Introduction
0/2 completed

Conversion between Polar and Cartesian Coordinates
0/7 completed

Tangent Lines in Polar Coordinates
0/3 completed

Arc Length in Polar Coordinates
0/3 completed

Area in Polar Coordinates
0/3 completed

Surface Area in Polar Coordinates
0/3 completed

{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"18m 3s","ChapterTopicVideoID":5995,"CourseChapterTopicPlaylistID":4005,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this clip, we\u0027ll be starting a new topic called polar coordinates,"},{"Start":"00:05.280 ","End":"00:08.970","Text":"as opposed to Cartesian coordinates,"},{"Start":"00:08.970 ","End":"00:11.070","Text":"which we know so far."},{"Start":"00:11.070 ","End":"00:15.870","Text":"The Cartesian coordinates are also called rectangular"},{"Start":"00:15.870 ","End":"00:22.540","Text":"coordinates and xy coordinates,"},{"Start":"00:22.940 ","End":"00:28.020","Text":"but mostly the old kind that we know so far,"},{"Start":"00:28.020 ","End":"00:29.820","Text":"I\u0027ll be calling them Cartesian."},{"Start":"00:29.820 ","End":"00:33.735","Text":"It\u0027s named after the French mathematician Descartes,"},{"Start":"00:33.735 ","End":"00:36.760","Text":"the guy who said, \"I think, therefore I am.\""},{"Start":"00:37.690 ","End":"00:43.415","Text":"In this diagram, if you ignore the colored bit, just the black,"},{"Start":"00:43.415 ","End":"00:45.470","Text":"what we see is that we\u0027re given a point in"},{"Start":"00:45.470 ","End":"00:49.720","Text":"the plane and we want to describe its position,"},{"Start":"00:49.720 ","End":"00:53.025","Text":"and you need 2 numbers because it\u0027s in 2 dimensions."},{"Start":"00:53.025 ","End":"01:00.600","Text":"The usual way is to take y-axis and an x-axis"},{"Start":"01:00.600 ","End":"01:10.535","Text":"and we measure how far east and how far north if you were thinking of it geographically."},{"Start":"01:10.535 ","End":"01:14.620","Text":"Those 2 numbers would tell me where the point is,"},{"Start":"01:14.620 ","End":"01:17.270","Text":"maybe it\u0027s like a latitude and longitude,"},{"Start":"01:17.270 ","End":"01:21.920","Text":"but that\u0027s not the only way of describing where a point is"},{"Start":"01:21.920 ","End":"01:27.110","Text":"relative to a certain origin and coordinates."},{"Start":"01:27.110 ","End":"01:29.450","Text":"You could also tell someone,"},{"Start":"01:29.450 ","End":"01:35.270","Text":"instead of saying go so many units east and so many units north,"},{"Start":"01:35.270 ","End":"01:41.200","Text":"you could say, go so many units in this direction given by this angle."},{"Start":"01:41.200 ","End":"01:43.655","Text":"That\u0027s what a polar coordinate is,"},{"Start":"01:43.655 ","End":"01:47.555","Text":"is just describing a point by an angle and a distance;"},{"Start":"01:47.555 ","End":"01:51.870","Text":"how far you have to travel and in which direction."},{"Start":"01:53.200 ","End":"01:57.770","Text":"With Cartesian coordinates, we took the convention that"},{"Start":"01:57.770 ","End":"02:01.640","Text":"there\u0027s an x-axis which is horizontal and facing to the right,"},{"Start":"02:01.640 ","End":"02:07.625","Text":"and a y-axis which is vertical and going upwards, and an origin."},{"Start":"02:07.625 ","End":"02:11.630","Text":"With polar coordinates, we\u0027re going to assume,"},{"Start":"02:11.630 ","End":"02:15.560","Text":"let me bring a sketch, here we are."},{"Start":"02:15.560 ","End":"02:18.230","Text":"In this case, we also have a point,"},{"Start":"02:18.230 ","End":"02:20.180","Text":"it\u0027s not called the origin anymore,"},{"Start":"02:20.180 ","End":"02:22.435","Text":"it\u0027s called the pole."},{"Start":"02:22.435 ","End":"02:30.994","Text":"This horizontal axis facing to the right or to the east is called the polar axis,"},{"Start":"02:30.994 ","End":"02:36.395","Text":"and the angle is conventionally measured counterclockwise."},{"Start":"02:36.395 ","End":"02:39.890","Text":"I don\u0027t know why mathematicians have to do things differently."},{"Start":"02:39.890 ","End":"02:43.340","Text":"In geography or navigation,"},{"Start":"02:43.340 ","End":"02:46.790","Text":"you would normally start from the north and go clockwise,"},{"Start":"02:46.790 ","End":"02:50.840","Text":"but mathematicians start from the east and go counterclockwise."},{"Start":"02:50.840 ","End":"02:52.625","Text":"It\u0027s just the way it is."},{"Start":"02:52.625 ","End":"02:54.350","Text":"When we have a point,"},{"Start":"02:54.350 ","End":"03:03.060","Text":"we just measure the distance to the pole and that distance maybe we use the letter r,"},{"Start":"03:03.060 ","End":"03:05.385","Text":"it\u0027s most commonly used."},{"Start":"03:05.385 ","End":"03:12.370","Text":"The counterclockwise angle from the polar axis is usually called Theta,"},{"Start":"03:12.370 ","End":"03:14.025","Text":"so instead of x and y,"},{"Start":"03:14.025 ","End":"03:15.960","Text":"we now have r and Theta."},{"Start":"03:15.960 ","End":"03:19.850","Text":"Also. 2 numbers, also written in brackets with a comma,"},{"Start":"03:19.850 ","End":"03:26.120","Text":"but we always understand whether we\u0027re in polar or Cartesian coordinates."},{"Start":"03:26.120 ","End":"03:30.550","Text":"It turns out that various situations are more"},{"Start":"03:30.550 ","End":"03:35.660","Text":"naturally described using polar rather than Cartesian."},{"Start":"03:35.790 ","End":"03:42.070","Text":"Initially, I\u0027m going to restrict r to being bigger or equal to 0."},{"Start":"03:42.070 ","End":"03:46.445","Text":"The distance is not going to be negative."},{"Start":"03:46.445 ","End":"03:51.715","Text":"Later, we\u0027ll generalize and we\u0027ll also see what it means if r is negative."},{"Start":"03:51.715 ","End":"04:00.580","Text":"Also, the angle will assume to be between 0 and 360 degrees if we\u0027re working in degrees."},{"Start":"04:00.580 ","End":"04:08.460","Text":"Sometimes we work in degrees and then I would say the angle Theta would be between 0,"},{"Start":"04:08.460 ","End":"04:15.840","Text":"and actually, I could exclude 360 degrees because 360 degrees would be back to 0."},{"Start":"04:15.840 ","End":"04:18.080","Text":"If we\u0027re working in radians,"},{"Start":"04:18.080 ","End":"04:25.120","Text":"I would say that Theta is between 0 and 2 Pi."},{"Start":"04:25.120 ","End":"04:27.260","Text":"Sometimes you write a radian sign."},{"Start":"04:27.260 ","End":"04:29.960","Text":"Actually, the radians sign is a little c here."},{"Start":"04:29.960 ","End":"04:31.670","Text":"As I say, this is just meanwhile,"},{"Start":"04:31.670 ","End":"04:37.325","Text":"later on we\u0027ll allow Theta to go more than a whole circle or even backwards,"},{"Start":"04:37.325 ","End":"04:40.045","Text":"but let\u0027s stick with this meanwhile."},{"Start":"04:40.045 ","End":"04:44.569","Text":"The pole is an exception because not only is r equal to 0,"},{"Start":"04:44.569 ","End":"04:46.580","Text":"but we don\u0027t really know what the angle is,"},{"Start":"04:46.580 ","End":"04:49.385","Text":"because the point is rotating on the spot."},{"Start":"04:49.385 ","End":"04:58.970","Text":"Actually, the coordinates would be 0 comma anything,"},{"Start":"04:58.970 ","End":"05:00.410","Text":"whatever you put here,"},{"Start":"05:00.410 ","End":"05:02.795","Text":"as long as the r is 0."},{"Start":"05:02.795 ","End":"05:05.915","Text":"The pole is determined basically by r equals 0,"},{"Start":"05:05.915 ","End":"05:08.555","Text":"Theta, we don\u0027t care about in this case."},{"Start":"05:08.555 ","End":"05:14.120","Text":"What I want to do now is show you some examples given a polar coordinate,"},{"Start":"05:14.120 ","End":"05:17.690","Text":"how I would locate it on the plane,"},{"Start":"05:17.690 ","End":"05:20.610","Text":"and the other way around."},{"Start":"05:21.050 ","End":"05:23.585","Text":"Here we have a plane,"},{"Start":"05:23.585 ","End":"05:29.195","Text":"and this is the pole and this is the polar axis and I put some units on."},{"Start":"05:29.195 ","End":"05:31.190","Text":"Let\u0027s go first of all,"},{"Start":"05:31.190 ","End":"05:32.990","Text":"in the direction of,"},{"Start":"05:32.990 ","End":"05:37.565","Text":"I\u0027ll give you a point and you find its polar coordinates."},{"Start":"05:37.565 ","End":"05:40.500","Text":"Let me take this point and I\u0027ll now call it"},{"Start":"05:40.500 ","End":"05:45.890","Text":"A. I joined the point A to the origin and then I measure"},{"Start":"05:45.890 ","End":"05:54.495","Text":"its length and I find that it\u0027s equal to 5 just by measurement."},{"Start":"05:54.495 ","End":"06:02.725","Text":"Then I look at the angle Theta and then measure it using an angle measurer, a protractor,"},{"Start":"06:02.725 ","End":"06:11.730","Text":"and it comes out to be about 37 degrees."},{"Start":"06:12.360 ","End":"06:15.475","Text":"If I knew its coordinates,"},{"Start":"06:15.475 ","End":"06:18.325","Text":"then I could use a conversion."},{"Start":"06:18.325 ","End":"06:21.760","Text":"In fact, let me give you a sneak preview."},{"Start":"06:21.760 ","End":"06:28.225","Text":"We\u0027ll soon be talking about how to convert from polar to Cartesian and vice versa."},{"Start":"06:28.225 ","End":"06:29.450","Text":"If we have r and Theta,"},{"Start":"06:29.450 ","End":"06:32.770","Text":"we\u0027d be using this pair of formulas that would give us x, y,"},{"Start":"06:32.770 ","End":"06:34.390","Text":"and if we had x and y,"},{"Start":"06:34.390 ","End":"06:35.710","Text":"we would use this formula,"},{"Start":"06:35.710 ","End":"06:37.835","Text":"but we\u0027ll get into that soon."},{"Start":"06:37.835 ","End":"06:40.020","Text":"I actually just cooked up this example,"},{"Start":"06:40.020 ","End":"06:47.985","Text":"I actually made this so that this point A was actually in Cartesian, the point 4,3."},{"Start":"06:47.985 ","End":"06:49.920","Text":"I just measured 1, 2,"},{"Start":"06:49.920 ","End":"06:51.735","Text":"3, 4 and then 1, 2,"},{"Start":"06:51.735 ","End":"06:55.990","Text":"3 and then r came out."},{"Start":"06:56.090 ","End":"06:58.770","Text":"We\u0027re semi learning this from now,"},{"Start":"06:58.770 ","End":"07:00.180","Text":"we\u0027ll come to this in more details."},{"Start":"07:00.180 ","End":"07:03.180","Text":"I plugged in 3 and 4 here and got 5,"},{"Start":"07:03.180 ","End":"07:08.220","Text":"and then I used the calculator to figure out the arctangent of y over x."},{"Start":"07:08.220 ","End":"07:11.580","Text":"Actually, the point A came out to be 5 and more"},{"Start":"07:11.580 ","End":"07:16.970","Text":"precisely to 2 decimal places, 36.87 degrees."},{"Start":"07:16.970 ","End":"07:19.640","Text":"Let\u0027s say we\u0027ll work in degrees now,"},{"Start":"07:19.640 ","End":"07:22.535","Text":"sometimes radians, sometimes degrees."},{"Start":"07:22.535 ","End":"07:24.230","Text":"Let\u0027s go on to another point."},{"Start":"07:24.230 ","End":"07:31.460","Text":"Suppose I give you the point B and let\u0027s say the point B is this point here."},{"Start":"07:31.460 ","End":"07:33.830","Text":"This time I wouldn\u0027t need any conversions or"},{"Start":"07:33.830 ","End":"07:36.740","Text":"anything because I can see that the distance is,"},{"Start":"07:36.740 ","End":"07:37.895","Text":"we\u0027ll count 1, 2,"},{"Start":"07:37.895 ","End":"07:43.509","Text":"it comes out to be 6 just by measuring the distance."},{"Start":"07:43.520 ","End":"07:46.490","Text":"Measure the distance 6,"},{"Start":"07:46.490 ","End":"07:49.820","Text":"and then the angle I can see is 90 degrees."},{"Start":"07:49.820 ","End":"07:52.910","Text":"Let\u0027s give another example."},{"Start":"07:52.910 ","End":"07:56.949","Text":"I\u0027ll now take a point C here."},{"Start":"07:56.949 ","End":"08:02.060","Text":"Again, I cooked it up with the numbers so I know what it\u0027s going to come out to be."},{"Start":"08:02.060 ","End":"08:03.680","Text":"But if you\u0027re just measuring,"},{"Start":"08:03.680 ","End":"08:07.980","Text":"you would just take the scale, measure it,"},{"Start":"08:07.980 ","End":"08:17.865","Text":"and you\u0027d find that the length from the pole to C would be 3 units and the angle,"},{"Start":"08:17.865 ","End":"08:21.330","Text":"it looked like it\u0027s going to hit the diagonal,"},{"Start":"08:21.330 ","End":"08:25.535","Text":"so it\u0027s 45 degrees from here plus another 90."},{"Start":"08:25.535 ","End":"08:31.050","Text":"The angle looks like it\u0027s 135 degrees."},{"Start":"08:31.820 ","End":"08:36.510","Text":"This wasn\u0027t 3, I measured it as 4.2,"},{"Start":"08:36.510 ","End":"08:39.380","Text":"in fact, 4.24 to 2 decimal places."},{"Start":"08:39.380 ","End":"08:43.540","Text":"But it\u0027s just the idea, you would just measure this approximately."},{"Start":"08:43.540 ","End":"08:50.520","Text":"Maybe another 1, let\u0027s say this point over here,"},{"Start":"08:50.520 ","End":"08:56.700","Text":"I\u0027ll call that point D. This point,"},{"Start":"08:56.700 ","End":"09:03.600","Text":"I can see that the point D is"},{"Start":"09:03.600 ","End":"09:11.940","Text":"2 units away from the pole and the angle from here to here is 270 degrees."},{"Start":"09:11.940 ","End":"09:13.710","Text":"That\u0027s just a few examples,"},{"Start":"09:13.710 ","End":"09:15.670","Text":"and then let\u0027s go the other way,"},{"Start":"09:15.670 ","End":"09:23.980","Text":"where I\u0027ll give you coordinates and then you\u0027ve got to plot it on the plane."},{"Start":"09:25.770 ","End":"09:28.210","Text":"1 more example."},{"Start":"09:28.210 ","End":"09:30.445","Text":"What happens at the pole itself?"},{"Start":"09:30.445 ","End":"09:32.995","Text":"We also used to call the origin."},{"Start":"09:32.995 ","End":"09:37.690","Text":"Let\u0027s call that point E. That point E,"},{"Start":"09:37.690 ","End":"09:39.520","Text":"we can say what r is,"},{"Start":"09:39.520 ","End":"09:41.470","Text":"but we can\u0027t say what Theta is."},{"Start":"09:41.470 ","End":"09:42.670","Text":"It doesn\u0027t even matter."},{"Start":"09:42.670 ","End":"09:44.005","Text":"I\u0027ll just leave it as a question mark."},{"Start":"09:44.005 ","End":"09:45.760","Text":"Any Theta will do."},{"Start":"09:45.760 ","End":"09:51.130","Text":"Usually, we just say r equals 0 because that says it all."},{"Start":"09:51.130 ","End":"09:52.780","Text":"If I travel 0 distance,"},{"Start":"09:52.780 ","End":"09:54.220","Text":"it doesn\u0027t matter the direction,"},{"Start":"09:54.220 ","End":"09:55.930","Text":"there is no direction."},{"Start":"09:55.930 ","End":"09:59.630","Text":"Now, the opposite task."},{"Start":"09:59.700 ","End":"10:03.730","Text":"I think I\u0027ll erase this stuff first."},{"Start":"10:03.730 ","End":"10:14.365","Text":"I\u0027ll take a first example and this time I\u0027ll take P in polar coordinates as 3 comma."},{"Start":"10:14.365 ","End":"10:20.230","Text":"I\u0027ll work in degrees because you\u0027re probably more used to degrees."},{"Start":"10:20.230 ","End":"10:27.910","Text":"That means I have to find a point whose distance from the pole is 3 units,"},{"Start":"10:27.910 ","End":"10:31.180","Text":"but it has to be at a 180 degrees."},{"Start":"10:31.180 ","End":"10:38.875","Text":"So I go all the way around a 180 degrees and I get to this point here,"},{"Start":"10:38.875 ","End":"10:42.670","Text":"the arc of a 180 degrees in this direction."},{"Start":"10:42.670 ","End":"10:45.745","Text":"Let\u0027s take another example Q."},{"Start":"10:45.745 ","End":"10:52.960","Text":"Let say it is the point 6"},{"Start":"10:52.960 ","End":"11:02.810","Text":"comma and let say 315 degrees."},{"Start":"11:02.880 ","End":"11:06.820","Text":"Well, this is 90, 180,"},{"Start":"11:06.820 ","End":"11:11.185","Text":"270, this is already 360. It\u0027s somewhere in between."},{"Start":"11:11.185 ","End":"11:18.520","Text":"You can easily see that this is either 270 plus 45 or 360 minus 45."},{"Start":"11:18.520 ","End":"11:25.134","Text":"We\u0027re talking about 45 degrees below the horizontal, 6 units away."},{"Start":"11:25.134 ","End":"11:30.280","Text":"What I just did was drew the 315 degrees."},{"Start":"11:30.280 ","End":"11:32.650","Text":"This was the easy part because as I said,"},{"Start":"11:32.650 ","End":"11:38.230","Text":"it\u0027s like 45 degrees from the horizontal and the 6 I just measured roughly from here."},{"Start":"11:38.230 ","End":"11:46.390","Text":"The same distance is 6 units along and I found that the point Q is somewhere here."},{"Start":"11:46.390 ","End":"11:49.090","Text":"Later on we\u0027ll be doing it with these formulas."},{"Start":"11:49.090 ","End":"11:52.060","Text":"I haven\u0027t formally introduced them, but you could do it."},{"Start":"11:52.060 ","End":"11:56.515","Text":"It turns out that if you plug in these values for r and Theta,"},{"Start":"11:56.515 ","End":"12:01.795","Text":"we get that the x coordinate is 4.24 and so is the y-coordinate,"},{"Start":"12:01.795 ","End":"12:04.390","Text":"but haven\u0027t introduced this yes."},{"Start":"12:04.390 ","End":"12:08.620","Text":"But we won\u0027t be doing it with the measurements or rulers or anything,"},{"Start":"12:08.620 ","End":"12:10.420","Text":"we\u0027ll be doing it with computation."},{"Start":"12:10.420 ","End":"12:11.680","Text":"Usually the question will say,"},{"Start":"12:11.680 ","End":"12:13.285","Text":"find the coordinates of?"},{"Start":"12:13.285 ","End":"12:15.760","Text":"I\u0027m just giving you the general idea that we take"},{"Start":"12:15.760 ","End":"12:18.925","Text":"an angle and a distance to get to a point."},{"Start":"12:18.925 ","End":"12:23.455","Text":"At this point I want to return to what I said earlier about these restrictions."},{"Start":"12:23.455 ","End":"12:26.125","Text":"I want to drop some of these restrictions."},{"Start":"12:26.125 ","End":"12:33.325","Text":"First of all, I want to drop the restriction on Theta being between 0 and 360 degrees."},{"Start":"12:33.325 ","End":"12:40.825","Text":"Erased. Let\u0027s see what happens if we have something bigger than 360 degrees."},{"Start":"12:40.825 ","End":"12:43.855","Text":"Let\u0027s take r as the point,"},{"Start":"12:43.855 ","End":"12:46.690","Text":"I don\u0027t know 4."},{"Start":"12:46.690 ","End":"12:53.360","Text":"Let say I have 765 degrees."},{"Start":"12:55.740 ","End":"12:59.815","Text":"We basically proceed as before,"},{"Start":"12:59.815 ","End":"13:05.515","Text":"but we note that 360 degrees is the whole circle."},{"Start":"13:05.515 ","End":"13:08.155","Text":"If I do 765 degrees,"},{"Start":"13:08.155 ","End":"13:10.660","Text":"1 whole circle is 360,"},{"Start":"13:10.660 ","End":"13:14.815","Text":"and another 1 makes 720, so really,"},{"Start":"13:14.815 ","End":"13:21.145","Text":"this is the same as 4, 45 degrees."},{"Start":"13:21.145 ","End":"13:23.545","Text":"What I do if it\u0027s over 360,"},{"Start":"13:23.545 ","End":"13:29.905","Text":"I just subtract multiples of 360 until I get it down below 360,"},{"Start":"13:29.905 ","End":"13:32.510","Text":"and then proceed as usual."},{"Start":"13:33.480 ","End":"13:40.165","Text":"I start with a 45 degree angle and then I measure of 4 units,"},{"Start":"13:40.165 ","End":"13:44.260","Text":"and I end up somewhere around here."},{"Start":"13:44.260 ","End":"13:47.560","Text":"That\u0027s what we do if it\u0027s bigger than 360 degrees,"},{"Start":"13:47.560 ","End":"13:49.795","Text":"and what do we do if it\u0027s less than 0?"},{"Start":"13:49.795 ","End":"13:51.310","Text":"Suppose I\u0027ll another point,"},{"Start":"13:51.310 ","End":"13:54.685","Text":"I\u0027ll just keep alphabetically P, Q, R, S,"},{"Start":"13:54.685 ","End":"14:04.520","Text":"which is 3 minus 60 degrees."},{"Start":"14:04.680 ","End":"14:07.510","Text":"Then what I do is,"},{"Start":"14:07.510 ","End":"14:14.635","Text":"I can use the same trick of adding 360 and making it go to 3."},{"Start":"14:14.635 ","End":"14:18.730","Text":"If I add 360, it\u0027s 300 degrees."},{"Start":"14:18.730 ","End":"14:20.695","Text":"Or I don\u0027t have to do that,"},{"Start":"14:20.695 ","End":"14:26.545","Text":"I could just change the direction and instead of going counterclockwise, go clockwise."},{"Start":"14:26.545 ","End":"14:36.560","Text":"I can circle of radius 3 and I go 60 degrees in the negative direction."},{"Start":"14:37.470 ","End":"14:42.340","Text":"It comes out somewhere around here,"},{"Start":"14:42.340 ","End":"14:47.515","Text":"where this would be say,"},{"Start":"14:47.515 ","End":"14:51.775","Text":"60 degrees in the clockwise direction because of the minus,"},{"Start":"14:51.775 ","End":"14:57.160","Text":"and this distance here would be 3 units that would"},{"Start":"14:57.160 ","End":"15:02.800","Text":"be the r distance from here to here approximately."},{"Start":"15:02.800 ","End":"15:05.530","Text":"Later on, of course, we\u0027ll be using these formulas,"},{"Start":"15:05.530 ","End":"15:13.180","Text":"but I just want to give you the idea that the angle can go above 360 or below 0."},{"Start":"15:13.180 ","End":"15:17.830","Text":"Once again, in this case we just subtract multiples of 360."},{"Start":"15:17.830 ","End":"15:21.790","Text":"In this case, we just go in the clockwise direction."},{"Start":"15:21.790 ","End":"15:24.550","Text":"Alternatively, you could also add 360,"},{"Start":"15:24.550 ","End":"15:26.410","Text":"however many times needed."},{"Start":"15:26.410 ","End":"15:31.030","Text":"That\u0027s 1 restriction. What about the restriction on r being non-negative?"},{"Start":"15:31.030 ","End":"15:34.780","Text":"Well, we can\u0027t take negative r also."},{"Start":"15:34.780 ","End":"15:41.890","Text":"As an example, our next letter is T. I\u0027ll take T to be the same as Q,"},{"Start":"15:41.890 ","End":"15:47.470","Text":"except I\u0027ll put a minus 6 here and 315 degrees."