start":"13:11.670 ","End":"13:15.435","Text":"This one, this one,"},{"Start":"13:15.435 ","End":"13:18.120","Text":"and this one, 14.5,"},{"Start":"13:18.120 ","End":"13:24.750","Text":"165.5 minus 90 up to multiples of 360."},{"Start":"13:24.750 ","End":"13:27.760","Text":"That\u0027s the end of part B."}],"ID":5418},{"Watched":false,"Name":"Exercise 26","Duration":"12m 3s","ChapterTopicVideoID":5420,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5420.jpeg","UploadDate":"2016-03-10T21:36:49.6670000","DurationForVideoObject":"PT12M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this exercise, we have 2 parts,"},{"Start":"00:02.700 ","End":"00:05.475","Text":"each of them, a trigonometric equation."},{"Start":"00:05.475 ","End":"00:08.595","Text":"It\u0027s in the chapter on substitution."},{"Start":"00:08.595 ","End":"00:10.830","Text":"We know that\u0027s what we\u0027re supposed to do,"},{"Start":"00:10.830 ","End":"00:14.265","Text":"but it\u0027s not immediately clear what to substitute,"},{"Start":"00:14.265 ","End":"00:19.470","Text":"because here I have cosine x and I also have cosine 2x."},{"Start":"00:19.470 ","End":"00:21.150","Text":"Here I have sine x,"},{"Start":"00:21.150 ","End":"00:23.085","Text":"but I also have cosine 2x."},{"Start":"00:23.085 ","End":"00:26.580","Text":"The cosine 2x is nuisance-some,"},{"Start":"00:26.580 ","End":"00:30.450","Text":"so we\u0027re going to have to use a trigonometric identity."},{"Start":"00:30.450 ","End":"00:36.645","Text":"Actually there are 2 versions of cosine 2x."},{"Start":"00:36.645 ","End":"00:39.405","Text":"Actually there\u0027s more than 2 but anyway,"},{"Start":"00:39.405 ","End":"00:43.170","Text":"1 version says that cosine"},{"Start":"00:43.170 ","End":"00:51.930","Text":"2x=2 cosine^2(x)"},{"Start":"00:51.930 ","End":"00:54.000","Text":"minus 1."},{"Start":"00:54.000 ","End":"00:59.825","Text":"I\u0027m going to use this 1 here because I need to get cosine x."},{"Start":"00:59.825 ","End":"01:02.390","Text":"But there\u0027s another version which I\u0027ll use in part b,"},{"Start":"01:02.390 ","End":"01:04.520","Text":"but I might as well write it already,"},{"Start":"01:04.520 ","End":"01:12.330","Text":"is that cosine 2x is 1 minus 2 sine^2(x)."},{"Start":"01:13.330 ","End":"01:15.890","Text":"Usually I\u0027ll write it in terms of Alpha,"},{"Start":"01:15.890 ","End":"01:17.690","Text":"some Greek letter,"},{"Start":"01:17.690 ","End":"01:18.995","Text":"but since we have x,"},{"Start":"01:18.995 ","End":"01:20.720","Text":"I\u0027ll write it in terms of x."},{"Start":"01:20.720 ","End":"01:23.960","Text":"If we use these identities,"},{"Start":"01:23.960 ","End":"01:27.140","Text":"then we should be able to use substitution."},{"Start":"01:27.140 ","End":"01:31.460","Text":"Let\u0027s see how this works for part a. I"},{"Start":"01:31.460 ","End":"01:37.675","Text":"let t=cosine x here."},{"Start":"01:37.675 ","End":"01:41.090","Text":"Already I know I might as well just write it that in the second 1 I\u0027m"},{"Start":"01:41.090 ","End":"01:45.080","Text":"going to use t=sine x."},{"Start":"01:45.080 ","End":"01:49.190","Text":"Let\u0027s just stick with part a. Cosine 2x,"},{"Start":"01:49.190 ","End":"01:52.265","Text":"I\u0027m really looking at this instead of it,"},{"Start":"01:52.265 ","End":"01:56.905","Text":"is 2t^2 minus 1."},{"Start":"01:56.905 ","End":"02:00.100","Text":"That\u0027s for this bit, and the rest of it,"},{"Start":"02:00.100 ","End":"02:04.780","Text":"cosine x is just t plus 2=0."},{"Start":"02:04.780 ","End":"02:09.620","Text":"Quadratic equation, I just need to collect like terms,"},{"Start":"02:09.960 ","End":"02:13.445","Text":"2t^2 minus 3t,"},{"Start":"02:13.445 ","End":"02:19.100","Text":"the minus 1 combines with the plus 2 to give us plus 1=0."},{"Start":"02:19.100 ","End":"02:26.450","Text":"Now we can use the formula and say that t=minus b plus or minus the square root"},{"Start":"02:26.450 ","End":"02:35.570","Text":"of b^2 minus 4ac/2a is 4."},{"Start":"02:35.570 ","End":"02:42.840","Text":"Let\u0027s see, 9 minus 8 is 1 and the square root of 1 is 1."},{"Start":"02:42.840 ","End":"02:47.605","Text":"We have 3 plus or minus 1/4."},{"Start":"02:47.605 ","End":"02:49.499","Text":"If we take the plus,"},{"Start":"02:49.499 ","End":"02:53.775","Text":"we get 3 plus 1/4 is 1."},{"Start":"02:53.775 ","End":"03:01.650","Text":"We take the minus, 3 minus 1/4 is 1/2 and t is cosine x."},{"Start":"03:01.650 ","End":"03:08.170","Text":"We have the cosine x=either 1 or 1/2,"},{"Start":"03:08.170 ","End":"03:10.475","Text":"and we\u0027ll solve each of these separately."},{"Start":"03:10.475 ","End":"03:13.924","Text":"Take the case of 1 first."},{"Start":"03:13.924 ","End":"03:21.120","Text":"Cosine x=1."},{"Start":"03:22.050 ","End":"03:24.070","Text":"We\u0027ve done this before,"},{"Start":"03:24.070 ","End":"03:26.545","Text":"but anyway we\u0027ll to do it again."},{"Start":"03:26.545 ","End":"03:33.480","Text":"This 1 can be written as the cosine of 0."},{"Start":"03:33.480 ","End":"03:35.685","Text":"Cosine of 0 is 1."},{"Start":"03:35.685 ","End":"03:41.910","Text":"We get that x=plus or"},{"Start":"03:41.910 ","End":"03:50.470","Text":"minus 0 plus 360k."},{"Start":"03:50.470 ","End":"03:53.590","Text":"But plus or minus 0,"},{"Start":"03:53.590 ","End":"03:56.595","Text":"either way it\u0027s 0."},{"Start":"03:56.595 ","End":"03:59.340","Text":"The solution is, we just need to get 1 family,"},{"Start":"03:59.340 ","End":"04:02.750","Text":"not 2 families of solutions, we just get 360k."},{"Start":"04:04.650 ","End":"04:10.750","Text":"That\u0027s for the cosine x=1 bit."},{"Start":"04:10.750 ","End":"04:19.945","Text":"Now the other bit, the cosine x=1/2."},{"Start":"04:19.945 ","End":"04:23.400","Text":"1/2 is also famous,"},{"Start":"04:23.400 ","End":"04:24.930","Text":"I use the word famous,"},{"Start":"04:24.930 ","End":"04:27.880","Text":"it\u0027s a known cosine,"},{"Start":"04:27.880 ","End":"04:31.370","Text":"it\u0027s the cosine of 60 degrees."},{"Start":"04:31.370 ","End":"04:35.450","Text":"This is cosine 60 and"},{"Start":"04:35.450 ","End":"04:40.130","Text":"that means that in this case we really do get 2 families of solution."},{"Start":"04:40.590 ","End":"04:45.400","Text":"We get that x=plus or"},{"Start":"04:45.400 ","End":"04:55.080","Text":"minus 60 plus 360k."},{"Start":"04:55.080 ","End":"04:56.940","Text":"I wrote them combined though sometimes,"},{"Start":"04:56.940 ","End":"04:58.590","Text":"if you like, you can write them separately."},{"Start":"04:58.590 ","End":"05:02.115","Text":"60 plus 360k and then minus 60."},{"Start":"05:02.115 ","End":"05:04.810","Text":"Just to save space and time."},{"Start":"05:04.810 ","End":"05:08.240","Text":"Those are our solution and I\u0027ll just highlight them."},{"Start":"05:08.240 ","End":"05:11.440","Text":"This is 1 family and this is family 2 and"},{"Start":"05:11.440 ","End":"05:17.165","Text":"3 with the plus or minus and k in each case is any integer."},{"Start":"05:17.165 ","End":"05:19.939","Text":"Onto part b."},{"Start":"05:19.939 ","End":"05:28.915","Text":"We\u0027ve already started by giving the formula we\u0027re going to use on the substitution."},{"Start":"05:28.915 ","End":"05:31.940","Text":"In this case we get since t is sine x,"},{"Start":"05:31.940 ","End":"05:35.570","Text":"the first term is 6t^2 plus,"},{"Start":"05:35.570 ","End":"05:38.630","Text":"now the cosine 2x, instead of that,"},{"Start":"05:38.630 ","End":"05:40.925","Text":"I\u0027m going to write what\u0027s written here,"},{"Start":"05:40.925 ","End":"05:44.455","Text":"which is 1 minus,"},{"Start":"05:44.455 ","End":"05:50.930","Text":"now this is 2t^2 because this sine x is t,"},{"Start":"05:50.930 ","End":"05:53.206","Text":"so sine^2(x) is t^2,"},{"Start":"05:53.206 ","End":"05:56.820","Text":"and then from here, minus 2t=2."