},{"Start":"15:47.470 ","End":"15:50.320","Text":"Now, because of this minus,"},{"Start":"15:50.320 ","End":"15:52.810","Text":"the rule is that when it\u0027s a minus,"},{"Start":"15:52.810 ","End":"15:57.830","Text":"you take the opposite point on the opposite side of the pole."},{"Start":"16:02.880 ","End":"16:07.195","Text":"If I reflect this through the pole,"},{"Start":"16:07.195 ","End":"16:09.460","Text":"take the anti-pole so to speak,"},{"Start":"16:09.460 ","End":"16:13.090","Text":"just go right through here on the same distance."},{"Start":"16:13.090 ","End":"16:15.010","Text":"If this was 6, this is also 6,"},{"Start":"16:15.010 ","End":"16:20.365","Text":"that would be the point T. That\u0027s how we deal with negative values of r,"},{"Start":"16:20.365 ","End":"16:25.330","Text":"and it turns out that it\u0027s still the same formulas that we haven\u0027t quite learned yet."},{"Start":"16:25.330 ","End":"16:28.765","Text":"We\u0027ll also work with negative r\u0027s."},{"Start":"16:28.765 ","End":"16:35.590","Text":"But what this means from what we\u0027ve discussed,"},{"Start":"16:35.590 ","End":"16:43.075","Text":"is that the same point T could have several representations."},{"Start":"16:43.075 ","End":"16:50.035","Text":"For example, I could take T instead of being minus 6, 315 degrees."},{"Start":"16:50.035 ","End":"16:53.570","Text":"I could say this actually corresponds."},{"Start":"16:53.640 ","End":"16:56.440","Text":"Let see what the angle would this be?"},{"Start":"16:56.440 ","End":"17:01.880","Text":"This would be 135 degrees and a plus 6."},{"Start":"17:02.280 ","End":"17:06.340","Text":"In fact, I could even add 360."},{"Start":"17:06.340 ","End":"17:09.460","Text":"I could say that this is the same as 6."},{"Start":"17:09.460 ","End":"17:15.145","Text":"Let see 360, so that\u0027s 495 degrees."},{"Start":"17:15.145 ","End":"17:24.205","Text":"I can keep on adding multiples of 360 and if I reverse the value of r,"},{"Start":"17:24.205 ","End":"17:30.820","Text":"then I add or subtract a 180 degrees to go on the other side."},{"Start":"17:30.820 ","End":"17:33.670","Text":"A given point might have"},{"Start":"17:33.670 ","End":"17:39.520","Text":"more than 1 polar coordinate representation and this didn\u0027t happen with Cartesian."},{"Start":"17:39.520 ","End":"17:43.540","Text":"In Cartesian, each pair gave us a point,"},{"Start":"17:43.540 ","End":"17:46.240","Text":"and each point gave us a pair of numbers."},{"Start":"17:46.240 ","End":"17:52.540","Text":"But with polars, each pair of r and Theta gives you 1 exact point,"},{"Start":"17:52.540 ","End":"17:54.790","Text":"but each point there might be several r,"},{"Start":"17:54.790 ","End":"17:56.905","Text":"Theta that corresponds to it."},{"Start":"17:56.905 ","End":"18:02.600","Text":"That\u0027s an important point to remember and we\u0027ll take a break now."}],"Thumbnail":null,"ID":9936},{"Watched":false,"Name":"Exercise 1","Duration":"5m 3s","ChapterTopicVideoID":6854,"CourseChapterTopicPlaylistID":4005,"HasSubtitles":false,"VideoComments":[],"Subtitles":[],"Thumbnail":null,"ID":9937}],"ID":4005},{"Name":"Conversion between Polar and Cartesian Coordinates","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Conversion-Polar-Cartesian Coordinates","Duration":"19m 37s","ChapterTopicVideoID":10045,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"Continuing with polar coordinates."},{"Start":"00:03.120 ","End":"00:05.400","Text":"Now I want to talk about conversion."},{"Start":"00:05.400 ","End":"00:07.590","Text":"We have Cartesian and we have polar."},{"Start":"00:07.590 ","End":"00:11.320","Text":"We\u0027d like to go back and forth between them."},{"Start":"00:11.660 ","End":"00:14.940","Text":"Let\u0027s see which direction we will start first."},{"Start":"00:14.940 ","End":"00:18.375","Text":"It\u0027s easiest to go from polar to Cartesian."},{"Start":"00:18.375 ","End":"00:22.980","Text":"Let\u0027s start out, and suppose we have an r and a Theta."},{"Start":"00:22.980 ","End":"00:26.580","Text":"This diagram almost says it all."},{"Start":"00:26.580 ","End":"00:31.455","Text":"We can find x here using trigonometry."},{"Start":"00:31.455 ","End":"00:34.105","Text":"We have a right-angled triangle here."},{"Start":"00:34.105 ","End":"00:36.995","Text":"This is 90 degrees."},{"Start":"00:36.995 ","End":"00:39.515","Text":"If this is r and this is Theta,"},{"Start":"00:39.515 ","End":"00:43.530","Text":"this bit will be r cosine Theta."},{"Start":"00:44.240 ","End":"00:51.440","Text":"We can say that x is equal to r times cosine Theta."},{"Start":"00:51.440 ","End":"00:54.665","Text":"This to remind you even further,"},{"Start":"00:54.665 ","End":"00:57.110","Text":"because the cosine of an angle in"},{"Start":"00:57.110 ","End":"01:01.805","Text":"a right angle triangle is the adjacent over the hypotenuse."},{"Start":"01:01.805 ","End":"01:06.450","Text":"Cosine Theta is this part x over r,"},{"Start":"01:06.450 ","End":"01:08.280","Text":"so x is r cosine Theta."},{"Start":"01:08.280 ","End":"01:10.895","Text":"Similarly, this height here,"},{"Start":"01:10.895 ","End":"01:12.830","Text":"which is the same as y,"},{"Start":"01:12.830 ","End":"01:17.090","Text":"is going to be r times the sine of the angle,"},{"Start":"01:17.090 ","End":"01:19.480","Text":"and in this case sine Theta."},{"Start":"01:19.480 ","End":"01:22.340","Text":"This is what was written before,"},{"Start":"01:22.340 ","End":"01:24.035","Text":"we just gave it a preview."},{"Start":"01:24.035 ","End":"01:28.190","Text":"But here are the formulae and it just straight away,"},{"Start":"01:28.190 ","End":"01:29.720","Text":"immediate plugging in."},{"Start":"01:29.720 ","End":"01:31.190","Text":"There\u0027s nothing that can go wrong,"},{"Start":"01:31.190 ","End":"01:32.719","Text":"nothing that can be undefined."},{"Start":"01:32.719 ","End":"01:34.265","Text":"There\u0027s nothing ambiguous."},{"Start":"01:34.265 ","End":"01:39.030","Text":"It always, every pair of r and Theta,"},{"Start":"01:39.030 ","End":"01:41.630","Text":"you get an exactly 1, x and y,"},{"Start":"01:41.630 ","End":"01:43.040","Text":"which you compute from this."},{"Start":"01:43.040 ","End":"01:47.795","Text":"I could give an example or 2."},{"Start":"01:47.795 ","End":"01:53.705","Text":"Let\u0027s take an example in polar coordinates 4."},{"Start":"01:53.705 ","End":"01:56.899","Text":"I\u0027ll write it as Pi over 3,"},{"Start":"01:56.899 ","End":"01:58.955","Text":"so we don\u0027t forget radians,"},{"Start":"01:58.955 ","End":"02:01.420","Text":"but above it I\u0027ll write,"},{"Start":"02:01.420 ","End":"02:07.500","Text":"this is 60 degrees for those who really like degrees."},{"Start":"02:07.500 ","End":"02:13.065","Text":"What we do is, this is my r and this is my Theta,"},{"Start":"02:13.065 ","End":"02:14.805","Text":"and we just substitute."},{"Start":"02:14.805 ","End":"02:19.139","Text":"We get that the x of the point is"},{"Start":"02:19.139 ","End":"02:26.775","Text":"4 times cosine of Pi over 3,"},{"Start":"02:26.775 ","End":"02:35.500","Text":"and y equals 4 times sine Pi over 3."},{"Start":"02:35.680 ","End":"02:39.485","Text":"You can keep thinking of it as 60 degrees."},{"Start":"02:39.485 ","End":"02:47.530","Text":"Cosine of 60 degrees is 0.5,"},{"Start":"02:47.530 ","End":"02:51.535","Text":"and so we get that this"},{"Start":"02:51.535 ","End":"02:59.585","Text":"is 4 times 1.5, which is 2."},{"Start":"02:59.585 ","End":"03:03.680","Text":"Y is 4 sine Pi over 3,"},{"Start":"03:03.680 ","End":"03:05.420","Text":"and sine Pi over 3,"},{"Start":"03:05.420 ","End":"03:06.950","Text":"which is sine 60,"},{"Start":"03:06.950 ","End":"03:09.395","Text":"is root 3 over 2."},{"Start":"03:09.395 ","End":"03:17.785","Text":"Root 3 over 2 times 4 is root 3 times 2 or 2 root 3."},{"Start":"03:17.785 ","End":"03:21.230","Text":"I\u0027m not going to convert that to decimal."},{"Start":"03:21.230 ","End":"03:23.405","Text":"What I would say was that,"},{"Start":"03:23.405 ","End":"03:24.935","Text":"this is the polar,"},{"Start":"03:24.935 ","End":"03:32.420","Text":"so the Cartesian is 2, 2 root 3."},{"Start":"03:32.420 ","End":"03:35.360","Text":"They\u0027re both written with round brackets and comma."},{"Start":"03:35.360 ","End":"03:39.380","Text":"You just have to know from the context what we\u0027re speaking about."},{"Start":"03:39.380 ","End":"03:41.210","Text":"This is the x,"},{"Start":"03:41.210 ","End":"03:43.980","Text":"y corresponding to this."},{"Start":"03:45.170 ","End":"03:49.625","Text":"Let me take another example which is very similar."},{"Start":"03:49.625 ","End":"03:52.790","Text":"But I want to take a negative value of r,"},{"Start":"03:52.790 ","End":"03:55.830","Text":"to show that the formula still works."},{"Start":"03:57.410 ","End":"04:05.190","Text":"Instead of this, suppose I took minus 4,"},{"Start":"04:05.190 ","End":"04:10.570","Text":"but I want to get the same point but with a different representation."},{"Start":"04:10.570 ","End":"04:13.860","Text":"Now, if I take minus 4,"},{"Start":"04:13.860 ","End":"04:19.559","Text":"I\u0027ve to add or subtract 180 degrees or Pi."},{"Start":"04:19.559 ","End":"04:23.400","Text":"If you want to do it in degrees, it\u0027s 60 plus 180,"},{"Start":"04:23.400 ","End":"04:25.370","Text":"I could make it 240 degrees,"},{"Start":"04:25.370 ","End":"04:27.905","Text":"and that would be on the other side."},{"Start":"04:27.905 ","End":"04:35.100","Text":"If you did it with radians, then you would say,"},{"Start":"04:35.100 ","End":"04:38.250","Text":"Pi plus Pi over 3,"},{"Start":"04:40.940 ","End":"04:45.150","Text":"1 plus 1 over 3 would be 4 over 3,"},{"Start":"04:45.150 ","End":"04:49.980","Text":"so I get 4 Pi over 3."},{"Start":"04:49.980 ","End":"04:53.690","Text":"Now, this is our second example here."},{"Start":"04:53.690 ","End":"04:55.415","Text":"I\u0027ll do it in different color."},{"Start":"04:55.415 ","End":"04:57.650","Text":"Change the color on the previous 1,"},{"Start":"04:57.650 ","End":"04:59.660","Text":"so anyway, we can distinguish."},{"Start":"04:59.660 ","End":"05:02.579","Text":"This is a second problem."},{"Start":"05:02.960 ","End":"05:06.420","Text":"I\u0027m taking this point in polar coordinates,"},{"Start":"05:06.420 ","End":"05:10.785","Text":"where this is r and this is Theta."},{"Start":"05:10.785 ","End":"05:13.559","Text":"For those of you who like degrees,"},{"Start":"05:13.559 ","End":"05:21.640","Text":"I\u0027ll make a note that this is 240 degrees."},{"Start":"05:22.070 ","End":"05:28.270","Text":"We say, convert this to Cartesian."},{"Start":"05:29.990 ","End":"05:32.685","Text":"The formulas are here."},{"Start":"05:32.685 ","End":"05:36.560","Text":"I\u0027ve got that my x is r,"},{"Start":"05:36.560 ","End":"05:44.280","Text":"which is minus 4 cosine of 240 degrees,"},{"Start":"05:44.280 ","End":"05:58.025","Text":"and y is equal to minus 4 times sine of 240 degrees."},{"Start":"05:58.025 ","End":"06:02.890","Text":"You can do it on the calculator and it will give you a decimal answer,"},{"Start":"06:02.890 ","End":"06:04.435","Text":"but we don\u0027t need to."},{"Start":"06:04.435 ","End":"06:07.145","Text":"We can use trigonometric formulas,"},{"Start":"06:07.145 ","End":"06:11.590","Text":"that when you add 180 degrees to an angle,"},{"Start":"06:11.590 ","End":"06:14.020","Text":"as we did from here to here,"},{"Start":"06:14.020 ","End":"06:18.440","Text":"the sine and the cosine both change sign."},{"Start":"06:26.160 ","End":"06:30.820","Text":"Instead of cosine 60 being 0.5,"},{"Start":"06:30.820 ","End":"06:33.890","Text":"this would be minus 0.5."},{"Start":"06:33.890 ","End":"06:39.404","Text":"Basically what I get is minus 4 times minus 0.5,"},{"Start":"06:39.404 ","End":"06:41.490","Text":"and in this case,"},{"Start":"06:41.490 ","End":"06:45.635","Text":"for the sine of 240 degrees,"},{"Start":"06:45.635 ","End":"06:48.920","Text":"it\u0027s going to be minus the sine of 60 degrees,"},{"Start":"06:48.920 ","End":"06:54.480","Text":"so it\u0027s minus 4 times minus root 3 over 2."},{"Start":"06:56.030 ","End":"06:58.320","Text":"We have 2 minuses,"},{"Start":"06:58.320 ","End":"07:01.745","Text":"we can throw them out so we get the same answers as before."},{"Start":"07:01.745 ","End":"07:04.010","Text":"Basically what we get is that,"},{"Start":"07:04.010 ","End":"07:08.050","Text":"the xy is going to be,"},{"Start":"07:08.050 ","End":"07:11.429","Text":"minus 4 times minus 1 over 2 is 2,"},{"Start":"07:11.429 ","End":"07:14.400","Text":"and this times this is plus 4 over 2,"},{"Start":"07:14.400 ","End":"07:17.400","Text":"is 2, 2 root 3,"},{"Start":"07:17.400 ","End":"07:20.770","Text":"that\u0027s the x and that\u0027s the y."},{"Start":"07:20.770 ","End":"07:23.690","Text":"Now, we had to get the same answer."},{"Start":"07:23.690 ","End":"07:26.300","Text":"This had to come out the same as this."},{"Start":"07:26.300 ","End":"07:29.230","Text":"If it didn\u0027t, we\u0027d be in trouble,"},{"Start":"07:29.230 ","End":"07:32.810","Text":"because the formulas as I presented them"},{"Start":"07:32.810 ","End":"07:35.345","Text":"I claim work for positive and negative."},{"Start":"07:35.345 ","End":"07:38.420","Text":"This point is exactly the same point."},{"Start":"07:38.420 ","End":"07:41.280","Text":"These 2 are the same,"},{"Start":"07:42.170 ","End":"07:45.660","Text":"because I added 180 degrees,"},{"Start":"07:45.660 ","End":"07:47.000","Text":"but I made the r minus,"},{"Start":"07:47.000 ","End":"07:50.400","Text":"so I\u0027m back to the same place."},{"Start":"07:50.950 ","End":"07:55.200","Text":"Let me first draw a quick sketch here."},{"Start":"07:55.240 ","End":"07:58.490","Text":"Very roughly what it will look like"},{"Start":"07:58.490 ","End":"08:00.785","Text":"is that I have an axis,"},{"Start":"08:00.785 ","End":"08:04.580","Text":"I have 4 and 60 degrees,"},{"Start":"08:04.580 ","End":"08:09.005","Text":"so I take a 60 degree angle and 4 units."},{"Start":"08:09.005 ","End":"08:10.460","Text":"This is 60."},{"Start":"08:10.460 ","End":"08:12.080","Text":"This is 4."},{"Start":"08:12.080 ","End":"08:15.660","Text":"The other way, instead of 60,"},{"Start":"08:15.660 ","End":"08:21.195","Text":"I take 240 degrees, which is here,"},{"Start":"08:21.195 ","End":"08:29.300","Text":"and this distance of minus 4 means I don\u0027t take this point,"},{"Start":"08:29.300 ","End":"08:30.530","Text":"but it\u0027s opposite points,"},{"Start":"08:30.530 ","End":"08:33.020","Text":"so I\u0027m back to here."},{"Start":"08:33.020 ","End":"08:38.389","Text":"After all, there\u0027s 2 reversals at a 180 degrees,"},{"Start":"08:38.389 ","End":"08:40.055","Text":"but take the mirror image,"},{"Start":"08:40.055 ","End":"08:41.585","Text":"so back to the same point."},{"Start":"08:41.585 ","End":"08:43.820","Text":"Obviously, the same answer."},{"Start":"08:43.820 ","End":"08:47.120","Text":"You can just blindly follow the formulas"},{"Start":"08:47.120 ","End":"08:51.330","Text":"and they do work for negative r\u0027s as well,"},{"Start":"08:51.330 ","End":"08:57.200","Text":"and of course they work for angles bigger than 360 and all that."},{"Start":"08:57.200 ","End":"08:59.920","Text":"That was 1 direction."},{"Start":"08:59.920 ","End":"09:04.070","Text":"Now let\u0027s go in the other direction."},{"Start":"09:04.070 ","End":"09:07.430","Text":"We did the polar to Cartesian,"},{"Start":"09:07.430 ","End":"09:11.420","Text":"now let\u0027s do Cartesian to polar."},{"Start":"09:11.420 ","End":"09:14.725","Text":"The drawing is going to get in the way."},{"Start":"09:14.725 ","End":"09:18.725","Text":"I\u0027m not so happy with it anyway, there it is."},{"Start":"09:18.725 ","End":"09:22.370","Text":"I\u0027m going to give you the formulas the other way around."},{"Start":"09:22.370 ","End":"09:23.870","Text":"I\u0027m not just going to copy them from here,"},{"Start":"09:23.870 ","End":"09:26.160","Text":"I\u0027m going to give a word of explanation."},{"Start":"09:26.160 ","End":"09:28.995","Text":"Here it doesn\u0027t need explaining."},{"Start":"09:28.995 ","End":"09:32.729","Text":"To get r is easy."},{"Start":"09:33.860 ","End":"09:39.295","Text":"We know that sine squared plus cosine squared is 1."},{"Start":"09:39.295 ","End":"09:44.720","Text":"Basically, if I do x squared plus y squared,"},{"Start":"09:44.720 ","End":"09:46.940","Text":"I\u0027m going to get r squared."},{"Start":"09:46.940 ","End":"09:50.085","Text":"Lets do that as a side computation."},{"Start":"09:50.085 ","End":"09:54.890","Text":"Lets see, x squared plus y squared is"},{"Start":"09:54.890 ","End":"10:03.465","Text":"r squared cosine squared Theta plus r squared sine squared Theta."},{"Start":"10:03.465 ","End":"10:06.785","Text":"If I take r squared outside the brackets,"},{"Start":"10:06.785 ","End":"10:11.635","Text":"I get cosine squared plus sine squared."},{"Start":"10:11.635 ","End":"10:13.570","Text":"We know this is equal to 1,"},{"Start":"10:13.570 ","End":"10:17.070","Text":"so this is just equal to r squared."},{"Start":"10:17.240 ","End":"10:21.080","Text":"R is going to equal plus"},{"Start":"10:21.080 ","End":"10:26.405","Text":"or minus the square root of x squared plus y squared."},{"Start":"10:26.405 ","End":"10:28.775","Text":"But since I have the choice,"},{"Start":"10:28.775 ","End":"10:30.695","Text":"I\u0027ll go with the plus."},{"Start":"10:30.695 ","End":"10:33.665","Text":"If I chose the minus,"},{"Start":"10:33.665 ","End":"10:35.675","Text":"I\u0027d have to change the angle."},{"Start":"10:35.675 ","End":"10:38.870","Text":"Also it would be a 180 degrees more."},{"Start":"10:38.870 ","End":"10:44.375","Text":"But you can always choose to take the positive square root and I prefer it."},{"Start":"10:44.375 ","End":"10:46.850","Text":"If you have a choice of a positive or negative r,"},{"Start":"10:46.850 ","End":"10:48.170","Text":"that\u0027s just equally difficult,"},{"Start":"10:48.170 ","End":"10:50.135","Text":"then go with the positive."},{"Start":"10:50.135 ","End":"10:55.080","Text":"We say that r is equal to x squared plus y squared."},{"Start":"10:57.420 ","End":"11:05.260","Text":"To remind you, when we have a point in polar coordinates,"},{"Start":"11:05.260 ","End":"11:07.405","Text":"it might have more than 1 representation."},{"Start":"11:07.405 ","End":"11:09.969","Text":"You could, if you want to be different,"},{"Start":"11:09.969 ","End":"11:12.920","Text":"take r as minus that."},{"Start":"11:13.230 ","End":"11:18.879","Text":"The other variable, Theta comes because"},{"Start":"11:18.879 ","End":"11:26.350","Text":"if you divide from here, y over x,"},{"Start":"11:26.350 ","End":"11:41.590","Text":"we see that y over x is equal to r sine Theta over r cosine Theta"},{"Start":"11:41.590 ","End":"11:44.530","Text":"and this is equal to,"},{"Start":"11:44.530 ","End":"11:49.130","Text":"r cancels with r, it\u0027s tangent Theta."},{"Start":"11:50.460 ","End":"12:00.910","Text":"I know that tangent Theta equals y over x and that implies, or does it?"},{"Start":"12:00.910 ","End":"12:03.955","Text":"Let me put a question mark here because there are some issues,"},{"Start":"12:03.955 ","End":"12:08.750","Text":"that Theta is the arctangent,"},{"Start":"12:09.050 ","End":"12:14.985","Text":"sometimes written tan with a minus 1 of y over x."},{"Start":"12:14.985 ","End":"12:19.380","Text":"But this is not quite right."},{"Start":"12:19.380 ","End":"12:23.800","Text":"This could need to be adjusted."},{"Start":"12:23.800 ","End":"12:26.620","Text":"I\u0027ll put an asterisk."},{"Start":"12:26.620 ","End":"12:30.370","Text":"Needs adjustments and I\u0027ll explain what that means."},{"Start":"12:30.370 ","End":"12:34.520","Text":"It\u0027s almost right but not quite right."},{"Start":"12:34.830 ","End":"12:39.460","Text":"See arctangent when you do it on a calculator,"},{"Start":"12:39.460 ","End":"12:44.770","Text":"always gives you something between minus 90 and 90."},{"Start":"12:44.770 ","End":"12:49.345","Text":"We might have to add multiples of 180,"},{"Start":"12:49.345 ","End":"12:52.060","Text":"to get it into the right range."},{"Start":"12:52.060 ","End":"12:55.345","Text":"I\u0027ll give an example."},{"Start":"12:55.345 ","End":"12:57.250","Text":"Maybe more than 1."},{"Start":"12:57.250 ","End":"13:00.580","Text":"Note that the formulas we got are"},{"Start":"13:00.580 ","End":"13:05.080","Text":"the same as the preview I gave you earlier,"},{"Start":"13:05.080 ","End":"13:06.700","Text":"they\u0027re exactly the same formulas."},{"Start":"13:06.700 ","End":"13:10.615","Text":"But I still would like to put an asterisk next to the second,"},{"Start":"13:10.615 ","End":"13:12.910","Text":"because this is not quite right."},{"Start":"13:12.910 ","End":"13:15.160","Text":"It needs adjustment."},{"Start":"13:15.160 ","End":"13:21.640","Text":"Might need to be adjusted by 180 or even 360 degrees."},{"Start":"13:21.640 ","End":"13:24.260","Text":"We\u0027ll see from the examples."},{"Start":"13:24.350 ","End":"13:28.605","Text":"First of all, take a very basic example."},{"Start":"13:28.605 ","End":"13:32.110","Text":"I\u0027ll take the Cartesian,"},{"Start":"13:33.240 ","End":"13:38.725","Text":"let\u0027s say 4, 3."},{"Start":"13:38.725 ","End":"13:46.540","Text":"That\u0027s Cartesian, this is x and this is y and I need to convert this to polar."},{"Start":"13:46.540 ","End":"13:49.465","Text":"First thing I would do,"},{"Start":"13:49.465 ","End":"13:51.265","Text":"would be to say,"},{"Start":"13:51.265 ","End":"14:02.720","Text":"r is equal to the square root of x squared plus y squared."},{"Start":"14:03.750 ","End":"14:07.540","Text":"Could have used some braces here."