},{"Start":"05:59.230 ","End":"06:02.030","Text":"Now this is a quadratic equation."},{"Start":"06:02.030 ","End":"06:03.575","Text":"We just have to tidy it up,"},{"Start":"06:03.575 ","End":"06:04.970","Text":"bring everything to the left,"},{"Start":"06:04.970 ","End":"06:06.965","Text":"and collect like terms."},{"Start":"06:06.965 ","End":"06:09.950","Text":"6t^2 minus 2t^2 is"},{"Start":"06:09.950 ","End":"06:18.615","Text":"4t^2 minus 2t plus 1 minus 2,"},{"Start":"06:18.615 ","End":"06:24.445","Text":"which is minus 1=0."},{"Start":"06:24.445 ","End":"06:28.370","Text":"Let\u0027s see, let\u0027s use the quadratic formula."},{"Start":"06:28.370 ","End":"06:33.635","Text":"We get that t is minus b plus or minus the square root"},{"Start":"06:33.635 ","End":"06:41.355","Text":"of b^2 minus 4 times a times c,"},{"Start":"06:41.355 ","End":"06:46.789","Text":"and all this over 2a,"},{"Start":"06:46.789 ","End":"06:52.515","Text":"and this=2 plus or minus,"},{"Start":"06:52.515 ","End":"06:57.925","Text":"now here we get 4 times 4 is 16 and it\u0027s plus because it\u0027s minus, minus,"},{"Start":"06:57.925 ","End":"07:00.205","Text":"16 plus 4 is 20,"},{"Start":"07:00.205 ","End":"07:01.930","Text":"it\u0027s not a whole number,"},{"Start":"07:01.930 ","End":"07:05.890","Text":"but we can do something anyway, over 8."},{"Start":"07:05.890 ","End":"07:13.690","Text":"Now, I\u0027ll do this at the side and say that the square root of 20,"},{"Start":"07:13.690 ","End":"07:18.955","Text":"I can write it as 4 times 5."},{"Start":"07:18.955 ","End":"07:22.940","Text":"Why would I want to write it as 4 times 5?"},{"Start":"07:23.100 ","End":"07:26.920","Text":"Because this is equal to square root of 4,"},{"Start":"07:26.920 ","End":"07:27.940","Text":"square root of 5,"},{"Start":"07:27.940 ","End":"07:29.775","Text":"twice square root of 5."},{"Start":"07:29.775 ","End":"07:31.490","Text":"I\u0027m not sure that it will really help,"},{"Start":"07:31.490 ","End":"07:39.020","Text":"but it\u0027s good practice anyway to work with the square roots and simplify them."},{"Start":"07:39.020 ","End":"07:41.090","Text":"Let\u0027s see what we do get."},{"Start":"07:41.090 ","End":"07:48.450","Text":"We get in this case 2 plus or minus 2 root 5/8."},{"Start":"07:49.670 ","End":"07:53.195","Text":"Then if we divide top and bottom by 2,"},{"Start":"07:53.195 ","End":"08:01.040","Text":"we get 1 plus or minus root 5/4."},{"Start":"08:01.040 ","End":"08:07.790","Text":"That means 1 plus root 5/4 and"},{"Start":"08:07.790 ","End":"08:16.160","Text":"1 minus root 5/4."},{"Start":"08:16.160 ","End":"08:20.040","Text":"Now, these values are of course,"},{"Start":"08:20.040 ","End":"08:27.575","Text":"for sine x. I\u0027m substituting back from t, I need to find x."},{"Start":"08:27.575 ","End":"08:30.844","Text":"Sine x is 1 of 2 things."},{"Start":"08:30.844 ","End":"08:36.080","Text":"Could be, let me write this."},{"Start":"08:36.080 ","End":"08:40.820","Text":"I have a reason, root 5 plus 1/4."},{"Start":"08:40.820 ","End":"08:48.200","Text":"The other 1 I\u0027m going to write as root 5 minus 1/4."},{"Start":"08:48.200 ","End":"08:49.940","Text":"But since I\u0027ve reversed the order,"},{"Start":"08:49.940 ","End":"08:51.320","Text":"I\u0027ll put a minus."},{"Start":"08:51.320 ","End":"08:57.230","Text":"Now, these are not famous, they\u0027re semi famous."},{"Start":"08:57.230 ","End":"08:58.745","Text":"I\u0027ve encountered these before."},{"Start":"08:58.745 ","End":"09:01.610","Text":"I recognize that this is the sine of"},{"Start":"09:01.610 ","End":"09:06.590","Text":"54 degrees and that this bit is the sine of minus 18."},{"Start":"09:06.590 ","End":"09:08.510","Text":"But of course,"},{"Start":"09:08.510 ","End":"09:11.220","Text":"you would use the calculator."},{"Start":"09:12.790 ","End":"09:19.790","Text":"What you would do is do a square root of 5 plus 1/4 and then shift sine or inverse sine."},{"Start":"09:19.790 ","End":"09:22.505","Text":"You\u0027ll see that if we take the first case,"},{"Start":"09:22.505 ","End":"09:32.620","Text":"we get that sine x=sine of 54 degrees exactly."},{"Start":"09:32.630 ","End":"09:36.105","Text":"We will solve this in a second."},{"Start":"09:36.105 ","End":"09:41.235","Text":"The other case, I would say that sine x"},{"Start":"09:41.235 ","End":"09:51.110","Text":"is minus the sine of 18."},{"Start":"09:51.110 ","End":"09:54.556","Text":"I don\u0027t really want to write the degree sign,"},{"Start":"09:54.556 ","End":"09:56.995","Text":"it\u0027s just force of habit."},{"Start":"09:56.995 ","End":"10:00.770","Text":"In this case, we put the minus inside because that\u0027s what we do"},{"Start":"10:00.770 ","End":"10:04.515","Text":"for sine if it\u0027s negative."},{"Start":"10:04.515 ","End":"10:10.030","Text":"This is sine of minus 18."},{"Start":"10:11.470 ","End":"10:14.750","Text":"I\u0027m assuming you don\u0027t know these values,"},{"Start":"10:14.750 ","End":"10:17.720","Text":"so you would just do it right from the beginning from the calculator."},{"Start":"10:17.720 ","End":"10:20.915","Text":"You would do this, shift sine, get 54."},{"Start":"10:20.915 ","End":"10:28.105","Text":"You would compute this as is and do shift sine or inverse sine and get minus 18,"},{"Start":"10:28.105 ","End":"10:29.750","Text":"and then we\u0027re at the same place."},{"Start":"10:29.750 ","End":"10:35.585","Text":"I just happen to recognize this because in time you learn these things."},{"Start":"10:35.585 ","End":"10:39.055","Text":"But that\u0027s what we have calculators for."},{"Start":"10:39.055 ","End":"10:44.540","Text":"Continuing, in each case we get 2 families of solutions."},{"Start":"10:44.540 ","End":"10:50.540","Text":"In the first 1 we get that x= I know there\u0027s going be 2 of them,"},{"Start":"10:50.540 ","End":"10:55.400","Text":"so that\u0027s 54 plus multiples of 360,"},{"Start":"10:55.400 ","End":"10:57.335","Text":"k is an integer,"},{"Start":"10:57.335 ","End":"11:00.110","Text":"and the other 1 we subtract from 180."},{"Start":"11:00.110 ","End":"11:01.850","Text":"We can do this in our heads."},{"Start":"11:01.850 ","End":"11:08.190","Text":"This becomes 126 plus 360k."},{"Start":"11:08.890 ","End":"11:14.435","Text":"For the other 1, we get that x"},{"Start":"11:14.435 ","End":"11:21.740","Text":"is either minus 18 plus 360k."},{"Start":"11:21.740 ","End":"11:26.280","Text":"Here we subtract from"},{"Start":"11:27.100 ","End":"11:35.010","Text":"180, it becomes 198."},{"Start":"11:35.010 ","End":"11:39.880","Text":"Because 180 minus, minus 18 is plus 18, 360k."},{"Start":"11:40.040 ","End":"11:46.455","Text":"There are 4 families of solutions; the 54,"},{"Start":"11:46.455 ","End":"11:49.485","Text":"the 126, the minus 18,"},{"Start":"11:49.485 ","End":"11:55.330","Text":"and the 198, each of them with multiples of 360."},{"Start":"11:56.600 ","End":"12:03.490","Text":"That\u0027s part b and we are done."}],"ID":5419},{"Watched":false,"Name":"Exercise 27","Duration":"15m 54s","ChapterTopicVideoID":5421,"CourseChapterTopicPlaylistID":257207,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5421.jpeg","UploadDate":"2016-03-10T21:39:04.8470000","DurationForVideoObject":"PT15M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"This exercise has 2 parts,"},{"Start":"00:02.550 ","End":"00:06.705","Text":"and in each of them, we have to solve a trigonometric equation."