},{"Start":"14:07.540 ","End":"14:09.310","Text":"Although, they\u0027re not next to each other,"},{"Start":"14:09.310 ","End":"14:11.140","Text":"but these are the 2 formulas,"},{"Start":"14:11.140 ","End":"14:13.345","Text":"this one and this one."},{"Start":"14:13.345 ","End":"14:15.640","Text":"Perhaps I should highlight."},{"Start":"14:15.640 ","End":"14:20.350","Text":"These are conversions from polar to Cartesian"},{"Start":"14:20.350 ","End":"14:26.780","Text":"and this is from Cartesian to polar."},{"Start":"14:27.030 ","End":"14:32.545","Text":"Either this or better still this,"},{"Start":"14:32.545 ","End":"14:34.885","Text":"but with the asterisk,"},{"Start":"14:34.885 ","End":"14:37.150","Text":"because we will be making adjustments,"},{"Start":"14:37.150 ","End":"14:39.808","Text":"otherwise we\u0027ll be getting the wrong angles,"},{"Start":"14:39.808 ","End":"14:41.950","Text":"that needs adjustment."},{"Start":"14:41.950 ","End":"14:43.750","Text":"I\u0027ll highlight that too."},{"Start":"14:43.750 ","End":"14:45.505","Text":"It\u0027s important."},{"Start":"14:45.505 ","End":"14:51.320","Text":"Using the formulas now we got r here equals that"},{"Start":"14:51.330 ","End":"14:58.780","Text":"and that is equal to the square root of 4, 3"},{"Start":"14:58.780 ","End":"15:03.550","Text":"so it\u0027s 4 squared plus 3 squared,"},{"Start":"15:03.550 ","End":"15:08.635","Text":"16 plus 9, 25, square root of 25."},{"Start":"15:08.635 ","End":"15:12.985","Text":"The answer is 5, r equals 5."},{"Start":"15:12.985 ","End":"15:15.920","Text":"Now what about Theta?"},{"Start":"15:23.940 ","End":"15:30.070","Text":"Well first of all, figure out what is the arctangent of y over x,"},{"Start":"15:30.070 ","End":"15:35.740","Text":"which is 4 over 3."},{"Start":"15:35.740 ","End":"15:37.690","Text":"We do this on the calculator."},{"Start":"15:37.690 ","End":"15:47.080","Text":"It\u0027s 3 over 4."},{"Start":"15:47.080 ","End":"15:47.740","Text":"Sorry."},{"Start":"15:47.740 ","End":"15:56.725","Text":"On the calculator, it comes out to be 36.87 in degrees."},{"Start":"15:56.725 ","End":"15:58.795","Text":"I\u0027ll leave it in degrees."},{"Start":"15:58.795 ","End":"16:02.979","Text":"Now, why might we need to make adjustments?"},{"Start":"16:02.979 ","End":"16:05.440","Text":"We have to make sure we\u0027re in the right quadrant."},{"Start":"16:05.440 ","End":"16:07.150","Text":"We know which quadrant we\u0027re in,"},{"Start":"16:07.150 ","End":"16:08.560","Text":"because we\u0027ve got x, y."},{"Start":"16:08.560 ","End":"16:10.810","Text":"We see x is positive as y is positive,"},{"Start":"16:10.810 ","End":"16:12.475","Text":"that\u0027s the first quadrant."},{"Start":"16:12.475 ","End":"16:21.580","Text":"In the first quadrant then Theta is going to be between 0 and 90 degrees."},{"Start":"16:21.580 ","End":"16:22.990","Text":"Remember quadrants?"},{"Start":"16:22.990 ","End":"16:27.580","Text":"Just roughly,"},{"Start":"16:27.580 ","End":"16:30.490","Text":"this is the first quadrant with Roman letters,"},{"Start":"16:30.490 ","End":"16:35.245","Text":"we often use second quadrant, third quadrant,"},{"Start":"16:35.245 ","End":"16:41.740","Text":"and fourth quadrant, and that\u0027s the x-axis and that is the y-axis."},{"Start":"16:41.740 ","End":"16:45.160","Text":"In the first quadrant x is positive and y is positive,"},{"Start":"16:45.160 ","End":"16:48.985","Text":"and the angle is between 0 and 90 and so on."},{"Start":"16:48.985 ","End":"16:52.600","Text":"We are already between 0 and 90."},{"Start":"16:52.600 ","End":"16:54.940","Text":"I\u0027ll just put a checkmark,"},{"Start":"16:54.940 ","End":"16:57.250","Text":"meaning a check doesn\u0027t need adjusting,"},{"Start":"16:57.250 ","End":"16:57.865","Text":"it doesn\u0027t need adjusting,"},{"Start":"16:57.865 ","End":"17:00.160","Text":"it\u0027s okay as is."},{"Start":"17:00.160 ","End":"17:04.010","Text":"I\u0027ll give an example that does need adjusting."},{"Start":"17:04.440 ","End":"17:08.380","Text":"Let\u0027s see, for the second example,"},{"Start":"17:08.380 ","End":"17:18.265","Text":"I\u0027ll take x, y to be minus 1, minus 1."},{"Start":"17:18.265 ","End":"17:19.870","Text":"That\u0027s the x, that\u0027s the y."},{"Start":"17:19.870 ","End":"17:21.505","Text":"This is in Cartesian."},{"Start":"17:21.505 ","End":"17:24.850","Text":"I need to convert it to polar."},{"Start":"17:24.850 ","End":"17:30.030","Text":"The easier one is the r,"},{"Start":"17:30.030 ","End":"17:36.760","Text":"r is equal to the square root of x squared plus y squared,"},{"Start":"17:36.760 ","End":"17:40.495","Text":"which is 1 squared plus 1 squared."},{"Start":"17:40.495 ","End":"17:49.285","Text":"That means that basically we have r is the square root of 2 and we\u0027ll leave it like that."},{"Start":"17:49.285 ","End":"17:52.000","Text":"Now we need the angle Theta"},{"Start":"17:52.000 ","End":"17:59.125","Text":"and that is equal to the arctangent of y over x,"},{"Start":"17:59.125 ","End":"18:03.350","Text":"which is minus 1 over minus 1."},{"Start":"18:04.950 ","End":"18:08.245","Text":"But minus 1 over minus 1 is 1."},{"Start":"18:08.245 ","End":"18:13.555","Text":"If you do arctangent of 1 on the calculator,"},{"Start":"18:13.555 ","End":"18:17.395","Text":"you will get 45 degrees."},{"Start":"18:17.395 ","End":"18:20.125","Text":"The question is, does this make sense?"},{"Start":"18:20.125 ","End":"18:24.385","Text":"I say no because minus 1, minus 1,"},{"Start":"18:24.385 ","End":"18:26.140","Text":"when x and y are both negative,"},{"Start":"18:26.140 ","End":"18:28.015","Text":"we\u0027re in the third quadrant,"},{"Start":"18:28.015 ","End":"18:31.870","Text":"but this 45 degrees is in the first quadrant."},{"Start":"18:31.870 ","End":"18:37.495","Text":"To adjust, we can add multiples of 180 degrees."},{"Start":"18:37.495 ","End":"18:41.230","Text":"This is because the tangent has a period of 180."},{"Start":"18:41.230 ","End":"18:47.770","Text":"Adding k times 180 degrees or k times Pi radians won\u0027t change it."},{"Start":"18:47.770 ","End":"18:52.810","Text":"This is where I\u0027m going to say, put the asterisk,"},{"Start":"18:52.810 ","End":"18:56.720","Text":"I need to add 180 degrees."},{"Start":"19:02.610 ","End":"19:09.010","Text":"The correct answer is 225 degrees."},{"Start":"19:09.010 ","End":"19:13.190","Text":"It\u0027s somewhere here, the minus 1, minus 1."},{"Start":"19:17.130 ","End":"19:19.585","Text":"I think I\u0027ll leave it at that."},{"Start":"19:19.585 ","End":"19:21.775","Text":"There will be also exercises."},{"Start":"19:21.775 ","End":"19:25.570","Text":"Just take heed that it may need adjustment"},{"Start":"19:25.570 ","End":"19:31.220","Text":"for the angle to the right quadrant by multiples of 180."},{"Start":"19:33.330 ","End":"19:36.590","Text":"We\u0027ll take a break here."}],"Thumbnail":null,"ID":9938},{"Watched":false,"Name":"Exercise 2","Duration":"9m 10s","ChapterTopicVideoID":10047,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, we have 1, 2, 3,"},{"Start":"00:02.430 ","End":"00:05.219","Text":"4 sets of Cartesian coordinates."},{"Start":"00:05.219 ","End":"00:09.585","Text":"You want to convert each one to polar coordinates."},{"Start":"00:09.585 ","End":"00:13.065","Text":"I brought in my formulas."},{"Start":"00:13.065 ","End":"00:15.435","Text":"These are the formulas for both directions."},{"Start":"00:15.435 ","End":"00:18.080","Text":"We\u0027re not going to need these."},{"Start":"00:18.080 ","End":"00:22.215","Text":"We\u0027re going to need these from Cartesian to polar."},{"Start":"00:22.215 ","End":"00:24.660","Text":"I also want to remind you,"},{"Start":"00:24.660 ","End":"00:28.425","Text":"in the tutorial there was an asterisk on the second one."},{"Start":"00:28.425 ","End":"00:31.875","Text":"You can\u0027t simply blindly do arctangent."},{"Start":"00:31.875 ","End":"00:33.600","Text":"You might have to adjust it."},{"Start":"00:33.600 ","End":"00:35.690","Text":"You have to make sure you\u0027re in the right quadrant."},{"Start":"00:35.690 ","End":"00:38.090","Text":"So this has to be done intelligently."},{"Start":"00:38.090 ","End":"00:41.074","Text":"Also, if x is 0 and there is just y,"},{"Start":"00:41.074 ","End":"00:45.930","Text":"then it\u0027s going to be plus or minus 90 degrees or plus or minus Pi over 2."},{"Start":"00:45.930 ","End":"00:48.020","Text":"The last one is not an automatic one,"},{"Start":"00:48.020 ","End":"00:50.430","Text":"need to think about it."},{"Start":"00:51.080 ","End":"00:54.830","Text":"In a, certainly r is straightforward."},{"Start":"00:54.830 ","End":"00:58.865","Text":"r is going to be the square root of x squared,"},{"Start":"00:58.865 ","End":"01:04.195","Text":"that\u0027s 1 squared, plus root 3 squared."},{"Start":"01:04.195 ","End":"01:06.750","Text":"Now, 1 squared is 1,"},{"Start":"01:06.750 ","End":"01:08.370","Text":"root 3 squared is 3,"},{"Start":"01:08.370 ","End":"01:14.010","Text":"it\u0027s root 4, and we take the positive square roots, so that\u0027s 2."},{"Start":"01:14.020 ","End":"01:18.440","Text":"We have that Theta is going to,"},{"Start":"01:18.440 ","End":"01:23.450","Text":"for starters, I\u0027m going to write it as the arctangent of y/x."},{"Start":"01:23.450 ","End":"01:28.340","Text":"y/x is root 3 over 1, is root 3."},{"Start":"01:28.340 ","End":"01:30.890","Text":"I\u0027m not going to write that this as over 1."},{"Start":"01:30.890 ","End":"01:37.455","Text":"Now, the angle whose tangent is root 3, we know this,"},{"Start":"01:37.455 ","End":"01:39.420","Text":"you could do it on the calculator,"},{"Start":"01:39.420 ","End":"01:42.020","Text":"but it\u0027s one of these famous angles,"},{"Start":"01:42.020 ","End":"01:45.155","Text":"it happens to be 60 degrees."},{"Start":"01:45.155 ","End":"01:47.870","Text":"I\u0027m not going to write 60 degrees,"},{"Start":"01:47.870 ","End":"01:50.955","Text":"I\u0027m going to write Pi over 3,"},{"Start":"01:50.955 ","End":"01:56.645","Text":"but in case it helps,"},{"Start":"01:56.645 ","End":"02:00.530","Text":"I\u0027ll write 60 degrees because that\u0027s how I used to remember them."},{"Start":"02:00.530 ","End":"02:03.350","Text":"Tangent of 60 degrees is root 3."},{"Start":"02:03.350 ","End":"02:05.390","Text":"Now that we have r and Theta,"},{"Start":"02:05.390 ","End":"02:09.295","Text":"we just write the answer as 2,"},{"Start":"02:09.295 ","End":"02:14.775","Text":"Pi over 3, and that\u0027s the answer for part a."},{"Start":"02:14.775 ","End":"02:17.975","Text":"Now part b, once again,"},{"Start":"02:17.975 ","End":"02:25.530","Text":"r is going to be the square root of minus 2 squared and minus 2 squared,"},{"Start":"02:25.530 ","End":"02:29.780","Text":"is going to be 4 plus 4,"},{"Start":"02:29.780 ","End":"02:32.880","Text":"which is the square root of 8."},{"Start":"02:33.420 ","End":"02:35.970","Text":"Of course, we could take 4 out,"},{"Start":"02:35.970 ","End":"02:38.540","Text":"and if you take the 4 out, it comes out as 2."},{"Start":"02:38.540 ","End":"02:41.195","Text":"This is equal to 2 root 2."},{"Start":"02:41.195 ","End":"02:43.650","Text":"You could leave it like this."},{"Start":"02:43.650 ","End":"02:55.355","Text":"The Theta is the inverse tangent of y/x is,"},{"Start":"02:55.355 ","End":"02:57.255","Text":"well this time I\u0027ll write it,"},{"Start":"02:57.255 ","End":"03:00.085","Text":"it\u0027s minus 2 over minus 2."},{"Start":"03:00.085 ","End":"03:09.315","Text":"Now, this thing is 1, and the arctangent of 1 is normally 45 degrees,"},{"Start":"03:09.315 ","End":"03:12.765","Text":"which is Pi over 4."},{"Start":"03:12.765 ","End":"03:15.869","Text":"Now, in part a, I didn\u0027t do any adjustment,"},{"Start":"03:15.869 ","End":"03:22.800","Text":"and I should have mentioned Pi over 3 is between 0 and Pi over 2."},{"Start":"03:22.800 ","End":"03:24.450","Text":"It\u0027s in the 1st quadrant."},{"Start":"03:24.450 ","End":"03:26.280","Text":"This point in the 1st quadrant,"},{"Start":"03:26.280 ","End":"03:27.480","Text":"x and y are plus."},{"Start":"03:27.480 ","End":"03:29.025","Text":"So I should have said, okay,"},{"Start":"03:29.025 ","End":"03:30.595","Text":"I leave it as is."},{"Start":"03:30.595 ","End":"03:33.290","Text":"In this case, I need to apply the asterisk."},{"Start":"03:33.290 ","End":"03:39.440","Text":"This is not good because Pi over 4 is in the 1st quadrant,"},{"Start":"03:39.440 ","End":"03:41.990","Text":"it\u0027s between 0 and Pi over 2,"},{"Start":"03:41.990 ","End":"03:45.095","Text":"but our point is in the 3rd quadrant,"},{"Start":"03:45.095 ","End":"03:48.000","Text":"so we need to adjust it."},{"Start":"03:48.860 ","End":"03:55.215","Text":"We adjust it, usually just adding Pi will do it."},{"Start":"03:55.215 ","End":"03:58.065","Text":"I\u0027ll add Pi."},{"Start":"03:58.065 ","End":"04:01.450","Text":"That will bring it from the 1st to the 3rd quadrant."},{"Start":"04:01.450 ","End":"04:04.970","Text":"If I add Pi, I\u0027ve got 1/4 plus 1 is 1 1/4,"},{"Start":"04:04.970 ","End":"04:09.775","Text":"5/4Pi, 5Pi over 4."},{"Start":"04:09.775 ","End":"04:16.970","Text":"Then we write the answer as the r is 2 root 2 or root 8,"},{"Start":"04:16.970 ","End":"04:21.535","Text":"and 5Pi over 4."},{"Start":"04:21.535 ","End":"04:30.909","Text":"Part c, 0, minus 5."},{"Start":"04:30.909 ","End":"04:41.340","Text":"I have no problems with r. r is the square root of 0 squared and minus 5 squared."},{"Start":"04:41.690 ","End":"04:45.585","Text":"Well, this is 25 and this is 0."},{"Start":"04:45.585 ","End":"04:49.545","Text":"Square root of 25, this comes out to be 5."},{"Start":"04:49.545 ","End":"04:52.615","Text":"The problem is with the Theta,"},{"Start":"04:52.615 ","End":"05:00.040","Text":"we can\u0027t take the arctangent of minus 5 over 0."},{"Start":"05:00.040 ","End":"05:03.110","Text":"That\u0027s no good. I said to you,"},{"Start":"05:03.110 ","End":"05:06.530","Text":"when x is 0, we can\u0027t use this formula."},{"Start":"05:06.530 ","End":"05:08.285","Text":"But if you think about it,"},{"Start":"05:08.285 ","End":"05:13.195","Text":"this is just on the y-axis, 5 units down."},{"Start":"05:13.195 ","End":"05:17.565","Text":"Quick sketch here, y, x."},{"Start":"05:17.565 ","End":"05:26.055","Text":"This would be the point 0, minus 5."},{"Start":"05:26.055 ","End":"05:30.530","Text":"What I can do is take the angle as in degrees,"},{"Start":"05:30.530 ","End":"05:32.990","Text":"that would be 270 degrees,"},{"Start":"05:32.990 ","End":"05:40.520","Text":"but in radians, that\u0027s 3 Pi over 2."},{"Start":"05:40.520 ","End":"05:45.880","Text":"I\u0027ll write the answer as 5, 3Pi/2."},{"Start":"05:45.880 ","End":"05:49.190","Text":"It\u0027s not the only answer,"},{"Start":"05:49.190 ","End":"05:54.390","Text":"I could actually also take a negative angle,"},{"Start":"05:54.390 ","End":"06:01.955","Text":"minus Pi/2 and go at it the other way in the negative direction, which is clockwise."},{"Start":"06:01.955 ","End":"06:10.080","Text":"But anyway, this is a good enough answer, probably the best."},{"Start":"06:10.330 ","End":"06:13.510","Text":"But notice that we had to use some intelligence,"},{"Start":"06:13.510 ","End":"06:15.835","Text":"not blindly follow formula."},{"Start":"06:15.835 ","End":"06:17.890","Text":"This last formula, as I say,"},{"Start":"06:17.890 ","End":"06:19.930","Text":"is something has to be used with caution."},{"Start":"06:19.930 ","End":"06:21.940","Text":"You have to see what you\u0027re doing."},{"Start":"06:21.940 ","End":"06:25.880","Text":"Part d, again straightforward,"},{"Start":"06:25.880 ","End":"06:28.920","Text":"at least as far as r goes."},{"Start":"06:28.920 ","End":"06:32.145","Text":"Well minus 3 squared is 9,"},{"Start":"06:32.145 ","End":"06:34.500","Text":"4 squared is 16,"},{"Start":"06:34.500 ","End":"06:38.730","Text":"square root of 25, 5 again this time."},{"Start":"06:38.730 ","End":"06:43.215","Text":"We have that Theta is the arctangent"},{"Start":"06:43.215 ","End":"06:51.110","Text":"of y/x is minus 4/3."},{"Start":"06:51.110 ","End":"06:58.130","Text":"My calculator gives it as minus 0.927, that\u0027s in radians."},{"Start":"06:58.130 ","End":"07:00.365","Text":"If you want it in degrees,"},{"Start":"07:00.365 ","End":"07:03.780","Text":"it comes out to be minus"},{"Start":"07:03.780 ","End":"07:10.815","Text":"53.13 degrees, I\u0027m just rounding off."},{"Start":"07:10.815 ","End":"07:12.785","Text":"But this is not good for us."},{"Start":"07:12.785 ","End":"07:20.429","Text":"The inverse tangent usually gives it in the 1st or 4th quadrant but our point,"},{"Start":"07:20.429 ","End":"07:29.390","Text":"this minus 3, 4 is going to be somewhere over here."},{"Start":"07:29.390 ","End":"07:31.955","Text":"Let\u0027s say this is minus 3 and this is 4,"},{"Start":"07:31.955 ","End":"07:35.615","Text":"and that\u0027s in the 2nd quadrant."},{"Start":"07:35.615 ","End":"07:38.990","Text":"It gave me the angle in the 4th quadrant."},{"Start":"07:38.990 ","End":"07:44.280","Text":"So what I have to do, one way of doing it is add 180 degrees."},{"Start":"07:44.280 ","End":"07:48.305","Text":"That\u0027s this asterisk which tells us we may have to adjust."},{"Start":"07:48.305 ","End":"07:50.510","Text":"If I was doing it in radians,"},{"Start":"07:50.510 ","End":"07:52.250","Text":"and that\u0027s what we\u0027re supposed to be doing,"},{"Start":"07:52.250 ","End":"07:56.160","Text":"I would add Pi."},{"Start":"07:59.680 ","End":"08:02.360","Text":"If I was doing it in degrees,"},{"Start":"08:02.360 ","End":"08:05.675","Text":"I would add 180 degrees,"},{"Start":"08:05.675 ","End":"08:09.230","Text":"but we\u0027re working in radians,"},{"Start":"08:09.230 ","End":"08:11.570","Text":"so I won\u0027t continue with this."},{"Start":"08:11.570 ","End":"08:15.420","Text":"Just to say, we expect to get something like,"},{"Start":"08:15.970 ","End":"08:20.260","Text":"if it was 53 from 180,"},{"Start":"08:20.260 ","End":"08:22.305","Text":"then it would be a 127 degrees."},{"Start":"08:22.305 ","End":"08:24.780","Text":"Anyway, we\u0027re adding Pi,"},{"Start":"08:24.780 ","End":"08:27.615","Text":"so we shall get,"},{"Start":"08:27.615 ","End":"08:30.305","Text":"let\u0027s see on the calculator."},{"Start":"08:30.305 ","End":"08:34.080","Text":"Anyway, I make this 2.214."},{"Start":"08:34.930 ","End":"08:37.525","Text":"It\u0027s bigger than Pi/2,"},{"Start":"08:37.525 ","End":"08:40.245","Text":"and it\u0027s less than Pi clearly."},{"Start":"08:40.245 ","End":"08:43.005","Text":"I take this 5 with this,"},{"Start":"08:43.005 ","End":"08:46.095","Text":"and I can say that one possible answer is"},{"Start":"08:46.095 ","End":"08:53.370","Text":"5, and 2.214 radians."},{"Start":"08:53.370 ","End":"08:57.705","Text":"This is approximate, it\u0027s a rounded computation."},{"Start":"08:57.705 ","End":"09:00.890","Text":"Lastly, I really should have done this all along,"},{"Start":"09:00.890 ","End":"09:03.355","Text":"just to highlight the results for a,"},{"Start":"09:03.355 ","End":"09:08.094","Text":"for b, c, and for d,"},{"Start":"09:08.094 ","End":"09:10.850","Text":"and we are done."}],"Thumbnail":null,"ID":9939},{"Watched":false,"Name":"Exercise 3","Duration":"7m 49s","ChapterTopicVideoID":10048,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, which is really 4 exercises,"},{"Start":"00:03.990 ","End":"00:09.105","Text":"we have to convert from polar to Cartesian coordinates."},{"Start":"00:09.105 ","End":"00:13.515","Text":"Don\u0027t forget that the angles that we use,"},{"Start":"00:13.515 ","End":"00:16.785","Text":"mostly is radians, not degrees."},{"Start":"00:16.785 ","End":"00:19.455","Text":"There are some standard formulas."},{"Start":"00:19.455 ","End":"00:21.270","Text":"I carry this with me."},{"Start":"00:21.270 ","End":"00:23.970","Text":"We don\u0027t need the second bit,"},{"Start":"00:23.970 ","End":"00:25.379","Text":"that was for the other direction."},{"Start":"00:25.379 ","End":"00:30.190","Text":"This was what we need from r Theta to xy."},{"Start":"00:31.130 ","End":"00:34.125","Text":"Let\u0027s start in part"},{"Start":"00:34.125 ","End":"00:40.590","Text":"a. I don\u0027t know why I wrote the y above the x, should have been the other way around."},{"Start":"00:40.590 ","End":"00:42.250","Text":"It doesn\u0027t really matter."},{"Start":"00:42.250 ","End":"00:50.235","Text":"We get that r is root 8 and Theta is 3Pi over 4."},{"Start":"00:50.235 ","End":"01:00.990","Text":"The x will be root 8 times cosine of 3Pi over 4."},{"Start":"01:00.990 ","End":"01:06.720","Text":"That\u0027s the x. The y is going to be our sine Theta."},{"Start":"01:06.720 ","End":"01:13.230","Text":"It\u0027s root 8 Sine of 3Pi over 4."},{"Start":"01:13.230 ","End":"01:16.890","Text":"Now, 3Pi over 4,"},{"Start":"01:16.890 ","End":"01:21.300","Text":"I think of it as a 135 degrees."},{"Start":"01:21.300 ","End":"01:25.050","Text":"Just make a note of that, that\u0027s 135 degrees."},{"Start":"01:25.050 ","End":"01:29.855","Text":"That\u0027s second quadrant, where x is negative and y is positive,"},{"Start":"01:29.855 ","End":"01:32.000","Text":"if it was cosine Pi over 4,"},{"Start":"01:32.000 ","End":"01:33.590","Text":"that\u0027ll be 45 degrees."},{"Start":"01:33.590 ","End":"01:37.345","Text":"The cosine and sine are both 1 over root 2."},{"Start":"01:37.345 ","End":"01:39.659","Text":"But in the 2nd quadrant,"},{"Start":"01:39.659 ","End":"01:44.625","Text":"this one is minus 1 over root 2 and this one is 1 over root 2."},{"Start":"01:44.625 ","End":"01:47.060","Text":"If I wasn\u0027t using the calculator,"},{"Start":"01:47.060 ","End":"01:53.025","Text":"I could say this is equal to root 8."},{"Start":"01:53.025 ","End":"01:58.730","Text":"Now, root 8 actually is equal to 2 root 2."