},{"Start":"00:06.705 ","End":"00:08.670","Text":"In each case we\u0027re going to try,"},{"Start":"00:08.670 ","End":"00:11.100","Text":"and use the method of substitution."},{"Start":"00:11.100 ","End":"00:15.405","Text":"But, we can\u0027t use it immediately because each of them is mixed."},{"Start":"00:15.405 ","End":"00:16.493","Text":"We have a sin x,"},{"Start":"00:16.493 ","End":"00:18.105","Text":"and we have a sin 2x."},{"Start":"00:18.105 ","End":"00:21.210","Text":"Here we have a sin x, and cos 2x."},{"Start":"00:21.210 ","End":"00:24.030","Text":"For substitution, we have to have it all in terms of"},{"Start":"00:24.030 ","End":"00:27.780","Text":"one trigonometric function or expression."},{"Start":"00:27.780 ","End":"00:32.025","Text":"Let\u0027s go with one at a time and see."},{"Start":"00:32.025 ","End":"00:35.235","Text":"In part A,"},{"Start":"00:35.235 ","End":"00:37.245","Text":"you have to think a bit,"},{"Start":"00:37.245 ","End":"00:43.185","Text":"and my idea is to use the formula for sin 2x."},{"Start":"00:43.185 ","End":"00:45.860","Text":"Now, you should have these formulas handy,"},{"Start":"00:45.860 ","End":"00:48.335","Text":"or better still memorize them."},{"Start":"00:48.335 ","End":"00:51.480","Text":"In general, sin of 2 something,"},{"Start":"00:51.480 ","End":"00:58.605","Text":"let\u0027s say Alpha is 2 sin Alpha cos Alpha."},{"Start":"00:58.605 ","End":"01:01.995","Text":"In our case, Alpha will be x."},{"Start":"01:01.995 ","End":"01:05.540","Text":"The idea is that if we do that substitution,"},{"Start":"01:05.540 ","End":"01:07.025","Text":"and we square everything,"},{"Start":"01:07.025 ","End":"01:08.998","Text":"then we\u0027ll have sin^2,"},{"Start":"01:08.998 ","End":"01:15.980","Text":"and cos^2, and we already know how to get from cos^2 squared to sin^2."},{"Start":"01:15.980 ","End":"01:21.960","Text":"Let\u0027s begin. What we have here is sin^2 x so,"},{"Start":"01:21.960 ","End":"01:23.895","Text":"I\u0027ll leave as it is,"},{"Start":"01:23.895 ","End":"01:29.550","Text":"plus now sin(2x) is 2 sin"},{"Start":"01:29.550 ","End":"01:38.170","Text":"x cos(x)^2 and this equals 1."},{"Start":"01:38.680 ","End":"01:41.675","Text":"Just open the brackets here."},{"Start":"01:41.675 ","End":"01:43.445","Text":"I have to square everything."},{"Start":"01:43.445 ","End":"01:45.685","Text":"2^2 is 4."},{"Start":"01:45.685 ","End":"01:49.415","Text":"Oh, I forgot the x here. Sorry about that."},{"Start":"01:49.415 ","End":"01:51.275","Text":"We\u0027ll fix it in the next line,"},{"Start":"01:51.275 ","End":"02:01.460","Text":"where we have sin^2(x) and cos^2 (x)=1."},{"Start":"02:01.460 ","End":"02:04.615","Text":"At this point, I have sin^2."},{"Start":"02:04.615 ","End":"02:06.800","Text":"I just need to do something about this,"},{"Start":"02:06.800 ","End":"02:09.755","Text":"and that\u0027s where I\u0027ll use another formula,"},{"Start":"02:09.755 ","End":"02:18.390","Text":"which is that cos^2 Alpha is 1 minus sin^2 Alpha."},{"Start":"02:19.060 ","End":"02:26.110","Text":"At this point we get sin^2x plus"},{"Start":"02:26.110 ","End":"02:29.645","Text":"4 sin^2 x times"},{"Start":"02:29.645 ","End":"02:37.460","Text":"1 minus sin^2 x=1."},{"Start":"02:37.460 ","End":"02:39.200","Text":"Now if you notice,"},{"Start":"02:39.200 ","End":"02:42.170","Text":"sure everything\u0027s in terms of sine x,"},{"Start":"02:42.170 ","End":"02:45.725","Text":"but even better, everything\u0027s in terms of sin^2 x."},{"Start":"02:45.725 ","End":"02:50.480","Text":"So we can get it even simpler if we make our substitution"},{"Start":"02:50.480 ","End":"02:57.590","Text":"for sin^2 x = t. Then we won\u0027t have so many exponents,"},{"Start":"02:57.590 ","End":"03:00.320","Text":"and I won\u0027t have something like t^2, 1 minus t^2,"},{"Start":"03:00.320 ","End":"03:03.140","Text":"if I let t be sin^2 x,"},{"Start":"03:03.140 ","End":"03:06.690","Text":"we\u0027ll get a lower exponent."},{"Start":"03:07.990 ","End":"03:15.425","Text":"Here we get just T because the whole sin^2 is t, plus 4,"},{"Start":"03:15.425 ","End":"03:17.668","Text":"this is t,"},{"Start":"03:17.668 ","End":"03:22.980","Text":"and this is 1 minus t. This equals 1."},{"Start":"03:22.980 ","End":"03:26.870","Text":"This is going to be a quadratic equation."},{"Start":"03:26.870 ","End":"03:29.720","Text":"Let\u0027s see if we can do it all in our heads."},{"Start":"03:29.720 ","End":"03:32.930","Text":"I\u0027m going to move everything to the right-hand side because I want"},{"Start":"03:32.930 ","End":"03:37.025","Text":"the t^2 bit to come up positive. On the right."},{"Start":"03:37.025 ","End":"03:40.195","Text":"I have 4t^2,"},{"Start":"03:40.195 ","End":"03:43.118","Text":"that\u0027s the only place to get t^2 squared from,"},{"Start":"03:43.118 ","End":"03:45.170","Text":"and then t I can get in 2 ways,"},{"Start":"03:45.170 ","End":"03:50.430","Text":"minus t, and then minus 4t."},{"Start":"03:50.430 ","End":"03:53.335","Text":"I get minus 5t."},{"Start":"03:53.335 ","End":"03:55.550","Text":"Of course afterwards I\u0027m putting what\u0027s on the right,"},{"Start":"03:55.550 ","End":"03:58.595","Text":"on the left, but meanwhile we\u0027re thinking of it as the right."},{"Start":"03:58.595 ","End":"04:05.690","Text":"Then we have the plus 1 that already was there on the right,"},{"Start":"04:05.690 ","End":"04:08.915","Text":"and then we put it on the left equals 0."},{"Start":"04:08.915 ","End":"04:14.810","Text":"This is now a quadratic equation in t. Let\u0027s solve what t is."},{"Start":"04:14.810 ","End":"04:22.270","Text":"T is using the formula minus b plus or minus the square root of b^2,"},{"Start":"04:22.270 ","End":"04:28.155","Text":"25 minus 4 times a times"},{"Start":"04:28.155 ","End":"04:35.880","Text":"c over 2a is 8."},{"Start":"04:35.880 ","End":"04:39.520","Text":"Let\u0027s see what that gives us."},{"Start":"04:40.690 ","End":"04:43.400","Text":"Under the square root,"},{"Start":"04:43.400 ","End":"04:47.395","Text":"it\u0027s 25- 16 is 9."},{"Start":"04:47.395 ","End":"04:52.290","Text":"I\u0027ve got 5 plus 3 over 2,"},{"Start":"04:52.290 ","End":"05:00.060","Text":"the 3 is from the square root of 9 and 5 minus 3 over 2."},{"Start":"05:00.490 ","End":"05:08.660","Text":"The plus 1 gives me 8/2 and is 4."},{"Start":"05:08.660 ","End":"05:15.680","Text":"5 minus 3 over 2 is, 2/2 is 1."},{"Start":"05:15.680 ","End":"05:21.250","Text":"Now remember that t was sin^2 x."},{"Start":"05:21.250 ","End":"05:25.040","Text":"When I want to substitute back from here,"},{"Start":"05:25.040 ","End":"05:27.930","Text":"I\u0027m going to get that"},{"Start":"05:29.390 ","End":"05:39.920","Text":"sin^2 x could be either 4 or 1,"},{"Start":"05:39.920 ","End":"05:45.800","Text":"and so sine x has"},{"Start":"05:45.800 ","End":"05:53.915","Text":"actually 4 possibilities because it could be from the 4,"},{"Start":"05:53.915 ","End":"06:04.215","Text":"I can get 2 or minus 2 when I take the square root plus or minus and from the 1,"},{"Start":"06:04.215 ","End":"06:08.050","Text":"I can get 1, or minus 1."},{"Start":"06:08.050 ","End":"06:11.840","Text":"If that\u0027s not clear, let me just explain it again."},{"Start":"06:11.840 ","End":"06:15.860","Text":"If sin^2 x is 4,"},{"Start":"06:15.860 ","End":"06:18.049","Text":"when something squared is something,"},{"Start":"06:18.049 ","End":"06:20.630","Text":"then the thing itself is plus,"},{"Start":"06:20.630 ","End":"06:24.975","Text":"or minus the square root of that and that\u0027s plus or minus 2."},{"Start":"06:24.975 ","End":"06:26.390","Text":"That gives us these two."