},{"Start":"01:58.730 ","End":"02:03.625","Text":"I\u0027ve got 2 root 2, the cosine."},{"Start":"02:03.625 ","End":"02:08.430","Text":"You could do it on the calculator of course. I\u0027ll do it without."},{"Start":"02:08.430 ","End":"02:12.845","Text":"1 over root 2, because that\u0027s the cosine of 135 degrees,"},{"Start":"02:12.845 ","End":"02:16.355","Text":"minus 1 over root 2, sorry."},{"Start":"02:16.355 ","End":"02:19.370","Text":"Second quadrant cosine is negative."},{"Start":"02:19.370 ","End":"02:24.380","Text":"Second quadrant sine is positive,"},{"Start":"02:24.380 ","End":"02:28.070","Text":"so here I get 2 root 2 from the root 8,"},{"Start":"02:28.070 ","End":"02:32.260","Text":"times 1 over root 2 positive."},{"Start":"02:32.260 ","End":"02:35.240","Text":"This comes out to be the root 2 cancels."},{"Start":"02:35.240 ","End":"02:38.815","Text":"This comes out to be minus 2,"},{"Start":"02:38.815 ","End":"02:46.100","Text":"2 and this is the Cartesian and it is in the second quadrant and as you can see,"},{"Start":"02:46.100 ","End":"02:48.380","Text":"because of the minus 2,"},{"Start":"02:48.380 ","End":"02:51.230","Text":"2, they\u0027re equal somewhere here,"},{"Start":"02:51.230 ","End":"02:57.515","Text":"minus 2, 2, it is 135 degrees and everything\u0027s fine."},{"Start":"02:57.515 ","End":"03:00.290","Text":"You don\u0027t need these little sketches,"},{"Start":"03:00.290 ","End":"03:03.390","Text":"but they\u0027re sometimes helpful."},{"Start":"03:04.340 ","End":"03:08.570","Text":"Part b, I think I\u0027ll organize it a bit differently."},{"Start":"03:08.570 ","End":"03:13.245","Text":"Let\u0027s start with the angle with my 2Pi over 3,"},{"Start":"03:13.245 ","End":"03:17.370","Text":"which is actually 120 degrees."},{"Start":"03:17.370 ","End":"03:21.720","Text":"Let\u0027s just figure out what are the sine and cosine of these."},{"Start":"03:21.720 ","End":"03:24.475","Text":"If I take the sine,"},{"Start":"03:24.475 ","End":"03:32.770","Text":"sine of 120 degrees is the same as sine of 60 is root 3 over 2,"},{"Start":"03:32.770 ","End":"03:37.880","Text":"and cosine 120 is minus cosine 60,"},{"Start":"03:37.880 ","End":"03:39.589","Text":"we expect it to be minus."},{"Start":"03:39.589 ","End":"03:41.750","Text":"It\u0027s also in the second quadrant,"},{"Start":"03:41.750 ","End":"03:45.810","Text":"like a, that would be minus 1/2."},{"Start":"03:45.810 ","End":"03:48.175","Text":"Cosine of 60 is 1/2."},{"Start":"03:48.175 ","End":"03:50.330","Text":"But as I said earlier,"},{"Start":"03:50.330 ","End":"03:53.915","Text":"you can also just do it on the calculator and get numerical answers."},{"Start":"03:53.915 ","End":"03:57.965","Text":"I try to avoid the calculator and get things exact if possible."},{"Start":"03:57.965 ","End":"04:00.260","Text":"Now that I have the sine and the cosine,"},{"Start":"04:00.260 ","End":"04:09.210","Text":"I just have to multiply them by r. Well,"},{"Start":"04:09.210 ","End":"04:10.350","Text":"this one\u0027s a bit different,"},{"Start":"04:10.350 ","End":"04:12.170","Text":"this r is negative,"},{"Start":"04:12.170 ","End":"04:17.970","Text":"which means that we\u0027re actually going to come out not in the 2nd quadrant,"},{"Start":"04:17.970 ","End":"04:19.605","Text":"but in the 4th quadrant."},{"Start":"04:19.605 ","End":"04:26.550","Text":"What I get is minus 4 times the cosine,"},{"Start":"04:26.550 ","End":"04:31.860","Text":"which is minus 1/2."},{"Start":"04:31.860 ","End":"04:41.310","Text":"The other coordinate is minus 4 times 1/2."},{"Start":"04:41.310 ","End":"04:49.330","Text":"This comes out to be minus 4 times minus 1/2 is plus 2."},{"Start":"04:54.140 ","End":"04:57.465","Text":"Sorry, I did something wrong here."},{"Start":"04:57.465 ","End":"05:00.790","Text":"This is root 3 over 2."},{"Start":"05:01.520 ","End":"05:07.560","Text":"Sorry, root 3."},{"Start":"05:07.560 ","End":"05:08.850","Text":"2 into 4 goes twice,"},{"Start":"05:08.850 ","End":"05:13.050","Text":"so it\u0027s minus 2 root 3."},{"Start":"05:13.050 ","End":"05:16.100","Text":"I should really be highlighting them as I go."},{"Start":"05:16.100 ","End":"05:17.870","Text":"This was the answer to a,"},{"Start":"05:17.870 ","End":"05:21.440","Text":"this is our answer to b."},{"Start":"05:21.440 ","End":"05:24.845","Text":"This was the odd one out because we had a negative r."},{"Start":"05:24.845 ","End":"05:29.930","Text":"The angle that it comes out is actually opposite the angle that we\u0027d expect."},{"Start":"05:29.930 ","End":"05:33.125","Text":"We\u0027d expect 120 degrees second quadrant."},{"Start":"05:33.125 ","End":"05:36.140","Text":"We actually got a point where x is positive,"},{"Start":"05:36.140 ","End":"05:39.295","Text":"y is negative, it\u0027s the fourth quadrant."},{"Start":"05:39.295 ","End":"05:45.830","Text":"C, not exactly a trick question,"},{"Start":"05:45.830 ","End":"05:48.065","Text":"but when r is 0,"},{"Start":"05:48.065 ","End":"05:50.030","Text":"if you look at it here also in the formula,"},{"Start":"05:50.030 ","End":"05:52.565","Text":"if r is 0, x and y are both zeros."},{"Start":"05:52.565 ","End":"05:55.555","Text":"R equals 0 is the origin."},{"Start":"05:55.555 ","End":"05:58.215","Text":"It really doesn\u0027t matter what Theta is,"},{"Start":"05:58.215 ","End":"06:00.975","Text":"0, anything is the origin."},{"Start":"06:00.975 ","End":"06:03.105","Text":"I\u0027ll just write the answer as 0,"},{"Start":"06:03.105 ","End":"06:08.190","Text":"0 because I ignore the angle."},{"Start":"06:08.420 ","End":"06:11.239","Text":"The distance is 0 from the origin."},{"Start":"06:11.239 ","End":"06:13.790","Text":"The angle doesn\u0027t make any difference which way I\u0027m looking,"},{"Start":"06:13.790 ","End":"06:16.590","Text":"I\u0027m stuck at the origin."},{"Start":"06:16.990 ","End":"06:21.955","Text":"D, again, use of the formula,"},{"Start":"06:21.955 ","End":"06:26.625","Text":"I just need the minus Pi over 3,"},{"Start":"06:26.625 ","End":"06:33.960","Text":"which is in degrees minus 60 degrees."},{"Start":"06:33.960 ","End":"06:40.560","Text":"I need the sine and the cosine, whatever order."},{"Start":"06:40.560 ","End":"06:44.340","Text":"The sine of minus 60 degrees,"},{"Start":"06:44.340 ","End":"06:48.020","Text":"sine\u0027s a naught function is minus the sine of 60 degrees."},{"Start":"06:48.020 ","End":"06:55.050","Text":"That would be minus root 3 over 2."},{"Start":"06:55.050 ","End":"07:01.235","Text":"The cosine of minus 60 is the same as the cosine of plus 60 because cosine is even,"},{"Start":"07:01.235 ","End":"07:08.050","Text":"and that comes out to be 1/2."},{"Start":"07:08.050 ","End":"07:12.930","Text":"The x would be 6 times the cosine comes first,"},{"Start":"07:12.930 ","End":"07:15.135","Text":"times 1/2, and then,"},{"Start":"07:15.135 ","End":"07:20.310","Text":"6 times minus root 3 over 2."},{"Start":"07:20.310 ","End":"07:24.255","Text":"That comes out to be 3,"},{"Start":"07:24.255 ","End":"07:29.355","Text":"minus 3 root 3."},{"Start":"07:29.355 ","End":"07:34.575","Text":"Notice that the minus 60 degrees"},{"Start":"07:34.575 ","End":"07:40.735","Text":"makes it come out in the fourth quadrant where x is positive and y is negative."},{"Start":"07:40.735 ","End":"07:43.080","Text":"This appears to be okay."},{"Start":"07:43.080 ","End":"07:45.060","Text":"I forgot to highlight this one."},{"Start":"07:45.060 ","End":"07:46.670","Text":"We\u0027ve completed a, b, c,"},{"Start":"07:46.670 ","End":"07:49.680","Text":"and d, and we\u0027re done."}],"Thumbnail":null,"ID":9940},{"Watched":false,"Name":"Conversion-Polar-Cartesian Equations","Duration":"19m 10s","ChapterTopicVideoID":10046,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"We\u0027re continuing with polar coordinates."},{"Start":"00:04.290 ","End":"00:08.490","Text":"Now, we\u0027re going to talk about polar equations."},{"Start":"00:08.490 ","End":"00:11.490","Text":"The old equations we used to have with x and"},{"Start":"00:11.490 ","End":"00:14.175","Text":"y are going to be called Cartesian equations."},{"Start":"00:14.175 ","End":"00:17.700","Text":"In general, Cartesian equations are equations with y and x,"},{"Start":"00:17.700 ","End":"00:21.134","Text":"and typically, they come in 1 of 3 forms."},{"Start":"00:21.134 ","End":"00:27.020","Text":"You either have that y is some function of x,"},{"Start":"00:27.020 ","End":"00:30.830","Text":"or sometimes x is some other function,"},{"Start":"00:30.830 ","End":"00:33.585","Text":"say g of y,"},{"Start":"00:33.585 ","End":"00:36.035","Text":"and sometimes you get it an implicit form."},{"Start":"00:36.035 ","End":"00:39.140","Text":"You get some function of x and"},{"Start":"00:39.140 ","End":"00:44.705","Text":"y usually bring everything to the left and they leave 0 on the right."},{"Start":"00:44.705 ","End":"00:53.440","Text":"Similarly, with polar, you could have that r would be a function,"},{"Start":"00:53.440 ","End":"00:55.010","Text":"I\u0027ll re-use the letters,"},{"Start":"00:55.010 ","End":"00:59.750","Text":"they\u0027re not the same, some function of Theta."},{"Start":"00:59.750 ","End":"01:05.430","Text":"More rarely, we will get Theta or some other function"},{"Start":"01:05.430 ","End":"01:11.390","Text":"of r and fairly frequently we would get again,"},{"Start":"01:11.390 ","End":"01:15.500","Text":"an implicit form of some relationship between r and Theta."},{"Start":"01:15.500 ","End":"01:21.315","Text":"A function of r and Theta equals 0."},{"Start":"01:21.315 ","End":"01:22.430","Text":"Maybe let\u0027s make a note."},{"Start":"01:22.430 ","End":"01:30.790","Text":"This is the polar and this is the Cartesian."},{"Start":"01:30.790 ","End":"01:33.320","Text":"Then at least in the old days,"},{"Start":"01:33.320 ","End":"01:37.730","Text":"what was commonly done is you\u0027d make a table of values x in 1 column,"},{"Start":"01:37.730 ","End":"01:39.065","Text":"y in another column,"},{"Start":"01:39.065 ","End":"01:42.815","Text":"and plot point-by-point on some graph paper."},{"Start":"01:42.815 ","End":"01:44.360","Text":"The same thing with polar,"},{"Start":"01:44.360 ","End":"01:47.135","Text":"there is actually polar graph paper."},{"Start":"01:47.135 ","End":"01:50.150","Text":"It\u0027s much more intricate I just brought you a simple example,"},{"Start":"01:50.150 ","End":"01:53.570","Text":"but it could be very elaborate."},{"Start":"01:53.570 ","End":"01:57.410","Text":"This 1 happens to be in radians,"},{"Start":"01:57.410 ","End":"02:01.930","Text":"but you can also get them in degrees."},{"Start":"02:01.930 ","End":"02:06.840","Text":"The important degrees would be 0 is 0 degrees."},{"Start":"02:06.840 ","End":"02:09.680","Text":"This would be 30 degrees,"},{"Start":"02:09.680 ","End":"02:12.920","Text":"45 degrees, 60 degrees,"},{"Start":"02:12.920 ","End":"02:14.840","Text":"90 degrees and so on."},{"Start":"02:14.840 ","End":"02:18.710","Text":"The important, the common angles."},{"Start":"02:19.190 ","End":"02:21.870","Text":"Each of these lines,"},{"Start":"02:21.870 ","End":"02:26.165","Text":"red lines, has the same angle, the same Theta."},{"Start":"02:26.165 ","End":"02:29.680","Text":"On the blue lines we have the same r,"},{"Start":"02:29.680 ","End":"02:31.800","Text":"and usually there\u0027s a scale."},{"Start":"02:31.800 ","End":"02:33.390","Text":"In the middle, r is 0 here,"},{"Start":"02:33.390 ","End":"02:36.405","Text":"r is 1, r is 2, 3,"},{"Start":"02:36.405 ","End":"02:41.760","Text":"4, 5, 6, and so on."},{"Start":"02:41.760 ","End":"02:43.275","Text":"Again, we can also,"},{"Start":"02:43.275 ","End":"02:44.970","Text":"if we have r as a function of Theta,"},{"Start":"02:44.970 ","End":"02:49.460","Text":"make a table and get a lot of points and put"},{"Start":"02:49.460 ","End":"02:54.080","Text":"the points on and plot them but that\u0027s not a modern way of doing things."},{"Start":"02:54.080 ","End":"02:56.150","Text":"At least it\u0027s not done much anymore."},{"Start":"02:56.150 ","End":"03:02.960","Text":"In fact, usually we\u0027re so used to Cartesian that typically if we have a polar equation,"},{"Start":"03:02.960 ","End":"03:09.630","Text":"we\u0027ll convert it to Cartesian and use our sketching techniques for Cartesian functions."},{"Start":"03:09.650 ","End":"03:12.680","Text":"Let me get some examples of equations in"},{"Start":"03:12.680 ","End":"03:17.460","Text":"both forms and how to convert between 1 and the other."},{"Start":"03:17.540 ","End":"03:24.350","Text":"I drag this little tool kit with me in all the polar stuff of how to convert"},{"Start":"03:24.350 ","End":"03:31.070","Text":"Cartesian to polar point-by-point but I also like to add my asterisk here,"},{"Start":"03:31.070 ","End":"03:35.090","Text":"which remember, this last equation needs to be adjusted."},{"Start":"03:35.090 ","End":"03:38.030","Text":"That\u0027s what the asterisk is to remind me."},{"Start":"03:38.030 ","End":"03:44.960","Text":"Let\u0027s start with a polar equation and then convert it to Cartesian."},{"Start":"03:44.960 ","End":"03:46.819","Text":"For my first example,"},{"Start":"03:46.819 ","End":"03:49.475","Text":"I\u0027ll take something in implicit form,"},{"Start":"03:49.475 ","End":"03:56.460","Text":"x squared plus y squared equals 4y."},{"Start":"03:57.370 ","End":"04:03.765","Text":"Turns out it\u0027s an equation of a circle of radius 2 with center at 2,"},{"Start":"04:03.765 ","End":"04:06.360","Text":"0 but I\u0027m not going to prove that."},{"Start":"04:06.360 ","End":"04:08.460","Text":"You studied the circles,"},{"Start":"04:08.460 ","End":"04:10.640","Text":"you should be able to easily do that."},{"Start":"04:10.640 ","End":"04:14.515","Text":"My question is, what will be the equation in terms of r Theta,"},{"Start":"04:14.515 ","End":"04:20.310","Text":"the same circle with center there and radius 2."},{"Start":"04:20.310 ","End":"04:25.820","Text":"What we do is, this is the easy direction from Cartesian to polar."},{"Start":"04:25.820 ","End":"04:30.800","Text":"We just substitute from here that x equals r cosine Theta,"},{"Start":"04:30.800 ","End":"04:33.235","Text":"y equals r sine Theta."},{"Start":"04:33.235 ","End":"04:39.570","Text":"We get r cosine Theta squared plus"},{"Start":"04:39.570 ","End":"04:48.760","Text":"r sine Theta squared equals 4 times r sine Theta."},{"Start":"04:48.770 ","End":"04:55.635","Text":"Then because cosine squared plus sine squared equals 1."},{"Start":"04:55.635 ","End":"04:58.280","Text":"I forgot the Theta here, let me put it in."},{"Start":"04:58.280 ","End":"05:02.270","Text":"Yeah, so 2 cosine squared plus sine squared is 1."},{"Start":"05:02.270 ","End":"05:09.730","Text":"Here, we get just r squared equals 4 sine Theta."},{"Start":"05:09.730 ","End":"05:12.635","Text":"Then I\u0027m going to divide both sides by r,"},{"Start":"05:12.635 ","End":"05:14.585","Text":"and I\u0027ll justify it in a moment."},{"Start":"05:14.585 ","End":"05:16.250","Text":"Divide both sides by r,"},{"Start":"05:16.250 ","End":"05:22.400","Text":"I just get that r equals 4 sine of Theta."},{"Start":"05:22.400 ","End":"05:28.390","Text":"Now, normally, you can\u0027t just divide by something until you check that it\u0027s not 0."},{"Start":"05:28.390 ","End":"05:31.665","Text":"Sure, could be that r equals 0,"},{"Start":"05:31.665 ","End":"05:34.925","Text":"but I claim that r equals 0 is covered here anyway,"},{"Start":"05:34.925 ","End":"05:37.145","Text":"because if you take Theta equals 0,"},{"Start":"05:37.145 ","End":"05:38.825","Text":"you\u0027ll get r equals 0."},{"Start":"05:38.825 ","End":"05:43.490","Text":"Remember, r equals 0 is the pole and it doesn\u0027t really matter what Theta is,"},{"Start":"05:43.490 ","End":"05:44.900","Text":"r equals 0 is the middle,"},{"Start":"05:44.900 ","End":"05:48.230","Text":"so we get it anyway so we haven\u0027t lost any information."},{"Start":"05:48.230 ","End":"05:53.720","Text":"This, in fact, would be the answer and then I\u0027ll change its color to show that we\u0027re"},{"Start":"05:53.720 ","End":"06:00.005","Text":"now in polar coordinates or polar equation."},{"Start":"06:00.005 ","End":"06:03.120","Text":"Let\u0027s do 1 the other way round."},{"Start":"06:03.340 ","End":"06:06.365","Text":"For the example, going the other way,"},{"Start":"06:06.365 ","End":"06:13.774","Text":"I\u0027m going to show you a shape called a cardioid curve,"},{"Start":"06:13.774 ","End":"06:17.030","Text":"which is better described in polar coordinates."},{"Start":"06:17.030 ","End":"06:19.250","Text":"I\u0027ll show you even what it looks like."},{"Start":"06:19.250 ","End":"06:25.510","Text":"It\u0027s this red shape that is being traced when 1 circle rolls around another."},{"Start":"06:25.510 ","End":"06:27.845","Text":"That\u0027s enough of the animation."},{"Start":"06:27.845 ","End":"06:32.600","Text":"Here\u0027s a more static picture and the general equation is"},{"Start":"06:32.600 ","End":"06:39.195","Text":"r equals a times 1 minus cosine Theta,"},{"Start":"06:39.195 ","End":"06:40.579","Text":"a is some parameter,"},{"Start":"06:40.579 ","End":"06:44.490","Text":"I think it\u0027s a diameter of that little circle that we saw."},{"Start":"06:44.800 ","End":"06:48.035","Text":"We can work with a parameter or let\u0027s just"},{"Start":"06:48.035 ","End":"06:51.904","Text":"take a value of the parameters. Let\u0027s take a equals 3."},{"Start":"06:51.904 ","End":"06:59.610","Text":"There we are and let\u0027s try and convert this to rectangular Cartesian equation."},{"Start":"07:01.520 ","End":"07:05.850","Text":"I said rectangular and Cartesian same thing."},{"Start":"07:05.850 ","End":"07:11.660","Text":"We just go ahead and substitute from these formulas but this time,"},{"Start":"07:11.660 ","End":"07:14.610","Text":"we\u0027re going to have to be careful with Theta."},{"Start":"07:15.020 ","End":"07:23.300","Text":"The trick here is to multiply both sides by r and then we get that r"},{"Start":"07:23.300 ","End":"07:27.874","Text":"squared is equal to"},{"Start":"07:27.874 ","End":"07:34.840","Text":"3 times r minus r cosine Theta."},{"Start":"07:34.840 ","End":"07:38.700","Text":"First of all, multiplying by r, yeah, could be 0,"},{"Start":"07:38.700 ","End":"07:41.450","Text":"but r would be 0 anyway, and 1 of the solutions,"},{"Start":"07:41.450 ","End":"07:44.960","Text":"so we\u0027re not adding anything new or taking anything away."},{"Start":"07:44.960 ","End":"07:51.460","Text":"Now, it\u0027s much easier because we know that r squared is x squared plus y squared."},{"Start":"07:51.460 ","End":"07:55.040","Text":"Here, we get x squared plus y squared."},{"Start":"07:55.040 ","End":"07:58.225","Text":"Here, we get 3."},{"Start":"07:58.225 ","End":"08:00.450","Text":"Now, r is just r,"},{"Start":"08:00.450 ","End":"08:07.890","Text":"but r cosine Theta is equal to x."},{"Start":"08:07.890 ","End":"08:11.780","Text":"All we have to do is get rid of this are now."},{"Start":"08:12.050 ","End":"08:18.730","Text":"Just left the messy bit to the end where we write r as this so we get x"},{"Start":"08:18.730 ","End":"08:25.120","Text":"squared plus y squared equals 3 times and here we have this expression,"},{"Start":"08:25.120 ","End":"08:31.950","Text":"square root of x squared plus y squared minus x."},{"Start":"08:31.950 ","End":"08:33.810","Text":"Maybe it could be simplified,"},{"Start":"08:33.810 ","End":"08:40.750","Text":"I\u0027m not sure but this would be the Cartesian form."},{"Start":"08:40.750 ","End":"08:46.435","Text":"We have at least 1 example of convergent each of conversion, yeah,"},{"Start":"08:46.435 ","End":"08:53.605","Text":"from Cartesian to polar and here from polar to Cartesian."},{"Start":"08:53.605 ","End":"08:56.980","Text":"I\u0027m going to delete some stuff I don\u0027t need."},{"Start":"08:56.980 ","End":"09:03.505","Text":"What I wanted to do is to just do 1 example of how you would actually sketch"},{"Start":"09:03.505 ","End":"09:08.380","Text":"a polar equation using a table of values and"},{"Start":"09:08.380 ","End":"09:13.585","Text":"actually plotting on polar graph paper so to speak."},{"Start":"09:13.585 ","End":"09:16.750","Text":"In fact, I don\u0027t even need this stuff."},{"Start":"09:16.750 ","End":"09:27.655","Text":"Now, I\u0027m going to draw a table that\u0027s the vertical line and a horizontal line,"},{"Start":"09:27.655 ","End":"09:31.385","Text":"and I\u0027ll label this Theta."},{"Start":"09:31.385 ","End":"09:35.820","Text":"That appears to be the independent variable here and r will"},{"Start":"09:35.820 ","End":"09:40.525","Text":"compute in terms of Theta and I\u0027ll tell you what I\u0027m going to take the values of Theta,"},{"Start":"09:40.525 ","End":"09:43.150","Text":"not too many, but the ones that are easy,"},{"Start":"09:43.150 ","End":"09:46.240","Text":"the ones that I know the cosine for easily."},{"Start":"09:46.240 ","End":"09:49.420","Text":"I know cosine of 0,"},{"Start":"09:49.420 ","End":"09:55.425","Text":"I know what the cosine of 60 is,"},{"Start":"09:55.425 ","End":"09:59.735","Text":"I know what the cosine of 90 is very easily."},{"Start":"09:59.735 ","End":"10:02.070","Text":"Well, I\u0027ll write them in already."},{"Start":"10:02.070 ","End":"10:04.655","Text":"Cosine of 0 is 1,"},{"Start":"10:04.655 ","End":"10:06.975","Text":"cosine of 60 is 1/2,"},{"Start":"10:06.975 ","End":"10:10.025","Text":"cosine of 90 is 0."},{"Start":"10:10.025 ","End":"10:12.070","Text":"This is not r yet."},{"Start":"10:12.070 ","End":"10:15.335","Text":"I\u0027ll make a third column here."},{"Start":"10:15.335 ","End":"10:18.670","Text":"This will just be cosine of Theta,"},{"Start":"10:18.670 ","End":"10:22.740","Text":"and afterwards we\u0027ll plug it into r. Let\u0027s see,"},{"Start":"10:22.740 ","End":"10:26.505","Text":"you got 90, which is easy to compute,"},{"Start":"10:26.505 ","End":"10:33.060","Text":"and then here, I have 120 degrees."},{"Start":"10:35.450 ","End":"10:38.895","Text":"Let\u0027s see what else is easy."},{"Start":"10:38.895 ","End":"10:44.500","Text":"Pi is easy, that\u0027s 180 degrees."},{"Start":"10:44.930 ","End":"10:50.730","Text":"Another easy thing is 4Pi over 3,"},{"Start":"10:50.730 ","End":"10:56.610","Text":"which is 240 degrees."},{"Start":"10:56.610 ","End":"11:07.395","Text":"Also, 300 degrees, which is this. Back to 0."},{"Start":"11:07.395 ","End":"11:08.940","Text":"These angles will be easy."},{"Start":"11:08.940 ","End":"11:11.610","Text":"I don\u0027t know if I can fit all of them in here."},{"Start":"11:11.610 ","End":"11:15.430","Text":"Let\u0027s just continue."},{"Start":"11:15.580 ","End":"11:23.620","Text":"See how many I can fit and maybe I\u0027ll squash a bit more, 120, 180,"},{"Start":"11:23.620 ","End":"11:34.810","Text":"240, oh yeah, I want this 1 too, 270 degrees."},{"Start":"11:37.220 ","End":"11:39.900","Text":"I\u0027m going to move up the table up a bit,"},{"Start":"11:39.900 ","End":"11:44.130","Text":"270, then 300 degrees."},{"Start":"11:44.130 ","End":"11:51.870","Text":"I\u0027ll just write this is in degrees not radians, 300. What next?"},{"Start":"11:53.050 ","End":"11:56.100","Text":"That\u0027s about it."},{"Start":"11:59.230 ","End":"12:01.740","Text":"I got room now. Let\u0027s see now."},{"Start":"12:01.740 ","End":"12:06.255","Text":"Cosine of 120 is minus 1/2."},{"Start":"12:06.255 ","End":"12:11.025","Text":"Cosine of 180 is minus 1."},{"Start":"12:11.025 ","End":"12:16.185","Text":"Then here we have again minus 1/2."},{"Start":"12:16.185 ","End":"12:21.090","Text":"Here we get 0."},{"Start":"12:21.090 ","End":"12:24.420","Text":"For 300, it\u0027s like minus 60,"},{"Start":"12:24.420 ","End":"12:29.715","Text":"so it would be the same as 1/2."},{"Start":"12:29.715 ","End":"12:36.450","Text":"As for r, I can compute 1 minus cosine Theta and then multiply by 3."},{"Start":"12:36.450 ","End":"12:40.665","Text":"So 1 minus 1 is 0, multiply by 3,"},{"Start":"12:40.665 ","End":"12:46.380","Text":"that\u0027s 0, 1 minus 1/2 times 3 is 1 and 1/2,"},{"Start":"12:46.380 ","End":"12:51.810","Text":"1 minus 0 times 3 is 3,"},{"Start":"12:51.810 ","End":"12:56.445","Text":"and then for 1 minus 1/2 is 1 and 1/2,"},{"Start":"12:56.445 ","End":"12:58.545","Text":"this is going to be 4 and 1/2,"},{"Start":"12:58.545 ","End":"13:01.845","Text":"1 minus minus 1 is 2, so that\u0027s 6."},{"Start":"13:01.845 ","End":"13:04.050","Text":"All the values start to repeat,"},{"Start":"13:04.050 ","End":"13:07.710","Text":"4 and 1/2, then 3,"},{"Start":"13:07.710 ","End":"13:09.660","Text":"and then 1 and 1/2."},{"Start":"13:09.660 ","End":"13:12.885","Text":"If we continue to 360,"},{"Start":"13:12.885 ","End":"13:15.780","Text":"we\u0027ll be back to,"},{"Start":"13:15.780 ","End":"13:19.090","Text":"anyway, we\u0027re going to run out of room for that."},{"Start":"13:20.570 ","End":"13:23.535","Text":"I changed the colors a bit."},{"Start":"13:23.535 ","End":"13:25.860","Text":"The ones in green are the Theta and the r,"},{"Start":"13:25.860 ","End":"13:26.955","Text":"and I\u0027m going to plot them."},{"Start":"13:26.955 ","End":"13:30.915","Text":"When r is 0, it\u0027s the pole."},{"Start":"13:30.915 ","End":"13:34.545","Text":"Then we have 60 degrees, 1 and 1/2,"},{"Start":"13:34.545 ","End":"13:41.950","Text":"so this is 60 degrees and 1 and 1/2 somewhere here."},{"Start":"13:44.180 ","End":"13:48.150","Text":"Then we have at 90 degrees,"},{"Start":"13:48.150 ","End":"13:55.500","Text":"3, this is 90 degrees and 1, 2, 3."},{"Start":"13:55.500 ","End":"14:00.255","Text":"Then 120 degrees, 4 and 1/2,"},{"Start":"14:00.255 ","End":"14:02.715","Text":"so 1, 2, 3,"},{"Start":"14:02.715 ","End":"14:05.910","Text":"4, 4 and 1/2 is here."},{"Start":"14:05.910 ","End":"14:13.500","Text":"Then 180, 6, so at 180, we\u0027re here."},{"Start":"14:13.500 ","End":"14:18.990","Text":"Then at 240, we\u0027re 4 and 1/2 again."},{"Start":"14:18.990 ","End":"14:21.850","Text":"That would be here."},{"Start":"14:22.160 ","End":"14:30.520","Text":"270 is 3, 1, 2, 3."},{"Start":"14:34.130 ","End":"14:38.610","Text":"300 degrees, 1 and 1/2 here."},{"Start":"14:38.610 ","End":"14:41.910","Text":"Then back to 360, 0."},{"Start":"14:41.910 ","End":"14:46.390","Text":"We imagine that we would join them through."},{"Start":"14:46.400 ","End":"14:48.840","Text":"The more points you would do,"},{"Start":"14:48.840 ","End":"14:50.340","Text":"the better you would get."},{"Start":"14:50.340 ","End":"14:53.565","Text":"Mine\u0027s come out looking terrible, just freehand,"},{"Start":"14:53.565 ","End":"14:57.850","Text":"but it should come out looking like a cardioid."},{"Start":"14:59.120 ","End":"15:02.670","Text":"This is what you get freehand."},{"Start":"15:02.670 ","End":"15:09.555","Text":"That\u0027s just the idea of how to make plots and sketches with polar coordinates."},{"Start":"15:09.555 ","End":"15:13.890","Text":"There\u0027s 1 more small topic I want to mention."},{"Start":"15:13.890 ","End":"15:16.319","Text":"I\u0027ll clear the board first."},{"Start":"15:16.319 ","End":"15:23.010","Text":"I\u0027d like to just say what are the polar equations for these colored lines."},{"Start":"15:23.010 ","End":"15:26.955","Text":"For the straight lines that are going through the pole,"},{"Start":"15:26.955 ","End":"15:29.700","Text":"and for the circles around them,"},{"Start":"15:29.700 ","End":"15:31.455","Text":"the red and the blue here."},{"Start":"15:31.455 ","End":"15:37.270","Text":"Just want to draw an analogy with the Cartesian case."},{"Start":"15:37.730 ","End":"15:43.815","Text":"In the Cartesian case, in Cartesian coordinates,"},{"Start":"15:43.815 ","End":"15:49.619","Text":"horizontal line of this form would be of the form y equals 3,"},{"Start":"15:49.619 ","End":"16:00.150","Text":"and a vertical line of this form would be described by x equals 4."},{"Start":"16:00.150 ","End":"16:04.170","Text":"In this 1, there is no x appearing."},{"Start":"16:04.170 ","End":"16:06.810","Text":"I\u0027ll just make a note of that, there is no x."},{"Start":"16:06.810 ","End":"16:08.805","Text":"In x equals 4,"},{"Start":"16:08.805 ","End":"16:12.210","Text":"there is no y appearing,"},{"Start":"16:12.210 ","End":"16:19.140","Text":"where x is a constant throughout this line and y is a constant throughout this line."},{"Start":"16:19.140 ","End":"16:22.785","Text":"We could generalize, horizontal lines,"},{"Start":"16:22.785 ","End":"16:27.765","Text":"would be, y equals some number, some parameter b."},{"Start":"16:27.765 ","End":"16:33.070","Text":"The vertical line would be something of the form x equals a."},{"Start":"16:35.240 ","End":"16:39.255","Text":"Let\u0027s first of all take, say,"},{"Start":"16:39.255 ","End":"16:42.885","Text":"a blue 1 and a blue 1,"},{"Start":"16:42.885 ","End":"16:50.505","Text":"and lets say, this 1 is where r is equal to,"},{"Start":"16:50.505 ","End":"16:53.620","Text":"say this was 3."},{"Start":"16:56.210 ","End":"16:58.995","Text":"Perhaps I should highlight it,"},{"Start":"16:58.995 ","End":"17:00.855","Text":"just so you know what I mean."},{"Start":"17:00.855 ","End":"17:07.350","Text":"This whole circle would just be everywhere where r is 3,"},{"Start":"17:07.350 ","End":"17:15.195","Text":"r equals 3 and there is no Theta appears in this equation."},{"Start":"17:15.195 ","End":"17:19.005","Text":"Similarly, if I take 1 of these rays,"},{"Start":"17:19.005 ","End":"17:21.975","Text":"let\u0027s say this 1,"},{"Start":"17:21.975 ","End":"17:28.259","Text":"this is where the angle is 60 degrees and I would say,"},{"Start":"17:28.259 ","End":"17:33.570","Text":"Theta equals, well, we don\u0027t usually use degrees when we\u0027re writing equations."},{"Start":"17:33.570 ","End":"17:37.590","Text":"I\u0027ll say, Theta equals Pi over 3."},{"Start":"17:37.590 ","End":"17:41.280","Text":"There\u0027s no r in the equation."},{"Start":"17:41.280 ","End":"17:45.510","Text":"Now, strictly speaking, it\u0027s actually the whole line,"},{"Start":"17:45.510 ","End":"17:47.685","Text":"not just up to the middle."},{"Start":"17:47.685 ","End":"17:52.304","Text":"I\u0027m going to continue right through to the other side."},{"Start":"17:52.304 ","End":"17:57.015","Text":"The reason is that if we take r as being negative,"},{"Start":"17:57.015 ","End":"17:58.650","Text":"and since r is not mentioned here,"},{"Start":"17:58.650 ","End":"18:00.435","Text":"we take whatever r we want,"},{"Start":"18:00.435 ","End":"18:03.600","Text":"but we\u0027ll get altogether positive rs, this bit,"},{"Start":"18:03.600 ","End":"18:05.670","Text":"the negative rs, this bit, another totality,"},{"Start":"18:05.670 ","End":"18:09.550","Text":"we\u0027ll get the whole line through here."},{"Start":"18:09.860 ","End":"18:17.460","Text":"It\u0027s debatable whether the pole is included because the pole doesn\u0027t really have a Theta,"},{"Start":"18:17.460 ","End":"18:20.130","Text":"but we somehow counted it anyway."},{"Start":"18:20.130 ","End":"18:22.845","Text":"Every Theta is good for this center."},{"Start":"18:22.845 ","End":"18:25.560","Text":"We\u0027re not quibble on those fine details."},{"Start":"18:25.560 ","End":"18:33.000","Text":"In general, something like r equals some constant,"},{"Start":"18:33.000 ","End":"18:39.465","Text":"a might be in general a circle and Theta"},{"Start":"18:39.465 ","End":"18:46.110","Text":"equals some number b would be 1 of these radii."},{"Start":"18:46.110 ","End":"18:50.955","Text":"That\u0027s equivalent, constant x, constant y here."},{"Start":"18:50.955 ","End":"18:56.295","Text":"Along the circles we have a constant r and along these we have a constant Theta."},{"Start":"18:56.295 ","End":"19:03.510","Text":"That\u0027s really all I want to say about polar and Cartesian equations."},{"Start":"19:03.510 ","End":"19:05.220","Text":"There is a lot more that could be said,"},{"Start":"19:05.220 ","End":"19:10.270","Text":"but I think that this is enough and there will be exercises too."}],"Thumbnail":null,"ID":9941},{"Watched":false,"Name":"Exercise 4","Duration":"4m 13s","ChapterTopicVideoID":10049,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.775","Text":"This exercise is 4 in 1,"},{"Start":"00:02.775 ","End":"00:07.440","Text":"and in each one of them we have an equation a Cartesian with x and y."},{"Start":"00:07.440 ","End":"00:13.065","Text":"Want to convert it to an equation in r and Theta, in polar."},{"Start":"00:13.065 ","End":"00:16.540","Text":"Let\u0027s start with the first 1."},{"Start":"00:16.580 ","End":"00:19.485","Text":"I\u0027ll need my formulas."},{"Start":"00:19.485 ","End":"00:28.345","Text":"This is the set I\u0027ll need for converting from xy to r Theta."},{"Start":"00:28.345 ","End":"00:30.905","Text":"There\u0027s nothing much more to do than substituting."},{"Start":"00:30.905 ","End":"00:33.410","Text":"Where I see x, I put r cosine Theta,"},{"Start":"00:33.410 ","End":"00:36.080","Text":"where I see y I put r sine Theta."},{"Start":"00:36.080 ","End":"00:43.220","Text":"We get 2r cosine Theta"},{"Start":"00:43.220 ","End":"00:49.410","Text":"minus 5r cosine Theta cubed."},{"Start":"00:49.410 ","End":"00:57.975","Text":"You can write it as r cubed cosine cubed Theta is equal to 1 plus,"},{"Start":"00:57.975 ","End":"01:01.860","Text":"x, y is r sine Theta r cosine Theta."},{"Start":"01:01.860 ","End":"01:04.335","Text":"I can collect the r\u0027s together,"},{"Start":"01:04.335 ","End":"01:10.360","Text":"r squared cosine Theta sine Theta."},{"Start":"01:10.580 ","End":"01:15.845","Text":"We might be able to simplify and collect stuff together."},{"Start":"01:15.845 ","End":"01:18.680","Text":"I\u0027m not going to bother trying to do that."},{"Start":"01:18.680 ","End":"01:21.620","Text":"We\u0027ll just leave it like this, that\u0027s fine."},{"Start":"01:21.620 ","End":"01:29.030","Text":"In b, we could just blindly substitute as before."},{"Start":"01:29.030 ","End":"01:31.790","Text":"However, when you see x squared plus y squared,"},{"Start":"01:31.790 ","End":"01:36.155","Text":"there\u0027s actually an extra formula that is handy, it\u0027s not essential."},{"Start":"01:36.155 ","End":"01:39.830","Text":"But if you look at this 1 and you square it,"},{"Start":"01:39.830 ","End":"01:44.156","Text":"we see that x squared plus y squared is r squared"},{"Start":"01:44.156 ","End":"01:48.125","Text":"and it\u0027s almost considered 1 of the basic formulas."},{"Start":"01:48.125 ","End":"01:52.550","Text":"Of course, it comes from the trigonometric identity"},{"Start":"01:52.550 ","End":"01:56.420","Text":"that sine squared plus cosine squared is 1 and if you apply this."},{"Start":"01:56.420 ","End":"01:59.900","Text":"I\u0027m going to use this and say on the left-hand side,"},{"Start":"01:59.900 ","End":"02:02.280","Text":"we have r squared,"},{"Start":"02:02.280 ","End":"02:04.650","Text":"on the right-hand side, I\u0027ll just do the substitution,"},{"Start":"02:04.650 ","End":"02:06.465","Text":"y is r sine Theta."},{"Start":"02:06.465 ","End":"02:10.675","Text":"It\u0027s 6 r sine Theta."},{"Start":"02:10.675 ","End":"02:15.560","Text":"However, I see r on both sides and I can cancel by r"},{"Start":"02:15.560 ","End":"02:23.569","Text":"so I get just r equals 6 sine of Theta."},{"Start":"02:23.569 ","End":"02:26.990","Text":"Now, you might but I divided by r."},{"Start":"02:26.990 ","End":"02:29.375","Text":"How do I know r is not 0?"},{"Start":"02:29.375 ","End":"02:34.315","Text":"Well, r equals 0 will work."},{"Start":"02:34.315 ","End":"02:37.205","Text":"Here, it\u0027s r equals 0,"},{"Start":"02:37.205 ","End":"02:41.495","Text":"doesn\u0027t make any difference what the angle is, it\u0027s the origin."},{"Start":"02:41.495 ","End":"02:45.215","Text":"The origin is on this curve."},{"Start":"02:45.215 ","End":"02:48.050","Text":"For 1 thing, the origin is on this curve."},{"Start":"02:48.050 ","End":"02:49.250","Text":"If I put x equals 0,"},{"Start":"02:49.250 ","End":"02:51.730","Text":"y equals 0, it works."},{"Start":"02:51.730 ","End":"02:55.480","Text":"Here, therefore, it should also work."},{"Start":"02:55.480 ","End":"03:01.255","Text":"In fact, if you let Theta equals 0 or Pi or any multiple of Pi,"},{"Start":"03:01.255 ","End":"03:03.715","Text":"then we\u0027ll get that r equals 0."},{"Start":"03:03.715 ","End":"03:06.880","Text":"There are values of Theta for which r is 0."},{"Start":"03:06.880 ","End":"03:08.525","Text":"That\u0027s fine."},{"Start":"03:08.525 ","End":"03:11.560","Text":"This also covers the origin,"},{"Start":"03:11.560 ","End":"03:16.795","Text":"so we can divide by r and this would be the answer."},{"Start":"03:16.795 ","End":"03:20.955","Text":"Moving on to c and c,"},{"Start":"03:20.955 ","End":"03:23.440","Text":"it is an equation in x and y,"},{"Start":"03:23.440 ","End":"03:25.689","Text":"just y happens to be missing."},{"Start":"03:25.689 ","End":"03:31.030","Text":"This is an equation of a vertical line through x equals 3."},{"Start":"03:31.030 ","End":"03:33.625","Text":"Nothing to do."},{"Start":"03:33.625 ","End":"03:36.900","Text":"Just replace x by r cosine Theta,"},{"Start":"03:36.900 ","End":"03:40.535","Text":"we\u0027ve got r cosine Theta equals 3."},{"Start":"03:40.535 ","End":"03:43.310","Text":"There a vertical line in polar coordinates"},{"Start":"03:43.310 ","End":"03:49.700","Text":"and d likewise is a horizontal line below the x-axis,"},{"Start":"03:49.700 ","End":"03:51.199","Text":"it\u0027s parallel to the x-axis,"},{"Start":"03:51.199 ","End":"03:56.225","Text":"4 units down the horizontal line through y equals minus 4."},{"Start":"03:56.225 ","End":"03:59.960","Text":"Again, just put in instead of y r sine Theta."},{"Start":"03:59.960 ","End":"04:03.110","Text":"R sine Theta equals minus 4,"},{"Start":"04:03.110 ","End":"04:07.505","Text":"and we have a straight line in polar coordinates."},{"Start":"04:07.505 ","End":"04:10.820","Text":"Left the 2 easy ones for last."},{"Start":"04:10.820 ","End":"04:13.560","Text":"That\u0027s it."}],"Thumbnail":null,"ID":9942},{"Watched":false,"Name":"Exercise 5","Duration":"6m 13s","ChapterTopicVideoID":10050,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this exercise, we have 4 different equations,"},{"Start":"00:04.710 ","End":"00:08.265","Text":"each of them in polar form with r and Theta."},{"Start":"00:08.265 ","End":"00:11.940","Text":"We want to convert them to Cartesian form in x and y."},{"Start":"00:11.940 ","End":"00:14.670","Text":"I brought in some formulas that might help."},{"Start":"00:14.670 ","End":"00:16.725","Text":"Then there\u0027s that extra formula,"},{"Start":"00:16.725 ","End":"00:20.865","Text":"that x squared plus y squared equals r squared."},{"Start":"00:20.865 ","End":"00:23.295","Text":"See if we can manage with this."},{"Start":"00:23.295 ","End":"00:27.315","Text":"Now, the first one is not immediately obvious."},{"Start":"00:27.315 ","End":"00:29.805","Text":"Certainly we could blindly substitute."},{"Start":"00:29.805 ","End":"00:33.449","Text":"But notice that if I multiply both sides by r,"},{"Start":"00:33.449 ","End":"00:37.690","Text":"it will be a lot more convenient because if I have r cosine Theta,"},{"Start":"00:37.690 ","End":"00:39.300","Text":"then I know that that\u0027s x."},{"Start":"00:39.300 ","End":"00:41.580","Text":"Also I have a formula for r squared."},{"Start":"00:41.580 ","End":"00:51.085","Text":"Let\u0027s do that and say that r squared equals minus 8r cosine Theta."},{"Start":"00:51.085 ","End":"00:54.630","Text":"You might say that, multiplied by r,"},{"Start":"00:54.630 ","End":"00:55.970","Text":"if r was 0,"},{"Start":"00:55.970 ","End":"00:57.650","Text":"we get an extra solution."},{"Start":"00:57.650 ","End":"01:01.400","Text":"But the origin is also a solution here also,"},{"Start":"01:01.400 ","End":"01:03.170","Text":"because for some values of Theta,"},{"Start":"01:03.170 ","End":"01:04.955","Text":"the cosine is going to be 0,"},{"Start":"01:04.955 ","End":"01:06.750","Text":"and that will give us r equals 0,"},{"Start":"01:06.750 ","End":"01:08.310","Text":"which is the origin."},{"Start":"01:08.310 ","End":"01:11.795","Text":"We haven\u0027t added any new solutions and we\u0027re okay with this."},{"Start":"01:11.795 ","End":"01:18.165","Text":"Now we can substitute r squared is x squared plus y squared,"},{"Start":"01:18.165 ","End":"01:21.810","Text":"and r cosine Theta is x."},{"Start":"01:21.810 ","End":"01:25.605","Text":"We have this equals minus 8x."},{"Start":"01:25.605 ","End":"01:30.345","Text":"Now, it\u0027s not terribly relevant,"},{"Start":"01:30.345 ","End":"01:32.934","Text":"but just as an addition,"},{"Start":"01:32.934 ","End":"01:36.215","Text":"this is actually an equation of a circle."},{"Start":"01:36.215 ","End":"01:40.985","Text":"Because if you bring the 8x over to the other side and add 16 to both sides,"},{"Start":"01:40.985 ","End":"01:44.555","Text":"you can get using a bit of algebra,"},{"Start":"01:44.555 ","End":"01:53.890","Text":"x plus 4 squared plus y squared equals 16."},{"Start":"01:53.890 ","End":"01:56.825","Text":"This is actually an equation of a circle,"},{"Start":"01:56.825 ","End":"02:00.710","Text":"and it passes through the origin as we saw when x is 0, y is 0."},{"Start":"02:00.710 ","End":"02:01.490","Text":"It works anyway."},{"Start":"02:01.490 ","End":"02:03.320","Text":"I don\u0027t want to get too much into that."},{"Start":"02:03.320 ","End":"02:06.590","Text":"We\u0027ve done the conversion and you could leave it like this."},{"Start":"02:06.590 ","End":"02:11.970","Text":"In b, also it\u0027s a bit more involved."},{"Start":"02:11.970 ","End":"02:15.440","Text":"Here also there\u0027s may be more than one way of doing it."},{"Start":"02:15.440 ","End":"02:20.945","Text":"But again, the same trick could help us multiplying both sides by r,"},{"Start":"02:20.945 ","End":"02:24.510","Text":"and then we\u0027ll get 6r."},{"Start":"02:24.510 ","End":"02:26.685","Text":"Well, instead of saying r^4,"},{"Start":"02:26.685 ","End":"02:28.229","Text":"I\u0027ll leave it as r cubed"},{"Start":"02:28.229 ","End":"02:39.515","Text":"and then write r sine Theta equals 4r minus r cosine Theta."},{"Start":"02:39.515 ","End":"02:45.500","Text":"Then the r sine Theta and the r cosine Theta are taken care of by this."},{"Start":"02:45.500 ","End":"02:48.735","Text":"Then there\u0027s just the add r and the r cubed,"},{"Start":"02:48.735 ","End":"02:52.510","Text":"where we can just substitute r as the square root."},{"Start":"02:52.510 ","End":"02:53.440","Text":"Let\u0027s do that."},{"Start":"02:53.440 ","End":"02:55.760","Text":"We get 6."},{"Start":"02:55.760 ","End":"02:57.570","Text":"Now, if r is this,"},{"Start":"02:57.570 ","End":"03:06.375","Text":"then r cubed is just the square root of x squared plus y squared cubed."},{"Start":"03:06.375 ","End":"03:10.030","Text":"We could write it as x squared plus y squared to the power of 3 over 2,"},{"Start":"03:10.030 ","End":"03:12.070","Text":"if you like fractional or exponents."},{"Start":"03:12.070 ","End":"03:13.570","Text":"I\u0027ll leave it like this."},{"Start":"03:13.570 ","End":"03:16.450","Text":"R sine Theta is y."},{"Start":"03:16.450 ","End":"03:24.080","Text":"R, once again, is the square root of x squared plus y squared"},{"Start":"03:24.080 ","End":"03:28.830","Text":"and r cosine Theta is x."},{"Start":"03:28.830 ","End":"03:31.325","Text":"Possibly it could be simplified,"},{"Start":"03:31.325 ","End":"03:34.620","Text":"but let\u0027s just leave it like this."},{"Start":"03:35.120 ","End":"03:41.305","Text":"Number c, just r appears, no Theta."