},{"Start":"06:26.390 ","End":"06:28.160","Text":"Similarly, if sin^ x is 1,"},{"Start":"06:28.160 ","End":"06:30.260","Text":"sin x is plus 1 or minus 1."},{"Start":"06:30.260 ","End":"06:34.820","Text":"There\u0027s 4 possibilities for sin(x) by taking plus or"},{"Start":"06:34.820 ","End":"06:40.740","Text":"minus the square root of whatever this was."},{"Start":"06:43.010 ","End":"06:47.245","Text":"But, sine has a limit."},{"Start":"06:47.245 ","End":"06:50.340","Text":"Maximum is 1 and the minimum is minus 1."},{"Start":"06:50.340 ","End":"06:53.290","Text":"It has to be between 1 and minus 1."},{"Start":"06:53.840 ","End":"06:56.375","Text":"This is out of range,"},{"Start":"06:56.375 ","End":"06:58.525","Text":"and this is out of range."},{"Start":"06:58.525 ","End":"07:00.739","Text":"1 and minus 1 are borderline,"},{"Start":"07:00.739 ","End":"07:02.128","Text":"but they\u0027re in,"},{"Start":"07:02.128 ","End":"07:03.680","Text":"the border is included."},{"Start":"07:03.680 ","End":"07:06.280","Text":"We have 2 possibilities."},{"Start":"07:06.280 ","End":"07:11.315","Text":"The first possibility is sin(x)=1,"},{"Start":"07:11.315 ","End":"07:19.385","Text":"and the second possibility is sin(x)= -1."},{"Start":"07:19.385 ","End":"07:21.170","Text":"We\u0027ve seen these before,"},{"Start":"07:21.170 ","End":"07:23.315","Text":"so I\u0027ll just go over them quickly."},{"Start":"07:23.315 ","End":"07:30.455","Text":"1 is sin of 90 and so what we get is"},{"Start":"07:30.455 ","End":"07:37.490","Text":"that x=90"},{"Start":"07:37.490 ","End":"07:45.630","Text":"degrees plus 360 times k."},{"Start":"07:46.290 ","End":"07:48.460","Text":"No need to write the degrees."},{"Start":"07:48.460 ","End":"07:50.365","Text":"It\u0027s just a habit I have."},{"Start":"07:50.365 ","End":"07:54.670","Text":"And that\u0027s the solution for sin(x)=1."},{"Start":"07:54.670 ","End":"07:55.960","Text":"Now you might be thinking,"},{"Start":"07:55.960 ","End":"07:58.670","Text":"wait a minute, Don\u0027t we also do 180-?"},{"Start":"07:58.860 ","End":"08:01.180","Text":"Yeah, we normally do,"},{"Start":"08:01.180 ","End":"08:05.995","Text":"but 180-90=90, so we get the same thing again."},{"Start":"08:05.995 ","End":"08:12.055","Text":"So, in this particular case there\u0027s no need to do the other,180-,"},{"Start":"08:12.055 ","End":"08:14.875","Text":"doesn\u0027t add anything. So, that\u0027s for this."},{"Start":"08:14.875 ","End":"08:19.090","Text":"And now let\u0027s go for the sin(x)=-1."},{"Start":"08:19.090 ","End":"08:25.240","Text":"And this is the sin(-90)."},{"Start":"08:25.240 ","End":"08:26.515","Text":"So, in this case,"},{"Start":"08:26.515 ","End":"08:30.955","Text":"we get, for this one, that x equals,"},{"Start":"08:30.955 ","End":"08:33.710","Text":"I\u0027m just continuing over here,"},{"Start":"08:34.110 ","End":"08:41.995","Text":"-90+360k and this is the other set of solutions."},{"Start":"08:41.995 ","End":"08:44.635","Text":"So, we\u0027ll highlight that also."},{"Start":"08:44.635 ","End":"08:47.080","Text":"And once again you might be saying, wait a minute,"},{"Start":"08:47.080 ","End":"08:50.510","Text":"don\u0027t we also do a 180-?"},{"Start":"08:51.090 ","End":"08:53.380","Text":"And we\u0027ve been here before,"},{"Start":"08:53.380 ","End":"08:54.820","Text":"but I\u0027ll just repeat what we did last time,"},{"Start":"08:54.820 ","End":"09:01.360","Text":"then we said no because 180- this gives us 270 and 270 Et cetera,"},{"Start":"09:01.360 ","End":"09:05.245","Text":"is included in this because if k=1,"},{"Start":"09:05.245 ","End":"09:09.355","Text":"then -90+360=270 and so on."},{"Start":"09:09.355 ","End":"09:10.870","Text":"So, it turns out that these are"},{"Start":"09:10.870 ","End":"09:15.400","Text":"the only solutions in each of these extreme cases of 1 and"},{"Start":"09:15.400 ","End":"09:21.355","Text":"-1 and that basically is the answer to part A."