},{"Start":"03:41.305 ","End":"03:44.300","Text":"But it\u0027s still an equation in r and Theta."},{"Start":"03:44.300 ","End":"03:47.540","Text":"More to see if you remember that if r equals 2,"},{"Start":"03:47.540 ","End":"03:48.350","Text":"what could that mean?"},{"Start":"03:48.350 ","End":"03:50.600","Text":"The distance from the origin is 2,"},{"Start":"03:50.600 ","End":"03:53.900","Text":"so a circle of radius 2 around the origin."},{"Start":"03:53.900 ","End":"03:57.050","Text":"What we could do is square both sides"},{"Start":"03:57.050 ","End":"04:03.770","Text":"and say that r squared equals 4."},{"Start":"04:03.770 ","End":"04:11.695","Text":"Then from here we could get that x squared plus y squared equals 4."},{"Start":"04:11.695 ","End":"04:14.210","Text":"Now, you might say, yeah,"},{"Start":"04:14.210 ","End":"04:17.810","Text":"well this covers also r equals minus 2."},{"Start":"04:17.810 ","End":"04:22.760","Text":"But r equals minus 2 is also a circle of radius 2 around the origin,"},{"Start":"04:22.760 ","End":"04:27.170","Text":"just with the antipode, the opposite point."},{"Start":"04:27.170 ","End":"04:30.770","Text":"Distance of minus 2 is just on the other side of the origin,"},{"Start":"04:30.770 ","End":"04:32.120","Text":"also a radius 2."},{"Start":"04:32.120 ","End":"04:35.570","Text":"This was okay, and this is the equation that we get,"},{"Start":"04:35.570 ","End":"04:37.925","Text":"which is of a circle of radius 2."},{"Start":"04:37.925 ","End":"04:47.010","Text":"In d, we have only Theta appearing without r. The angle is Pi over 4."},{"Start":"04:47.010 ","End":"04:54.570","Text":"What we could do is use the formula here"},{"Start":"04:54.570 ","End":"05:00.405","Text":"that if Theta is the arctangent of y over x,"},{"Start":"05:00.405 ","End":"05:07.000","Text":"then y over x is the tangent of Theta."},{"Start":"05:08.300 ","End":"05:11.355","Text":"But Theta is Pi over 4."},{"Start":"05:11.355 ","End":"05:14.160","Text":"This is tangent of Pi over 4,"},{"Start":"05:14.160 ","End":"05:18.715","Text":"and this is tangent of 45 degrees is known to be 1."},{"Start":"05:18.715 ","End":"05:22.780","Text":"This actually gives us that y over x is 1,"},{"Start":"05:22.780 ","End":"05:25.075","Text":"that y equals x,"},{"Start":"05:25.075 ","End":"05:28.239","Text":"and it\u0027s a straight line through the origin."},{"Start":"05:28.239 ","End":"05:35.575","Text":"Now, I want to point out that y equals x is not just,"},{"Start":"05:35.575 ","End":"05:39.095","Text":"well, I\u0027ll show you what I mean in a picture."},{"Start":"05:39.095 ","End":"05:43.285","Text":"Pi over 4 is normally this angle here,"},{"Start":"05:43.285 ","End":"05:45.939","Text":"but it\u0027s only a half line."},{"Start":"05:45.939 ","End":"05:50.330","Text":"However, y equals x is the full line."},{"Start":"05:50.330 ","End":"05:54.260","Text":"That\u0027s okay because this other side"},{"Start":"05:54.260 ","End":"05:58.505","Text":"can also be considered to have Theta of just Pi over 4,"},{"Start":"05:58.505 ","End":"06:00.050","Text":"but with a negative r."},{"Start":"06:00.050 ","End":"06:01.999","Text":"It all works out okay"},{"Start":"06:01.999 ","End":"06:09.365","Text":"and this really is the equation of the line 45 degrees through the origin."},{"Start":"06:09.365 ","End":"06:12.990","Text":"All 4 parts are done."}],"Thumbnail":null,"ID":9943},{"Watched":false,"Name":"Exercise 6","Duration":"8m ","ChapterTopicVideoID":10051,"CourseChapterTopicPlaylistID":8896,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.374","Text":"In this exercise, we\u0027re going to sketch"},{"Start":"00:03.374 ","End":"00:07.725","Text":"a rough graph of an equation in polar coordinates."},{"Start":"00:07.725 ","End":"00:10.979","Text":"This is the graph paper."},{"Start":"00:10.979 ","End":"00:14.145","Text":"We have r equals 2 plus 4 sine Theta."},{"Start":"00:14.145 ","End":"00:16.260","Text":"I\u0027m just wondering about scale."},{"Start":"00:16.260 ","End":"00:19.250","Text":"The largest and sine Theta can be is 1,"},{"Start":"00:19.250 ","End":"00:22.190","Text":"so the most r can be is 6."},{"Start":"00:22.190 ","End":"00:23.840","Text":"Let\u0027s see, we have 1, 2,"},{"Start":"00:23.840 ","End":"00:25.670","Text":"3, 4, 5, 6."},{"Start":"00:25.670 ","End":"00:30.855","Text":"We\u0027ll take 1 unit here as 1."},{"Start":"00:30.855 ","End":"00:33.930","Text":"Though r is up to 6 here."},{"Start":"00:33.930 ","End":"00:36.255","Text":"Let\u0027s make a table,"},{"Start":"00:36.255 ","End":"00:38.040","Text":"we\u0027ll have 2 columns,"},{"Start":"00:38.040 ","End":"00:46.680","Text":"a column for Theta and a column for r. There\u0027s a lot of values to plot."},{"Start":"00:46.680 ","End":"00:48.515","Text":"I want to use some symmetry."},{"Start":"00:48.515 ","End":"00:53.000","Text":"I want to use the fact that the sine of"},{"Start":"00:53.000 ","End":"00:59.779","Text":"Pi minus an angle is the same as sine of the angle."},{"Start":"00:59.779 ","End":"01:03.200","Text":"In other words, if I have this angle here,"},{"Start":"01:03.200 ","End":"01:06.620","Text":"Pi minus that is the one over here,"},{"Start":"01:06.620 ","End":"01:08.440","Text":"we\u0027re going to get a certain symmetry,"},{"Start":"01:08.440 ","End":"01:10.790","Text":"so I can do some values together."},{"Start":"01:10.790 ","End":"01:15.705","Text":"For example, if Theta is 0 or Pi,"},{"Start":"01:15.705 ","End":"01:22.920","Text":"then what I get is that sine Theta is 0,"},{"Start":"01:22.920 ","End":"01:25.515","Text":"so r is equal to 2."},{"Start":"01:25.515 ","End":"01:28.155","Text":"That gives me already 2 points."},{"Start":"01:28.155 ","End":"01:32.700","Text":"At 0 it\u0027s equal to 1,"},{"Start":"01:32.700 ","End":"01:39.310","Text":"2, and at Pi it\u0027s also equal to 2."},{"Start":"01:41.780 ","End":"01:45.615","Text":"Next, I\u0027ll take Pi over 6,"},{"Start":"01:45.615 ","End":"01:47.570","Text":"but at the same time,"},{"Start":"01:47.570 ","End":"01:53.665","Text":"I\u0027ll also take 5 Pi over 6 because of this symmetry."},{"Start":"01:53.665 ","End":"01:58.805","Text":"Then the sine of Pi over 6, the sine of 30."},{"Start":"01:58.805 ","End":"02:03.005","Text":"Perhaps I should have made a column for the sine. You know what?"},{"Start":"02:03.005 ","End":"02:05.990","Text":"Here we are an extra column for sine Theta."},{"Start":"02:05.990 ","End":"02:08.050","Text":"We said here it was 0."},{"Start":"02:08.050 ","End":"02:10.850","Text":"We got 2 plus 4 times 0."},{"Start":"02:10.850 ","End":"02:14.575","Text":"Here, sine Theta is 1.5."},{"Start":"02:14.575 ","End":"02:21.425","Text":"We get 2 plus 4 times a 1/2 is 4."},{"Start":"02:21.425 ","End":"02:23.765","Text":"I\u0027ll write 4 in here,"},{"Start":"02:23.765 ","End":"02:25.730","Text":"that gives us another 2 points."},{"Start":"02:25.730 ","End":"02:29.330","Text":"Here it\u0027s equal to 1, 2, 3, 4,"},{"Start":"02:29.330 ","End":"02:32.119","Text":"and the same on the other side,"},{"Start":"02:32.119 ","End":"02:35.310","Text":"1, 2, 3, 4."},{"Start":"02:36.050 ","End":"02:40.200","Text":"Next angle Pi over 4."},{"Start":"02:40.200 ","End":"02:42.735","Text":"The one on the other side,"},{"Start":"02:42.735 ","End":"02:44.490","Text":"3 Pi over 4."},{"Start":"02:44.490 ","End":"02:46.755","Text":"They both have the same sine,"},{"Start":"02:46.755 ","End":"02:52.410","Text":"which is 1 over root 2 or root 2/2,"},{"Start":"02:52.410 ","End":"03:00.495","Text":"same thing, we get 2 plus 4 times this."},{"Start":"03:00.495 ","End":"03:04.840","Text":"It\u0027s 2 plus twice root 2."},{"Start":"03:05.000 ","End":"03:08.895","Text":"Roughly root 2 is about 1.4."},{"Start":"03:08.895 ","End":"03:12.529","Text":"Here we get 2.8 altogether,"},{"Start":"03:12.529 ","End":"03:15.905","Text":"about 4.8, it\u0027s not exact anyway."},{"Start":"03:15.905 ","End":"03:22.260","Text":"4.8 is a bit less than the 5 circle, somewhere here."},{"Start":"03:22.260 ","End":"03:26.399","Text":"On the other side, somewhere here."},{"Start":"03:26.930 ","End":"03:34.829","Text":"Pi over 3, as well as 2 Pi over 3,"},{"Start":"03:34.829 ","End":"03:37.220","Text":"60 and 120 degrees."},{"Start":"03:37.220 ","End":"03:41.020","Text":"The sine is root 3 over 2."},{"Start":"03:41.020 ","End":"03:47.235","Text":"What we get is 2 plus twice root 3."},{"Start":"03:47.235 ","End":"03:53.790","Text":"Root 3 is about 1.7, 3.4 altogether, 5.4."},{"Start":"03:53.790 ","End":"04:00.000","Text":"Somewhere here and here."},{"Start":"04:00.000 ","End":"04:04.625","Text":"Then we get to the middle Pi over 2."},{"Start":"04:04.625 ","End":"04:09.270","Text":"There the sine is equal to 1."},{"Start":"04:09.860 ","End":"04:15.015","Text":"We get 2 plus 4 is 6."},{"Start":"04:15.015 ","End":"04:20.790","Text":"We\u0027ve reached the maximum, the outer circle."},{"Start":"04:22.070 ","End":"04:26.170","Text":"Let\u0027s see what happens when we continue."},{"Start":"04:26.450 ","End":"04:31.770","Text":"Next, let\u0027s take 7 Pi over 6,"},{"Start":"04:31.770 ","End":"04:34.470","Text":"as well as the other side,"},{"Start":"04:34.470 ","End":"04:37.150","Text":"11 Pi over 6."},{"Start":"04:42.350 ","End":"04:45.515","Text":"These are all going to be negative."},{"Start":"04:45.515 ","End":"04:52.565","Text":"Sine of 7 Pi over 6 is going to be the minus of sine of Pi over 6, which was 1/2."},{"Start":"04:52.565 ","End":"04:55.085","Text":"We\u0027re going to get minus 1/2."},{"Start":"04:55.085 ","End":"04:58.945","Text":"You know what? Let me just put in some of the values here."},{"Start":"04:58.945 ","End":"05:04.205","Text":"I put in the remaining values and we\u0027re going to get just the minuses of these."},{"Start":"05:04.205 ","End":"05:07.550","Text":"We\u0027re going to get minus root 2 over 2."},{"Start":"05:07.550 ","End":"05:11.820","Text":"We\u0027re going to get minus root 3 over 2."},{"Start":"05:11.820 ","End":"05:16.930","Text":"Here we\u0027re going to get minus 1 at 270 degrees."},{"Start":"05:21.470 ","End":"05:25.690","Text":"I forgot to put the 6 here."},{"Start":"05:26.720 ","End":"05:33.510","Text":"Notice that here, 2 plus 4 sine Theta is 2 plus 4 times negative 1/2,"},{"Start":"05:33.510 ","End":"05:36.120","Text":"it comes out to be 0."},{"Start":"05:36.120 ","End":"05:38.714","Text":"If I start joining these,"},{"Start":"05:38.714 ","End":"05:43.995","Text":"let\u0027s see, here, here, here."},{"Start":"05:43.995 ","End":"05:47.080","Text":"It\u0027s just free hand."},{"Start":"05:47.270 ","End":"05:52.530","Text":"Next, over here comes to 0,"},{"Start":"05:52.530 ","End":"05:57.485","Text":"and it also comes to 0 here."},{"Start":"05:57.485 ","End":"06:01.320","Text":"But let\u0027s see what happens as we continue."},{"Start":"06:04.610 ","End":"06:06.985","Text":"Let\u0027s do the last one."},{"Start":"06:06.985 ","End":"06:15.280","Text":"The last one, minus 1 will give us 2 minus 4, is minus 2."},{"Start":"06:15.280 ","End":"06:17.565","Text":"Over here it\u0027s minus 2."},{"Start":"06:17.565 ","End":"06:18.750","Text":"We don\u0027t take 1, 2,"},{"Start":"06:18.750 ","End":"06:22.810","Text":"we take 1, 2 in the other direction."},{"Start":"06:24.140 ","End":"06:27.500","Text":"I\u0027ll just save time here and here."},{"Start":"06:27.500 ","End":"06:30.185","Text":"It comes out also to be negative."},{"Start":"06:30.185 ","End":"06:32.015","Text":"We\u0027re working on the inside."},{"Start":"06:32.015 ","End":"06:35.225","Text":"What we get, I\u0027ll just cut it short,"},{"Start":"06:35.225 ","End":"06:43.010","Text":"is it actually makes a loop and goes like this."},{"Start":"06:43.010 ","End":"06:45.590","Text":"Now it\u0027s a bit hard to follow this,"},{"Start":"06:45.590 ","End":"06:49.610","Text":"but there is a theory behind this which I could have taught,"},{"Start":"06:49.610 ","End":"06:52.105","Text":"but I\u0027ll show you now."},{"Start":"06:52.105 ","End":"06:57.855","Text":"This shape, r equals something plus something sine Theta or cosine"},{"Start":"06:57.855 ","End":"07:05.820","Text":"Theta gives what is called a limacon,"},{"Start":"07:05.820 ","End":"07:11.730","Text":"think it has a little cedilla in French."},{"Start":"07:11.730 ","End":"07:17.850","Text":"Where is it? Limacon. B plus a cosine Theta,"},{"Start":"07:17.850 ","End":"07:19.065","Text":"or in our case,"},{"Start":"07:19.065 ","End":"07:20.810","Text":"b plus a sine Theta,"},{"Start":"07:20.810 ","End":"07:22.160","Text":"one case it\u0027s horizontal,"},{"Start":"07:22.160 ","End":"07:23.540","Text":"one case it\u0027s vertical."},{"Start":"07:23.540 ","End":"07:25.160","Text":"You get one of these Pictures,"},{"Start":"07:25.160 ","End":"07:28.250","Text":"and depending on what b and a are, in our case,"},{"Start":"07:28.250 ","End":"07:33.265","Text":"b is 2 and a is 4."},{"Start":"07:33.265 ","End":"07:34.670","Text":"B is less than a,"},{"Start":"07:34.670 ","End":"07:36.200","Text":"then we get a loop on it."},{"Start":"07:36.200 ","End":"07:41.785","Text":"This is the illustration for cosine and the illustration for sine is vertical."},{"Start":"07:41.785 ","End":"07:43.475","Text":"We get one of these,"},{"Start":"07:43.475 ","End":"07:45.785","Text":"and this fits in with the theory."},{"Start":"07:45.785 ","End":"07:47.660","Text":"But I don\u0027t want to get too much into it,"},{"Start":"07:47.660 ","End":"07:52.595","Text":"I just wanted to give you the idea of how to plot points and then join them."},{"Start":"07:52.595 ","End":"07:55.925","Text":"It may be you would have missed out on the fact that there\u0027s a looping in here."},{"Start":"07:55.925 ","End":"07:59.280","Text":"Anyway. That\u0027s all I want to say for this."}],"Thumbnail":null,"ID":9948}],"ID":8896},{"Name":"Tangent Lines in Polar Coordinates","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tangent Lines with Polar Coordinates","Duration":"14m 23s","ChapterTopicVideoID":10255,"CourseChapterTopicPlaylistID":8897,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this section, we\u0027ll learn how to find tangent lines"},{"Start":"00:04.020 ","End":"00:07.860","Text":"of functions given in polar coordinates at a given point,"},{"Start":"00:07.860 ","End":"00:13.080","Text":"and we\u0027re going to assume that the function is given in the form that"},{"Start":"00:13.080 ","End":"00:20.280","Text":"r is equal to some function of Theta."},{"Start":"00:20.280 ","End":"00:25.560","Text":"Then, we\u0027ll be asked at a certain point Theta to find the tangent line."},{"Start":"00:25.560 ","End":"00:29.730","Text":"In fact, why don\u0027t I start right away with an example."},{"Start":"00:29.730 ","End":"00:32.805","Text":"My example is going to be,"},{"Start":"00:32.805 ","End":"00:42.330","Text":"that I\u0027m given that r is equal to 3 plus 8 sine Theta,"},{"Start":"00:42.330 ","End":"00:48.800","Text":"and what we have to do is to find the tangent line,"},{"Start":"00:48.800 ","End":"00:51.535","Text":"I\u0027ll just call it the tangent,"},{"Start":"00:51.535 ","End":"00:58.410","Text":"at the point where Theta equals Pi over 6."},{"Start":"00:58.410 ","End":"01:00.260","Text":"For those who like degrees,"},{"Start":"01:00.260 ","End":"01:02.705","Text":"that is equal to 180 over 6,"},{"Start":"01:02.705 ","End":"01:07.010","Text":"that\u0027s 30 degrees. How do we do this?"},{"Start":"01:07.010 ","End":"01:08.959","Text":"Well, for the tangent,"},{"Start":"01:08.959 ","End":"01:14.330","Text":"what we need is,"},{"Start":"01:14.960 ","End":"01:20.930","Text":"dy/dx and then we also have to figure out what is this point in terms of x and y."},{"Start":"01:20.930 ","End":"01:24.740","Text":"But in general, we need a formula for dy/dx."},{"Start":"01:24.740 ","End":"01:26.735","Text":"What we do is,"},{"Start":"01:26.735 ","End":"01:29.915","Text":"I\u0027ll bring out those formulas."},{"Start":"01:29.915 ","End":"01:32.030","Text":"Here are the formulas,"},{"Start":"01:32.030 ","End":"01:33.840","Text":"but actually I don\u0027t need all of them,"},{"Start":"01:33.840 ","End":"01:35.300","Text":"I only need the top 2,"},{"Start":"01:35.300 ","End":"01:39.065","Text":"so I\u0027m going to just write them out and bigger."},{"Start":"01:39.065 ","End":"01:42.200","Text":"These are the only 2 formulas I\u0027ll need."},{"Start":"01:42.200 ","End":"01:45.890","Text":"Now, I\u0027m going to find dy/dx in general."},{"Start":"01:45.890 ","End":"01:48.500","Text":"Let\u0027s return to the example later."},{"Start":"01:48.500 ","End":"01:50.060","Text":"I\u0027m going to do it in general,"},{"Start":"01:50.060 ","End":"01:52.880","Text":"where r is given as a function of Theta."},{"Start":"01:52.880 ","End":"01:58.740","Text":"What I\u0027m going to do, is first of all say in general that,"},{"Start":"01:58.750 ","End":"02:02.460","Text":"dy/dx is going to be,"},{"Start":"02:04.610 ","End":"02:08.000","Text":"in the Leibniz\u0027s notation with the d\u0027s,"},{"Start":"02:08.000 ","End":"02:09.470","Text":"you can treat them like fractions."},{"Start":"02:09.470 ","End":"02:13.880","Text":"This would be dy over d Theta divided"},{"Start":"02:13.880 ","End":"02:20.220","Text":"by dx over d Theta as if it was a fraction."},{"Start":"02:20.220 ","End":"02:23.565","Text":"We\u0027re going to compute each of these separately."},{"Start":"02:23.565 ","End":"02:28.280","Text":"For dy over d Theta, in general,"},{"Start":"02:28.280 ","End":"02:32.720","Text":"we have that y equals sine Theta,"},{"Start":"02:32.720 ","End":"02:36.320","Text":"but r is going to be given as a function of Theta,"},{"Start":"02:36.320 ","End":"02:43.690","Text":"so y is some function of Theta times sine Theta."},{"Start":"02:43.910 ","End":"02:50.960","Text":"Y prime, or in this case I\u0027m going to write it as dy over d Theta is equal to,"},{"Start":"02:50.960 ","End":"02:52.580","Text":"and using the product rule,"},{"Start":"02:52.580 ","End":"02:54.860","Text":"I hope you remember the product rule."},{"Start":"02:54.860 ","End":"02:59.695","Text":"Just in case you forgotten when we have a product of 2 things,"},{"Start":"02:59.695 ","End":"03:03.930","Text":"and we take the derivative or rather derivative with the prime notation,"},{"Start":"03:03.930 ","End":"03:07.215","Text":"it\u0027s the derivative of one times the other,"},{"Start":"03:07.215 ","End":"03:10.520","Text":"plus one times the derivative of the other."},{"Start":"03:10.520 ","End":"03:17.600","Text":"Getting back here, we\u0027ll take the derivative of the first f prime of"},{"Start":"03:17.600 ","End":"03:21.420","Text":"Theta times the other one as is"},{"Start":"03:21.420 ","End":"03:28.430","Text":"sine Theta plus the first one as is times the derivative of the second one,"},{"Start":"03:28.430 ","End":"03:31.270","Text":"which is cosine Theta,"},{"Start":"03:31.270 ","End":"03:33.975","Text":"That\u0027s dy over d Theta."},{"Start":"03:33.975 ","End":"03:40.515","Text":"Then dx over d Theta will be a similar thing,"},{"Start":"03:40.515 ","End":"03:43.755","Text":"x is equal to r cosine Theta,"},{"Start":"03:43.755 ","End":"03:49.125","Text":"but r is f of Theta times cosine Theta."},{"Start":"03:49.125 ","End":"03:51.660","Text":"Pretty much the same as this with cosine so,"},{"Start":"03:51.660 ","End":"03:53.655","Text":"dx over d Theta will equal,"},{"Start":"03:53.655 ","End":"03:55.245","Text":"again the product rule,"},{"Start":"03:55.245 ","End":"04:03.210","Text":"f prime of Theta times cosine Theta,"},{"Start":"04:03.210 ","End":"04:07.655","Text":"and then the first one times the derivative of the second."},{"Start":"04:07.655 ","End":"04:12.100","Text":"Now, another derivative of cosine is going to be minus sign."},{"Start":"04:12.100 ","End":"04:16.320","Text":"So the first one is f of the Theta, and then,"},{"Start":"04:16.320 ","End":"04:22.090","Text":"minus sine Theta has the minus and here is the sine Theta."},{"Start":"04:22.090 ","End":"04:24.440","Text":"Now, I\u0027ve got these 2 bits."},{"Start":"04:24.440 ","End":"04:28.370","Text":"I don\u0027t like mixing the Newton and Leibniz notations,"},{"Start":"04:28.370 ","End":"04:30.050","Text":"the prime and the d,"},{"Start":"04:30.050 ","End":"04:37.085","Text":"so let\u0027s just note that f prime of Theta is dr over d Theta,"},{"Start":"04:37.085 ","End":"04:44.180","Text":"because we don\u0027t really need the letter f just to show that r was a function of Theta,"},{"Start":"04:44.180 ","End":"04:47.800","Text":"so I\u0027m rewriting this as this f prime of Theta,"},{"Start":"04:47.800 ","End":"04:51.175","Text":"again, as dr over d Theta,"},{"Start":"04:51.175 ","End":"04:56.585","Text":"and now if I substitute this and this into this equation,"},{"Start":"04:56.585 ","End":"04:59.135","Text":"and maybe I can squeeze it in here."},{"Start":"04:59.135 ","End":"05:07.610","Text":"Then I can write that dy/dx is equal to dy over d Theta from"},{"Start":"05:07.610 ","End":"05:17.010","Text":"here is dr over d Theta times sine Theta plus,"},{"Start":"05:17.010 ","End":"05:19.725","Text":"and f of Theta,"},{"Start":"05:19.725 ","End":"05:24.989","Text":"or we can replace that by r here and here."},{"Start":"05:25.640 ","End":"05:30.090","Text":"Why am I being lazy? Why don\u0027t I just rewrite it here?"},{"Start":"05:30.550 ","End":"05:39.445","Text":"dr over d Theta sine Theta plus r cosine sine Theta."},{"Start":"05:39.445 ","End":"05:44.250","Text":"This is equal to dr over d Theta cosine Theta"},{"Start":"05:44.250 ","End":"05:50.465","Text":"minus r sine Theta and now I can complete this so,"},{"Start":"05:50.465 ","End":"05:58.290","Text":"plus r cosine Theta divided by and I\u0027m copying from here so,"},{"Start":"05:58.290 ","End":"06:05.010","Text":"dr over d Theta cosine Theta"},{"Start":"06:05.300 ","End":"06:12.850","Text":"minus r sine Theta."},{"Start":"06:12.890 ","End":"06:16.335","Text":"I\u0027ve put the formula in the box."},{"Start":"06:16.335 ","End":"06:19.820","Text":"I don\u0027t know if it needs to be memorized."},{"Start":"06:19.820 ","End":"06:22.115","Text":"You should be able to derive it yourself."},{"Start":"06:22.115 ","End":"06:27.290","Text":"This you will know or this you should know r cosine Theta and r sine Theta and actually,"},{"Start":"06:27.290 ","End":"06:31.685","Text":"we could have taken a shortcut without using the letter f and gone straight from"},{"Start":"06:31.685 ","End":"06:37.534","Text":"r cosine Theta to derivative of the first is dr d Theta times the second,"},{"Start":"06:37.534 ","End":"06:43.070","Text":"then, this one as is times minus cosine Theta and similarly for the other."},{"Start":"06:43.070 ","End":"06:48.595","Text":"From these 2, you can get these 2 yourself so don\u0027t memorize this formula."},{"Start":"06:48.595 ","End":"06:51.855","Text":"Yeah, hopefully you won\u0027t need to."},{"Start":"06:51.855 ","End":"06:54.439","Text":"Now that we\u0027ve got the general formula,"},{"Start":"06:54.439 ","End":"06:59.145","Text":"let\u0027s solve our particular example,"},{"Start":"06:59.145 ","End":"07:02.660","Text":"where we\u0027re given r as this function of"},{"Start":"07:02.660 ","End":"07:06.570","Text":"Theta and we have to find the tangent at the given point."},{"Start":"07:07.290 ","End":"07:11.215","Text":"Let\u0027s see, scroll down a bit,"},{"Start":"07:11.215 ","End":"07:21.250","Text":"and I\u0027ll just copy the function that r equals 3 plus 8 sine of Theta."},{"Start":"07:21.250 ","End":"07:23.079","Text":"So using this formula,"},{"Start":"07:23.079 ","End":"07:30.100","Text":"we have that dy over dx is equal to dr over d Theta."},{"Start":"07:30.100 ","End":"07:32.270","Text":"Differentiate this."},{"Start":"07:32.310 ","End":"07:35.095","Text":"I got an extra Theta there."},{"Start":"07:35.095 ","End":"07:37.090","Text":"So dr over d Theta,"},{"Start":"07:37.090 ","End":"07:39.175","Text":"this is a constant, so it\u0027s nothing."},{"Start":"07:39.175 ","End":"07:49.720","Text":"We just get 8 cosine Theta times sine Theta plus r,"},{"Start":"07:49.720 ","End":"07:55.000","Text":"which is 3 plus 8 sine Theta times"},{"Start":"07:55.000 ","End":"08:03.895","Text":"cosine Theta over dr over d Theta I just copy,"},{"Start":"08:03.895 ","End":"08:07.315","Text":"it\u0027s 8 cosine Theta."},{"Start":"08:07.315 ","End":"08:12.790","Text":"But this time, times cosine Theta minus r,"},{"Start":"08:12.790 ","End":"08:13.930","Text":"which is the same thing,"},{"Start":"08:13.930 ","End":"08:16.960","Text":"3 plus 8 sine Theta."},{"Start":"08:16.960 ","End":"08:19.720","Text":"Then sine Theta."},{"Start":"08:19.720 ","End":"08:21.640","Text":"Looks a mess,"},{"Start":"08:21.640 ","End":"08:25.915","Text":"we can\u0027t simplify it a bit before we start substituting."},{"Start":"08:25.915 ","End":"08:28.600","Text":"Because look, I have here 8 cosine Theta,"},{"Start":"08:28.600 ","End":"08:31.450","Text":"sine Theta, and here I have another 8 of them."},{"Start":"08:31.450 ","End":"08:35.590","Text":"So altogether I have 16 cosine Theta,"},{"Start":"08:35.590 ","End":"08:40.990","Text":"sine Theta, and then I have plus 3 cosine Theta,"},{"Start":"08:40.990 ","End":"08:43.900","Text":"so that\u0027s the numerator."},{"Start":"08:43.900 ","End":"08:53.810","Text":"Over the denominator, I have 8 and here I have cosine squared Theta,"},{"Start":"08:55.530 ","End":"09:00.160","Text":"and here I have minus sine squared Theta."},{"Start":"09:00.160 ","End":"09:01.960","Text":"I mean it\u0027s minus 8,"},{"Start":"09:01.960 ","End":"09:03.895","Text":"but the 8 I took outside here,"},{"Start":"09:03.895 ","End":"09:09.560","Text":"and then minus 3 sine Theta."},{"Start":"09:09.720 ","End":"09:13.420","Text":"I\u0027m not going to do any more simplification at this point."},{"Start":"09:13.420 ","End":"09:20.190","Text":"We should remember that Theta is equal to Pi over 6,"},{"Start":"09:20.190 ","End":"09:22.185","Text":"and I\u0027ll just put in brackets,"},{"Start":"09:22.185 ","End":"09:25.965","Text":"that is 30 degrees and we know the sine."},{"Start":"09:25.965 ","End":"09:32.760","Text":"Well, I\u0027ll remind you that sine of 30 degrees is a half"},{"Start":"09:32.760 ","End":"09:40.120","Text":"and cosine sine of 30 degrees is the square root of 3 over 2."},{"Start":"09:40.120 ","End":"09:45.190","Text":"If we go back here now and substitute Theta,"},{"Start":"09:45.190 ","End":"09:47.750","Text":"and I need a bit more space,"},{"Start":"09:47.750 ","End":"09:54.280","Text":"so we get 16."},{"Start":"09:54.280 ","End":"09:59.470","Text":"Now, cosine Theta is root 3 over 2,"},{"Start":"09:59.470 ","End":"10:08.860","Text":"sine Theta is a half plus 3 times cosine Theta is root"},{"Start":"10:08.860 ","End":"10:17.245","Text":"3 over 2 over 8 times cosine"},{"Start":"10:17.245 ","End":"10:20.695","Text":"squared Theta is 3 quarters."},{"Start":"10:20.695 ","End":"10:25.330","Text":"Sine squared Theta is 1 quarter"},{"Start":"10:25.330 ","End":"10:33.700","Text":"minus 3 times 1.5."},{"Start":"10:33.700 ","End":"10:35.350","Text":"Let\u0027s see what we can simplify."},{"Start":"10:35.350 ","End":"10:37.760","Text":"I\u0027ll continue over here."},{"Start":"10:40.920 ","End":"10:46.225","Text":"Let\u0027s see, 16 over 4,"},{"Start":"10:46.225 ","End":"10:54.580","Text":"2 times 2 is 4 times square root of 3 plus 3 over"},{"Start":"10:54.580 ","End":"10:58.910","Text":"2 times square root of 3"},{"Start":"10:59.220 ","End":"11:10.340","Text":"over 8 times a half is 4 minus 3 over 2."},{"Start":"11:12.270 ","End":"11:16.810","Text":"I\u0027m going to multiply top and bottom by 2."},{"Start":"11:16.810 ","End":"11:23.170","Text":"So I\u0027ve got 8 plus 3 times square root of 3,"},{"Start":"11:23.170 ","End":"11:28.315","Text":"so that\u0027s 11 square root of 3."},{"Start":"11:28.315 ","End":"11:32.430","Text":"On the denominator, also double that,"},{"Start":"11:32.430 ","End":"11:36.495","Text":"so it\u0027s 8 minus 3 is 5."},{"Start":"11:36.495 ","End":"11:40.470","Text":"This is dy over dx,"},{"Start":"11:40.470 ","End":"11:42.810","Text":"this is the slope of the tangent."},{"Start":"11:42.810 ","End":"11:49.165","Text":"This is my m, the slope."},{"Start":"11:49.165 ","End":"11:52.210","Text":"I have a slope, but I still don\u0027t have the point."},{"Start":"11:52.210 ","End":"11:53.785","Text":"I have the point,"},{"Start":"11:53.785 ","End":"11:55.450","Text":"I have the Theta of the point,"},{"Start":"11:55.450 ","End":"11:57.265","Text":"which is pi over 6."},{"Start":"11:57.265 ","End":"12:01.440","Text":"At this point, we\u0027ll need to get the x,"},{"Start":"12:01.440 ","End":"12:03.520","Text":"y of the point."},{"Start":"12:06.920 ","End":"12:14.954","Text":"We first of all, let\u0027s find r. R is equal to 3 plus 8 sine Theta,"},{"Start":"12:14.954 ","End":"12:18.780","Text":"which is sine Theta is a half."},{"Start":"12:18.780 ","End":"12:23.095","Text":"It\u0027s 3 plus 4,"},{"Start":"12:23.095 ","End":"12:27.340","Text":"because 8 times a half is 4, which is 7."},{"Start":"12:27.340 ","End":"12:31.690","Text":"Now that we have Theta and r,"},{"Start":"12:31.690 ","End":"12:33.460","Text":"we can get x and y."},{"Start":"12:33.460 ","End":"12:38.680","Text":"So x, which is equal to r times cosine Theta,"},{"Start":"12:38.680 ","End":"12:42.310","Text":"is 7 times the cosine,"},{"Start":"12:42.310 ","End":"12:45.145","Text":"which is 7 root 3 over 2,"},{"Start":"12:45.145 ","End":"12:49.075","Text":"7 over 2 root 3."},{"Start":"12:49.075 ","End":"12:56.125","Text":"Y is equal to r sine Theta is 7 times a half,"},{"Start":"12:56.125 ","End":"12:59.660","Text":"which is just 7 over 2."},{"Start":"13:00.150 ","End":"13:07.750","Text":"Now, we have x and y and m. So x,"},{"Start":"13:07.750 ","End":"13:12.320","Text":"y and m, now I can get the equation of the tangent."},{"Start":"13:13.710 ","End":"13:16.975","Text":"Let me just write it in general."},{"Start":"13:16.975 ","End":"13:24.220","Text":"General equation would be that y minus the y"},{"Start":"13:24.220 ","End":"13:30.880","Text":"of the point equals m times x minus the x of the point that we have."},{"Start":"13:30.880 ","End":"13:42.740","Text":"In our case, it becomes y minus 7 over 2 equals m,"},{"Start":"13:44.460 ","End":"13:51.925","Text":"11 root 3 over 5 times"},{"Start":"13:51.925 ","End":"13:59.875","Text":"x minus 7 over 2 root 3 from here."},{"Start":"13:59.875 ","End":"14:01.390","Text":"You could leave it at that."},{"Start":"14:01.390 ","End":"14:05.020","Text":"You could tidy it up a little bit more and say that y is"},{"Start":"14:05.020 ","End":"14:12.580","Text":"equal to 7 over 2 plus,"},{"Start":"14:12.580 ","End":"14:15.265","Text":"and I just copy pasted this."},{"Start":"14:15.265 ","End":"14:17.529","Text":"This would be the answer,"},{"Start":"14:17.529 ","End":"14:23.450","Text":"and I\u0027ll just highlight it and we are done."}],"Thumbnail":null,"ID":10590},{"Watched":false,"Name":"Exercise 7","Duration":"5m 34s","ChapterTopicVideoID":10054,"CourseChapterTopicPlaylistID":8897,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this exercise, we\u0027re given function r of Theta"},{"Start":"00:04.710 ","End":"00:10.215","Text":"in polar form and this is a polar graph."},{"Start":"00:10.215 ","End":"00:17.970","Text":"We want the equation of the tangent to this graph at the point where Theta is Pi over 6,"},{"Start":"00:17.970 ","End":"00:21.210","Text":"it could be more than 1 tangent theoretically,"},{"Start":"00:21.210 ","End":"00:24.630","Text":"because we\u0027ve seen curves that cross themselves in polar."},{"Start":"00:24.630 ","End":"00:25.920","Text":"So there might be more than 1."},{"Start":"00:25.920 ","End":"00:29.940","Text":"So to be safe, I put an s in brackets."},{"Start":"00:29.940 ","End":"00:35.490","Text":"The equation we\u0027re looking for I should have mentioned is in Cartesian form,"},{"Start":"00:35.490 ","End":"00:38.320","Text":"we want y as a function of x."},{"Start":"00:38.720 ","End":"00:40.940","Text":"For a tangent line,"},{"Start":"00:40.940 ","End":"00:43.115","Text":"we need a point and a slope."},{"Start":"00:43.115 ","End":"00:46.870","Text":"Let\u0027s get to the point so to speak."},{"Start":"00:46.870 ","End":"00:52.175","Text":"When Theta is equal to Pi over 6,"},{"Start":"00:52.175 ","End":"00:55.705","Text":"that gives us that r is equal to."},{"Start":"00:55.705 ","End":"01:00.259","Text":"Now, cosine of Pi over 6 is cosine 30,"},{"Start":"01:00.259 ","End":"01:08.070","Text":"is root 3 over 2 and 4 Theta is 2/3 of Pi,"},{"Start":"01:08.070 ","End":"01:10.245","Text":"which is 120 degrees."},{"Start":"01:10.245 ","End":"01:15.740","Text":"Sine of 120 is also root 3 over 2."},{"Start":"01:15.740 ","End":"01:17.585","Text":"It\u0027s the same as sine 60."},{"Start":"01:17.585 ","End":"01:20.760","Text":"So we\u0027ve got 3/4."},{"Start":"01:20.760 ","End":"01:27.770","Text":"So that means that we have the point is r is 3/4,"},{"Start":"01:28.160 ","End":"01:33.105","Text":"Theta is Pi over 6."},{"Start":"01:33.105 ","End":"01:40.095","Text":"That\u0027s the polar coordinates of the point."},{"Start":"01:40.095 ","End":"01:44.140","Text":"That gives us that the xy of the point,"},{"Start":"01:44.140 ","End":"01:47.240","Text":"well, let\u0027s see what the x and y are,"},{"Start":"01:49.130 ","End":"01:51.495","Text":"well that\u0027ll do it at the side,"},{"Start":"01:51.495 ","End":"01:56.380","Text":"r cosine Theta for this particular point is r,"},{"Start":"01:56.380 ","End":"02:02.955","Text":"which is 3/4 cosine of 30 degrees is root 3 over 2."},{"Start":"02:02.955 ","End":"02:10.155","Text":"So this gives us 3 root 3 over 8."},{"Start":"02:10.155 ","End":"02:19.665","Text":"R sine Theta is 3/4 times sine of Pi over 6 is sine 30 is 1/2."},{"Start":"02:19.665 ","End":"02:22.830","Text":"So here we have 3/8."},{"Start":"02:23.510 ","End":"02:25.940","Text":"We found the point."},{"Start":"02:25.940 ","End":"02:28.730","Text":"Now, the big thing is the slope."},{"Start":"02:28.730 ","End":"02:33.560","Text":"I brought in the formula for the derivative dy over dx."},{"Start":"02:33.560 ","End":"02:35.950","Text":"It\u0027s this expression here,"},{"Start":"02:35.950 ","End":"02:38.225","Text":"I\u0027m going to compute that now."},{"Start":"02:38.225 ","End":"02:42.760","Text":"But look in the numerator and in the denominator we need dr by dTheta."},{"Start":"02:42.760 ","End":"02:44.735","Text":"So let\u0027s do that first,"},{"Start":"02:44.735 ","End":"02:48.870","Text":"dr by dTheta is equal to, let\u0027s see,"},{"Start":"02:48.870 ","End":"02:50.225","Text":"there\u0027s the original function,"},{"Start":"02:50.225 ","End":"02:52.295","Text":"r as a function of Theta,"},{"Start":"02:52.295 ","End":"02:55.435","Text":"use the product rule here."},{"Start":"02:55.435 ","End":"03:04.940","Text":"The product, this is like u times v. So we need the derivative of this times this as is."},{"Start":"03:04.940 ","End":"03:07.915","Text":"So the derivative of sine for Theta,"},{"Start":"03:07.915 ","End":"03:11.895","Text":"we start off with cosine 4 Theta."},{"Start":"03:11.895 ","End":"03:15.530","Text":"But because 4 Theta has an inner derivative,"},{"Start":"03:15.530 ","End":"03:18.515","Text":"we have to multiply by that also."},{"Start":"03:18.515 ","End":"03:22.280","Text":"Then the cosine Theta from there as is."},{"Start":"03:22.280 ","End":"03:31.070","Text":"Then we need also this 1 as is sine 4 Theta."},{"Start":"03:31.070 ","End":"03:35.990","Text":"The derivative of cosine Theta is minus which I put in front."},{"Start":"03:35.990 ","End":"03:41.770","Text":"Sine Theta, let\u0027s put brackets everywhere."},{"Start":"03:42.320 ","End":"03:47.110","Text":"What we have to do now is plug this into this,"},{"Start":"03:47.110 ","End":"03:48.790","Text":"which will be a bit of a mess."},{"Start":"03:48.790 ","End":"03:53.085","Text":"So I\u0027ve just brought you the result and here it is."},{"Start":"03:53.085 ","End":"03:56.860","Text":"Just replace dr by dTheta with what\u0027s written here,"},{"Start":"03:56.860 ","End":"03:59.950","Text":"and replace r with what\u0027s written here."},{"Start":"03:59.950 ","End":"04:06.010","Text":"This is what we get and doesn\u0027t seem to be much to do in the way of simplification."},{"Start":"04:06.010 ","End":"04:12.895","Text":"What we need is dr by dTheta when Theta is"},{"Start":"04:12.895 ","End":"04:20.050","Text":"equal to Pi over 6, or 30 degrees."},{"Start":"04:20.050 ","End":"04:23.260","Text":"I\u0027m not going to do all the computations."},{"Start":"04:23.260 ","End":"04:25.585","Text":"I\u0027ll give you the answer here."},{"Start":"04:25.585 ","End":"04:30.995","Text":"Comes out to be 1/3 root 3."},{"Start":"04:30.995 ","End":"04:34.210","Text":"If you do all the calculations."},{"Start":"04:34.340 ","End":"04:45.010","Text":"The equation of the tangent is going to be y minus the y of the point, which is 3/8,"},{"Start":"04:45.010 ","End":"04:49.870","Text":"equals the slope /3 root 3 times x"},{"Start":"04:49.870 ","End":"04:55.680","Text":"minus the x of the point 3 root 3 over 8."},{"Start":"04:55.680 ","End":"05:00.460","Text":"If you bring the 3/8 over to the other side and simplify a bit,"},{"Start":"05:00.460 ","End":"05:09.045","Text":"we get y equals 1/3 root 3 times x."},{"Start":"05:09.045 ","End":"05:13.260","Text":"Then if you multiply this by this,"},{"Start":"05:13.260 ","End":"05:19.965","Text":"we\u0027ve got minus 1/8 plus 3/8,"},{"Start":"05:19.965 ","End":"05:23.590","Text":"which means plus 1/4."},{"Start":"05:23.660 ","End":"05:26.705","Text":"There would\u0027ve been a lot of work if we"},{"Start":"05:26.705 ","End":"05:29.795","Text":"written it out full hand and then all the computations."},{"Start":"05:29.795 ","End":"05:31.445","Text":"Anyway, get the idea,"},{"Start":"05:31.445 ","End":"05:35.160","Text":"and that\u0027s the tangent line."}],"Thumbnail":null,"ID":9949},{"Watched":false,"Name":"Exercise 8","Duration":"5m 47s","ChapterTopicVideoID":10055,"CourseChapterTopicPlaylistID":8897,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.850","Text":"In this exercise, we\u0027re given a polar equation and,"},{"Start":"00:03.850 ","End":"00:05.550","Text":"of course, it has a graph."},{"Start":"00:05.550 ","End":"00:10.260","Text":"The equation of the tangent line,"},{"Start":"00:10.260 ","End":"00:15.990","Text":"or lines, might be more than 1, of this curve at the origin,"},{"Start":"00:15.990 ","End":"00:19.275","Text":"or we\u0027ll show that it actually does go through the origin."},{"Start":"00:19.275 ","End":"00:29.100","Text":"The equation, we will put in Cartesian coordinates, as we learned. Now, let\u0027s see,"},{"Start":"00:29.100 ","End":"00:32.460","Text":"does this curve go through the origin?"},{"Start":"00:32.460 ","End":"00:33.885","Text":"Well, at the origin,"},{"Start":"00:33.885 ","End":"00:36.150","Text":"r is equal to 0."},{"Start":"00:36.150 ","End":"00:40.020","Text":"Let\u0027s see, what would Theta be in order for r to be 0?"},{"Start":"00:40.020 ","End":"00:44.565","Text":"That would give us the equation that this is 0,"},{"Start":"00:44.565 ","End":"00:52.120","Text":"so Sine Theta has to equal 1/2, if I set this to be 0."},{"Start":"00:58.190 ","End":"01:05.505","Text":"I only need to consider from 0 to 2 Pi, because sine Theta has a period of 2 Pi,"},{"Start":"01:05.505 ","End":"01:07.215","Text":"just for everything just repeats."},{"Start":"01:07.215 ","End":"01:11.320","Text":"So from 0-2 Pi, where sine is 1/2,"},{"Start":"01:11.320 ","End":"01:13.399","Text":"first of all, could be 30 degrees,"},{"Start":"01:13.399 ","End":"01:16.240","Text":"which is Pi over 6."},{"Start":"01:16.240 ","End":"01:19.880","Text":"Besides 30 degrees, could be 150 degrees,"},{"Start":"01:19.880 ","End":"01:23.420","Text":"in other words, 5 Pi over 6,"},{"Start":"01:23.420 ","End":"01:27.860","Text":"this is the 2 angles from 0-2 Pi."},{"Start":"01:27.860 ","End":"01:36.394","Text":"Now, I want to compute the derivative at this point."},{"Start":"01:36.394 ","End":"01:38.330","Text":"Normally, I\u0027d first of all,"},{"Start":"01:38.330 ","End":"01:40.655","Text":"compute the x, y of the point,"},{"Start":"01:40.655 ","End":"01:43.595","Text":"but we know what that is, because it\u0027s at the origin."},{"Start":"01:43.595 ","End":"01:46.410","Text":"So we know the x, y."},{"Start":"01:47.060 ","End":"01:50.130","Text":"The polar coordinates are 0,"},{"Start":"01:50.130 ","End":"01:54.080","Text":"Pi over 6, 0,5 Pi over 6, but we don\u0027t need that."},{"Start":"01:54.080 ","End":"01:58.610","Text":"I will just use them to convert to Cartesian,"},{"Start":"01:58.610 ","End":"02:00.920","Text":"but they both give me the same point and of course,"},{"Start":"02:00.920 ","End":"02:03.230","Text":"it has to be the origin."},{"Start":"02:03.230 ","End":"02:06.020","Text":"But we have a point, now we need the slope,"},{"Start":"02:06.020 ","End":"02:11.650","Text":"so that\u0027s what we need, dy over dx, and there\u0027s a formula."},{"Start":"02:11.650 ","End":"02:14.644","Text":"This is the formula we\u0027ve been using."},{"Start":"02:14.644 ","End":"02:21.545","Text":"What we need is dr over d Theta, for the numerator and for the denominator. Let\u0027s see,"},{"Start":"02:21.545 ","End":"02:26.465","Text":"dr by d Theta is equal to, from this,"},{"Start":"02:26.465 ","End":"02:33.425","Text":"just minus 2 cosine of Theta."},{"Start":"02:33.425 ","End":"02:36.530","Text":"Now, if I plug this in here,"},{"Start":"02:36.530 ","End":"02:41.760","Text":"what we get is that dy by dx is equal"},{"Start":"02:41.760 ","End":"02:47.490","Text":"to dr over d Theta minus 2 cosine Theta."},{"Start":"02:47.490 ","End":"02:53.640","Text":"Here, we have sine Theta plus r,"},{"Start":"02:53.640 ","End":"03:02.700","Text":"which is 1 minus 2 sine Theta times cosine Theta."},{"Start":"03:02.700 ","End":"03:11.255","Text":"All this over dr over d Theta again, minus 2 cosine Theta,"},{"Start":"03:11.255 ","End":"03:14.910","Text":"this time cosine Theta."},{"Start":"03:15.890 ","End":"03:19.350","Text":"No, not the same thing, there\u0027s a minus."},{"Start":"03:19.350 ","End":"03:22.800","Text":"There\u0027s 1 minus 2 sine Theta,"},{"Start":"03:22.800 ","End":"03:26.520","Text":"and this time, it\u0027s a sine Theta."},{"Start":"03:26.520 ","End":"03:29.450","Text":"Let\u0027s see what we can do with this."},{"Start":"03:29.450 ","End":"03:33.695","Text":"We have here, cosine Theta from here."},{"Start":"03:33.695 ","End":"03:37.415","Text":"The rest is minus 2 sine Theta cosine Theta, and the same again,"},{"Start":"03:37.415 ","End":"03:39.475","Text":"so it\u0027s minus 4,"},{"Start":"03:39.475 ","End":"03:44.345","Text":"I\u0027ll write it as sine Theta, cosine Theta."},{"Start":"03:44.345 ","End":"03:55.140","Text":"On the denominator, we\u0027re going to get minus 1 and the sine Theta,"},{"Start":"03:55.140 ","End":"03:59.730","Text":"so I\u0027ve got minus sine Theta,"},{"Start":"03:59.730 ","End":"04:09.405","Text":"and then, I have plus 2 sine squared Theta minus 2 cosine squared Theta."},{"Start":"04:09.405 ","End":"04:13.870","Text":"We can slightly simplify"},{"Start":"04:14.020 ","End":"04:20.765","Text":"the numerator using the formula that twice sine Theta cosine Theta is sine of 2 Theta."},{"Start":"04:20.765 ","End":"04:26.995","Text":"So we\u0027ve got cosine Theta minus 2 sine of 2 Theta."},{"Start":"04:26.995 ","End":"04:35.595","Text":"Here, I could add 2 sine squared Theta and subtract 2 sine squared Theta,"},{"Start":"04:35.595 ","End":"04:43.125","Text":"and using trigonometric identities, I\u0027d get 4 sine squared Theta,"},{"Start":"04:43.125 ","End":"04:49.264","Text":"then I\u0027ll get minus 2 minus sine Theta."