},{"Start":"09:21.355 ","End":"09:25.000","Text":"So, let\u0027s get on to part B now."},{"Start":"09:25.000 ","End":"09:29.740","Text":"And in part B, once again,"},{"Start":"09:29.740 ","End":"09:35.890","Text":"we want to use trigonometric identities to get us out of this."},{"Start":"09:35.890 ","End":"09:41.320","Text":"We\u0027re going to use formula for cos(2x)."},{"Start":"09:41.320 ","End":"09:43.510","Text":"There\u0027s more than one identity,"},{"Start":"09:43.510 ","End":"09:50.005","Text":"but the one that we want is the one that puts cos(2x) in terms of sine or rather sin^2."},{"Start":"09:50.005 ","End":"10:00.080","Text":"And that is that in general, cos(2α)=1-2sin(α)^2."},{"Start":"10:01.230 ","End":"10:04.345","Text":"In our case, Alpha would be X."},{"Start":"10:04.345 ","End":"10:13.015","Text":"And so, what we get is 4sin^2, sorry, that\u0027s sin^4."},{"Start":"10:13.015 ","End":"10:14.845","Text":"But you know what? That\u0027s okay."},{"Start":"10:14.845 ","End":"10:19.550","Text":"Because sin^4=(sin^2)^2."},{"Start":"10:19.740 ","End":"10:24.730","Text":"To the power of 4 and take to the power of 2 to the power of 2. This is actually good."},{"Start":"10:24.730 ","End":"10:25.660","Text":"And"},{"Start":"10:25.660 ","End":"10:27.050","Text":"then"},{"Start":"10:30.930 ","End":"10:37.345","Text":"+7(1-2sin(x)^2) from there,"},{"Start":"10:37.345 ","End":"10:41.905","Text":"with x being Alpha equals 1."},{"Start":"10:41.905 ","End":"10:44.605","Text":"Now we substitute."},{"Start":"10:44.605 ","End":"10:48.025","Text":"Everywhere we see sin(x)^2,"},{"Start":"10:48.025 ","End":"10:58.970","Text":"we\u0027re going to put t. And so, we get 4t^2+7(1-2t)=1."},{"Start":"11:02.790 ","End":"11:06.730","Text":"And that gives me a quadratic equation,"},{"Start":"11:06.730 ","End":"11:09.290","Text":"just bringing everything to the left."},{"Start":"11:21.420 ","End":"11:25.210","Text":"4t^2-14t+6=0. Everything to divide by 2."},{"Start":"11:25.210 ","End":"11:27.625","Text":"So, let\u0027s divide by 2."},{"Start":"11:27.625 ","End":"11:37.150","Text":"We get 2t^2-7t+3=0."},{"Start":"11:37.150 ","End":"11:39.340","Text":"Now a quadratic equation."},{"Start":"11:39.340 ","End":"11:41.065","Text":"Let\u0027s see what t is."},{"Start":"11:41.065 ","End":"11:45.760","Text":"It\u0027s minus b plus or minus the square root of b"},{"Start":"11:45.760 ","End":"11:51.940","Text":"squared minus 4 times a times c,"},{"Start":"11:51.940 ","End":"11:55.000","Text":"all this over 2a."},{"Start":"11:55.000 ","End":"12:00.340","Text":"Let\u0027s see, 4*2*3,8*3."},{"Start":"12:00.340 ","End":"12:08.410","Text":"24, 49-24=25, and√25=5."},{"Start":"12:08.410 ","End":"12:15.670","Text":"So I\u0027ve got 7±5 over 4."},{"Start":"12:15.670 ","End":"12:19.390","Text":"If I take the plus, I get,"},{"Start":"12:19.390 ","End":"12:24.310","Text":"let\u0027s see, 12/4 is 3."},{"Start":"12:24.310 ","End":"12:25.675","Text":"If I take the minus,"},{"Start":"12:25.675 ","End":"12:29.785","Text":"I get 2/4, which is 1.5."},{"Start":"12:29.785 ","End":"12:36.295","Text":"Now this is t. These 2 values are the possible values of t,"},{"Start":"12:36.295 ","End":"12:39.040","Text":"but t is sin^2."},{"Start":"12:39.040 ","End":"12:50.710","Text":"So, what this gives us is that sin(x)^2=3 or 1.5."},{"Start":"12:50.710 ","End":"12:52.810","Text":"Now as before, we can\u0027t"},{"Start":"12:52.810 ","End":"13:02.230","Text":"have sin(x)^2=3 because then sin(x) would be ±√3."},{"Start":"13:02.230 ","End":"13:03.970","Text":"And that\u0027s out of range,"},{"Start":"13:03.970 ","End":"13:05.770","Text":"√3 is 1.7 something,"},{"Start":"13:05.770 ","End":"13:07.600","Text":"definitely out of range.