},{"Start":"04:49.264 ","End":"04:52.295","Text":"Now, I have 2 candidate points,"},{"Start":"04:52.295 ","End":"04:54.260","Text":"let me plug in, in each of them."},{"Start":"04:54.260 ","End":"05:00.260","Text":"Let\u0027s try plugging Theta equals Pi over 6 first,"},{"Start":"05:00.260 ","End":"05:02.270","Text":"and then we\u0027ll do the other 1,"},{"Start":"05:02.270 ","End":"05:07.145","Text":"Theta equals 5 Pi over 6."},{"Start":"05:07.145 ","End":"05:09.230","Text":"I\u0027ll just give you the answer."},{"Start":"05:09.230 ","End":"05:12.470","Text":"This comes out to be root 3 over 3."},{"Start":"05:12.470 ","End":"05:17.300","Text":"This comes out to be minus root 3 over 3."},{"Start":"05:17.300 ","End":"05:26.104","Text":"Now, a line that goes through the origin is just y equals the slope times x,"},{"Start":"05:26.104 ","End":"05:28.020","Text":"there\u0027s no constant part,"},{"Start":"05:28.020 ","End":"05:35.280","Text":"so here, we get y equals root 3 over 3x,"},{"Start":"05:35.280 ","End":"05:37.370","Text":"and here, we get the other tangent,"},{"Start":"05:37.370 ","End":"05:42.170","Text":"y equals minus root 3 over 3x."},{"Start":"05:42.170 ","End":"05:47.260","Text":"Those are the 2 tangents, and we\u0027re done."}],"Thumbnail":null,"ID":9950}],"ID":8897},{"Name":"Arc Length in Polar Coordinates","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Arc Length with Polar Coordinates","Duration":"14m 55s","ChapterTopicVideoID":10258,"CourseChapterTopicPlaylistID":8898,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.380","Text":"Okay, so we\u0027ve finished area with polar coordinates now on to the next topic."},{"Start":"00:05.660 ","End":"00:14.280","Text":"This will be arc length also with polar coordinates and here\u0027s the setup."},{"Start":"00:14.280 ","End":"00:20.710","Text":"We assume that r is given as some function of theta."},{"Start":"00:20.810 ","End":"00:23.595","Text":"This is the curve,"},{"Start":"00:23.595 ","End":"00:32.715","Text":"or a portion of a curve given by theta between alpha and beta."},{"Start":"00:32.715 ","End":"00:42.389","Text":"We\u0027ll also going to assume that the curve is traced out only once."},{"Start":"00:42.389 ","End":"00:47.420","Text":"Because we\u0027ve seen cases where, in parametric,"},{"Start":"00:47.420 ","End":"00:49.370","Text":"when we travel from one end to the other,"},{"Start":"00:49.370 ","End":"00:54.185","Text":"that the curve might be traced either several times where we might go back and forth."},{"Start":"00:54.185 ","End":"00:57.950","Text":"We assume that it\u0027s traced just once and in one direction."},{"Start":"00:57.950 ","End":"01:04.010","Text":"Usually it\u0027s fairly clear from the given that this will be so just by the way,"},{"Start":"01:04.010 ","End":"01:08.070","Text":"arc length is sometimes called curve length."},{"Start":"01:08.070 ","End":"01:15.690","Text":"Both are equivalent, so if I want to give the curve length a letter,"},{"Start":"01:15.690 ","End":"01:22.510","Text":"let me call it L. Curve length L. I\u0027ll give you the formula for this."},{"Start":"01:22.510 ","End":"01:31.635","Text":"What this is equal to is the integral from alpha to beta of"},{"Start":"01:31.635 ","End":"01:36.330","Text":"the square root of r squared"},{"Start":"01:36.330 ","End":"01:44.905","Text":"plus dr over d theta squared d theta."},{"Start":"01:44.905 ","End":"01:48.110","Text":"Of course, since r is f of theta,"},{"Start":"01:48.110 ","End":"01:53.140","Text":"we could also write this as the integral,"},{"Start":"01:53.140 ","End":"01:59.120","Text":"if we want to an alternative form of we know this function f of"},{"Start":"01:59.120 ","End":"02:06.779","Text":"theta squared plus f prime of theta,"},{"Start":"02:06.779 ","End":"02:11.460","Text":"because that\u0027s what dr d theta is, squared d theta."},{"Start":"02:11.460 ","End":"02:13.960","Text":"Either one of these forms."},{"Start":"02:13.960 ","End":"02:17.210","Text":"Let\u0027s start with an example."},{"Start":"02:17.210 ","End":"02:21.170","Text":"My question is to find the formula for"},{"Start":"02:21.170 ","End":"02:25.465","Text":"the circumference of a circle in terms of the radius."},{"Start":"02:25.465 ","End":"02:27.610","Text":"Here I wrote it out."},{"Start":"02:27.610 ","End":"02:30.925","Text":"I used the capital R because that r is already taken."},{"Start":"02:30.925 ","End":"02:33.800","Text":"r and theta are already reserved."},{"Start":"02:34.530 ","End":"02:43.195","Text":"Let\u0027s see what is the equation of a circle with a certain radius in polar coordinates,"},{"Start":"02:43.195 ","End":"02:50.450","Text":"the equation of a circle is just r equals whatever it is radius,"},{"Start":"02:51.080 ","End":"02:54.465","Text":"theta doesn\u0027t appear here at all,"},{"Start":"02:54.465 ","End":"02:59.460","Text":"so f of theta is just this constant R,"},{"Start":"02:59.460 ","End":"03:06.910","Text":"and so dr over d theta equals 0."},{"Start":"03:06.920 ","End":"03:10.320","Text":"When we trace out the circle,"},{"Start":"03:10.320 ","End":"03:15.304","Text":"we have 0 less than or equal to theta."},{"Start":"03:15.304 ","End":"03:18.305","Text":"We go around the whole way to 2 pi."},{"Start":"03:18.305 ","End":"03:23.600","Text":"According to the formula to this one, this form,"},{"Start":"03:23.600 ","End":"03:27.540","Text":"I\u0027ve got that the length of curve"},{"Start":"03:28.150 ","End":"03:35.605","Text":"is equal to the integral from 0 to 2 pi,"},{"Start":"03:35.605 ","End":"03:39.345","Text":"the square root of R squared."},{"Start":"03:39.345 ","End":"03:42.120","Text":"The variable is R squared,"},{"Start":"03:42.120 ","End":"03:49.600","Text":"the constant plus 0 squared d theta."},{"Start":"03:49.630 ","End":"03:57.020","Text":"This square root of R squared is just equal to R because R is obviously positive."},{"Start":"03:57.020 ","End":"04:01.020","Text":"I\u0027m assuming that we\u0027ve got a positive radius."},{"Start":"04:01.880 ","End":"04:09.650","Text":"This is just equal to the integral from 0 to 2 Pi of R d theta."},{"Start":"04:09.650 ","End":"04:15.405","Text":"The integral of a constant is just R times theta."},{"Start":"04:15.405 ","End":"04:19.895","Text":"But we take it from 0 to 2 pi,"},{"Start":"04:19.895 ","End":"04:25.340","Text":"just to emphasize it\u0027s theta equals 0 to theta equals 2 Pi."},{"Start":"04:25.340 ","End":"04:28.310","Text":"If we substitute theta equals 2 pi,"},{"Start":"04:28.310 ","End":"04:30.655","Text":"we get 2 pi R,"},{"Start":"04:30.655 ","End":"04:32.700","Text":"theta equals 0 gives us nothing,"},{"Start":"04:32.700 ","End":"04:36.385","Text":"so minus 0, which is just 2 pi R,"},{"Start":"04:36.385 ","End":"04:41.165","Text":"which is the well-known formula for the circumference of a circle."},{"Start":"04:41.165 ","End":"04:44.225","Text":"It\u0027s also equal to Pi times the diameter."},{"Start":"04:44.225 ","End":"04:46.715","Text":"If I let 2R equals,"},{"Start":"04:46.715 ","End":"04:50.360","Text":"I could say that this equals Pi times the diameter,"},{"Start":"04:50.360 ","End":"04:53.550","Text":"where the diameter is twice the radius,"},{"Start":"04:53.550 ","End":"04:56.195","Text":"and that\u0027s actually Pi is defined."},{"Start":"04:56.195 ","End":"04:58.235","Text":"We have verified this."},{"Start":"04:58.235 ","End":"05:00.019","Text":"That\u0027s a trivial example."},{"Start":"05:00.019 ","End":"05:02.225","Text":"Let\u0027s take a better example."},{"Start":"05:02.225 ","End":"05:08.600","Text":"Okay, I erase the old example this time we also have to find the length of a curve."},{"Start":"05:08.600 ","End":"05:11.780","Text":"This time I\u0027ll also have a simple formula."},{"Start":"05:11.780 ","End":"05:19.380","Text":"Let\u0027s just take r equals theta and theta will go"},{"Start":"05:19.380 ","End":"05:26.850","Text":"between 0 and 1 and that\u0027s one radian of 57 degrees."},{"Start":"05:26.850 ","End":"05:33.210","Text":"A simple example because this formula often gives unpleasant integrals."},{"Start":"05:33.210 ","End":"05:37.425","Text":"I wanted it to come out simple. Let\u0027s see."},{"Start":"05:37.425 ","End":"05:41.670","Text":"Let\u0027s use this formula here."},{"Start":"05:41.670 ","End":"05:50.289","Text":"We get that L is equal to the integral from 0 to 1"},{"Start":"05:50.289 ","End":"06:00.300","Text":"of the square root of r squared,"},{"Start":"06:00.300 ","End":"06:04.710","Text":"which is theta squared."},{"Start":"06:04.930 ","End":"06:12.150","Text":"Now, dr over d theta is just equal to 1,"},{"Start":"06:12.150 ","End":"06:17.640","Text":"so it\u0027s theta squared plus 1 d theta."},{"Start":"06:17.640 ","End":"06:24.335","Text":"You might ask, how do we know this curve is traced out only once?"},{"Start":"06:24.335 ","End":"06:26.775","Text":"I\u0027ll show you a sketch of it."},{"Start":"06:26.775 ","End":"06:29.870","Text":"Although, there\u0027s a more formal way to do this."},{"Start":"06:29.870 ","End":"06:32.375","Text":"I don\u0027t want to get bogged down with these details."},{"Start":"06:32.375 ","End":"06:34.770","Text":"Here\u0027s the picture."},{"Start":"06:35.230 ","End":"06:40.100","Text":"What happens is because r equals theta as the angle grows,"},{"Start":"06:40.100 ","End":"06:41.210","Text":"so does the radius,"},{"Start":"06:41.210 ","End":"06:42.470","Text":"the distance from the origin,"},{"Start":"06:42.470 ","End":"06:44.495","Text":"and so we get a spiral."},{"Start":"06:44.495 ","End":"06:51.585","Text":"We actually only want a very small portion of the spiral between 0 and 1,"},{"Start":"06:51.585 ","End":"06:57.720","Text":"theta equals 1 would be like 57 point something degrees,"},{"Start":"06:57.720 ","End":"07:00.545","Text":"it\u0027s 1 radian is just part of it here."},{"Start":"07:00.545 ","End":"07:04.260","Text":"Anyway, lets get to it,"},{"Start":"07:05.810 ","End":"07:08.645","Text":"how do we do this kind of integral?"},{"Start":"07:08.645 ","End":"07:12.440","Text":"This needs a trigonometric substitution just from experience."},{"Start":"07:12.440 ","End":"07:20.950","Text":"What we\u0027re going to do, we\u0027ll substitute theta equals tangent x,"},{"Start":"07:20.950 ","End":"07:31.690","Text":"and then the derivative or d theta will equal the derivative of this dx."},{"Start":"07:31.690 ","End":"07:38.890","Text":"The derivative of tangent x is 1 over cosine squared x."},{"Start":"07:38.890 ","End":"07:41.845","Text":"This is sometimes written as secant squared x."},{"Start":"07:41.845 ","End":"07:45.310","Text":"Not sure if you\u0027ve studied secant and now you will have studied cosine."},{"Start":"07:45.310 ","End":"07:47.530","Text":"I\u0027ll leave it this way, and times dx,"},{"Start":"07:47.530 ","End":"07:51.635","Text":"I just put the dx in the numerator."},{"Start":"07:51.635 ","End":"07:58.000","Text":"That\u0027s one thing now we also have to substitute the limits."},{"Start":"07:58.610 ","End":"08:03.045","Text":"You could say that x is the arctangent of Theta,"},{"Start":"08:03.045 ","End":"08:13.935","Text":"where Theta equals 0 corresponds to x equals 0 and Theta equals 1,"},{"Start":"08:13.935 ","End":"08:17.040","Text":"if I know that tangent of something is 1,"},{"Start":"08:17.040 ","End":"08:20.190","Text":"that something is 45 degrees,"},{"Start":"08:20.190 ","End":"08:21.885","Text":"but we have to use radians,"},{"Start":"08:21.885 ","End":"08:23.655","Text":"it\u0027s Pi over 4."},{"Start":"08:23.655 ","End":"08:25.695","Text":"Or you could look on your calculator,"},{"Start":"08:25.695 ","End":"08:28.980","Text":"arctangent of 1 comes up Pi over 4."},{"Start":"08:28.980 ","End":"08:31.900","Text":"Remember to set it to radians."},{"Start":"08:34.730 ","End":"08:43.815","Text":"We substitute in here and we\u0027ll get the integral from 0 to 1 of"},{"Start":"08:43.815 ","End":"08:53.370","Text":"the square root of tangent squared x plus 1."},{"Start":"08:53.370 ","End":"08:55.530","Text":"It\u0027s not from 0 to 1, sorry,"},{"Start":"08:55.530 ","End":"09:01.544","Text":"it\u0027s from 0 to Pi over 4 because we\u0027re now in the world of x."},{"Start":"09:01.544 ","End":"09:07.365","Text":"0 to Pi over 4, and this would be dx."},{"Start":"09:07.365 ","End":"09:12.190","Text":"Let me do a little computation at the side as to what this is."},{"Start":"09:15.260 ","End":"09:22.030","Text":"I\u0027ll do a Take 2. Take 2 on the last bit."},{"Start":"09:22.580 ","End":"09:27.660","Text":"We get the integral not from 0 to 1."},{"Start":"09:27.660 ","End":"09:32.590","Text":"I\u0027m going to put everything in terms of x from 0 to Pi over 4."},{"Start":"09:32.960 ","End":"09:40.425","Text":"The square root of tangent squared x plus 1."},{"Start":"09:40.425 ","End":"09:44.910","Text":"D Theta is dx"},{"Start":"09:44.910 ","End":"09:53.220","Text":"over cosine squared x cosine squared x."},{"Start":"09:53.220 ","End":"09:56.865","Text":"Let me do this computation at the side."},{"Start":"09:56.865 ","End":"10:02.760","Text":"This becomes the square root of tangent squared x"},{"Start":"10:02.760 ","End":"10:09.705","Text":"is sine squared x over cosine squared x."},{"Start":"10:09.705 ","End":"10:13.590","Text":"Because tangent is sine over cosine and 1,"},{"Start":"10:13.590 ","End":"10:22.890","Text":"I can write to give it a common denominator as cosine squared x over cosine squared x,"},{"Start":"10:22.890 ","End":"10:24.150","Text":"that\u0027s what 1 equals,"},{"Start":"10:24.150 ","End":"10:26.940","Text":"I\u0027m making it with the same denominator."},{"Start":"10:26.940 ","End":"10:29.830","Text":"Let\u0027s extend the square root sign."},{"Start":"10:30.890 ","End":"10:33.060","Text":"This is equal to,"},{"Start":"10:33.060 ","End":"10:36.015","Text":"if I put it on a common denominator,"},{"Start":"10:36.015 ","End":"10:42.270","Text":"I get the square root of sine squared plus"},{"Start":"10:42.270 ","End":"10:49.665","Text":"cosine squared is 1 over cosine squared x."},{"Start":"10:49.665 ","End":"10:56.520","Text":"But cosine is positive everywhere from 0 to pi over 2,"},{"Start":"10:56.520 ","End":"11:02.400","Text":"that\u0027s 0 -1.57 something."},{"Start":"11:02.400 ","End":"11:04.965","Text":"It\u0027s certainly positive from 0 to 1."},{"Start":"11:04.965 ","End":"11:09.360","Text":"The square root of something squared is the absolute value."},{"Start":"11:09.360 ","End":"11:12.990","Text":"Just say here that thing is equal to this,"},{"Start":"11:12.990 ","End":"11:20.770","Text":"equals absolute value of"},{"Start":"11:20.840 ","End":"11:26.835","Text":"1 over cosine of x."},{"Start":"11:26.835 ","End":"11:28.589","Text":"But the absolute value,"},{"Start":"11:28.589 ","End":"11:30.375","Text":"because cosine x is positive,"},{"Start":"11:30.375 ","End":"11:33.420","Text":"I don\u0027t need the absolute value."},{"Start":"11:33.420 ","End":"11:38.100","Text":"I erase the bars and I move this aside."},{"Start":"11:38.100 ","End":"11:47.745","Text":"Back here, what we get is the integral from 0 to Pi over 4."},{"Start":"11:47.745 ","End":"11:50.010","Text":"This is 1 over cosine x."},{"Start":"11:50.010 ","End":"11:52.575","Text":"Here we have cosine squared x."},{"Start":"11:52.575 ","End":"11:59.070","Text":"We have dx over cosine cubed x."},{"Start":"11:59.070 ","End":"12:04.875","Text":"Now, we have to look this up in a table of integrals."},{"Start":"12:04.875 ","End":"12:07.185","Text":"It could be done. It\u0027s lengthy."},{"Start":"12:07.185 ","End":"12:12.700","Text":"The easiest is look it up in a table of integrals."},{"Start":"12:13.570 ","End":"12:16.805","Text":"You might find it not in this form,"},{"Start":"12:16.805 ","End":"12:21.200","Text":"but you might find it as instead of 1 over cosine,"},{"Start":"12:21.200 ","End":"12:22.430","Text":"we can put secant."},{"Start":"12:22.430 ","End":"12:32.610","Text":"You might look for secant cubed of x instead of 1 over cosine cubed of x. I looked"},{"Start":"12:32.610 ","End":"12:38.985","Text":"it up and it says that the integral of this is"},{"Start":"12:38.985 ","End":"12:46.545","Text":"secant x tangent x plus natural log"},{"Start":"12:46.545 ","End":"12:54.645","Text":"of absolute value of secant x plus tangent x plus a constant,"},{"Start":"12:54.645 ","End":"13:00.130","Text":"which we don\u0027t really need because we\u0027re going to put it back in here."},{"Start":"13:02.540 ","End":"13:05.400","Text":"I\u0027ll go back to the 1 over cosine,"},{"Start":"13:05.400 ","End":"13:06.915","Text":"which is what we\u0027re used to."},{"Start":"13:06.915 ","End":"13:17.060","Text":"We have tangent x over cosine x plus natural log"},{"Start":"13:17.060 ","End":"13:19.580","Text":"of 1 over"},{"Start":"13:19.580 ","End":"13:28.710","Text":"cosine x plus tangent x."},{"Start":"13:28.710 ","End":"13:30.510","Text":"We don\u0027t need the constant."},{"Start":"13:30.510 ","End":"13:38.910","Text":"All this we have to take between 0 and pi over 4."},{"Start":"13:38.910 ","End":"13:40.980","Text":"Let\u0027s see what we get."},{"Start":"13:40.980 ","End":"13:44.250","Text":"When x is Pi over 4,"},{"Start":"13:44.250 ","End":"13:48.315","Text":"the tangent of pi over 4 is 1 it\u0027s like 45 degrees."},{"Start":"13:48.315 ","End":"13:51.120","Text":"Cosine is 1 over square root of 2."},{"Start":"13:51.120 ","End":"13:54.810","Text":"This just comes out to be square root of 2."},{"Start":"13:54.810 ","End":"13:58.395","Text":"Tangent of Pi over 4 is still 1,"},{"Start":"13:58.395 ","End":"14:05.115","Text":"1 over the square root of 2 is also square root of 2."},{"Start":"14:05.115 ","End":"14:12.210","Text":"We have natural log of 1 plus square root of 2."},{"Start":"14:12.210 ","End":"14:13.710","Text":"Or is it the other way round?"},{"Start":"14:13.710 ","End":"14:15.585","Text":"It doesn\u0027t matter."},{"Start":"14:15.585 ","End":"14:17.160","Text":"Because this is positive,"},{"Start":"14:17.160 ","End":"14:19.770","Text":"I don\u0027t need absolute value."},{"Start":"14:19.770 ","End":"14:24.210","Text":"I claim that this is all because when we put in 0, we\u0027re going to get 0."},{"Start":"14:24.210 ","End":"14:30.030","Text":"Look, tangent of 0 is 0 and cosine of 0 is not 0,"},{"Start":"14:30.030 ","End":"14:32.895","Text":"so 0 over non-zero is 0."},{"Start":"14:32.895 ","End":"14:37.365","Text":"What do we get in here? This is 0, this is 1."},{"Start":"14:37.365 ","End":"14:42.764","Text":"Altogether 1 over 1 plus 0 is 1 natural log of 1 is 0."},{"Start":"14:42.764 ","End":"14:45.570","Text":"When we put in 0, we get 0."},{"Start":"14:45.570 ","End":"14:48.690","Text":"This is the answer."},{"Start":"14:48.690 ","End":"14:53.100","Text":"We are done. I\u0027m not going to give a numerical evaluation."},{"Start":"14:53.100 ","End":"14:55.270","Text":"I will leave it at that."}],"Thumbnail":null,"ID":10592},{"Watched":false,"Name":"Exercise 11","Duration":"11m 13s","ChapterTopicVideoID":10058,"CourseChapterTopicPlaylistID":8898,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.885","Text":"In this exercise, we\u0027re given a curve which is a cardioid,"},{"Start":"00:06.885 ","End":"00:10.465","Text":"r equals 1 plus cosine Theta."},{"Start":"00:10.465 ","End":"00:14.160","Text":"Here\u0027s a sketch though we don\u0027t really need it."},{"Start":"00:14.160 ","End":"00:20.480","Text":"By the way, cardioid of course you see the word from card,"},{"Start":"00:20.480 ","End":"00:24.180","Text":"meaning the root of heart and its heart-shaped,"},{"Start":"00:24.180 ","End":"00:25.665","Text":"a heart on its side."},{"Start":"00:25.665 ","End":"00:28.020","Text":"Anyway, that\u0027s neither here nor there."},{"Start":"00:28.020 ","End":"00:30.510","Text":"We want to find the circumference,"},{"Start":"00:30.510 ","End":"00:33.270","Text":"which is a question of arc length."},{"Start":"00:33.270 ","End":"00:38.034","Text":"Now the formula for arc length is going to work provided that we trace"},{"Start":"00:38.034 ","End":"00:43.325","Text":"the curve out exactly once and counterclockwise."},{"Start":"00:43.325 ","End":"00:48.415","Text":"In this case, you can easily see that Theta,"},{"Start":"00:48.415 ","End":"00:51.125","Text":"if we start from 0, which is here,"},{"Start":"00:51.125 ","End":"01:01.250","Text":"and go counterclockwise, we start at Theta equals 0 and we end at Theta equals 2Pi."},{"Start":"01:01.250 ","End":"01:03.270","Text":"As we go from 0-2 Pi,"},{"Start":"01:03.270 ","End":"01:06.540","Text":"we trace the cardioid out once."},{"Start":"01:06.540 ","End":"01:08.960","Text":"It doesn\u0027t happen with all closed curves,"},{"Start":"01:08.960 ","End":"01:11.405","Text":"it could be the Omega from 0-2 Pi,"},{"Start":"01:11.405 ","End":"01:14.990","Text":"we go around several times or less than 1s."},{"Start":"01:15.620 ","End":"01:18.270","Text":"I\u0027ll just remark that in this case,"},{"Start":"01:18.270 ","End":"01:23.940","Text":"0 less than or equal to Theta less than or equal to 2Pi."},{"Start":"01:23.940 ","End":"01:27.815","Text":"From 0-2 Pi, we go around once counterclockwise,"},{"Start":"01:27.815 ","End":"01:30.875","Text":"which is the positive mathematical direction."},{"Start":"01:30.875 ","End":"01:33.110","Text":"Now let\u0027s use the formula."},{"Start":"01:33.110 ","End":"01:36.305","Text":"The formula for the length of curve,"},{"Start":"01:36.305 ","End":"01:38.315","Text":"let\u0027s call it L,"},{"Start":"01:38.315 ","End":"01:43.100","Text":"is the integral from the lower limit,"},{"Start":"01:43.100 ","End":"01:50.030","Text":"in this case,0 to the upper limit on Theta of"},{"Start":"01:50.030 ","End":"01:56.210","Text":"the square root of r squared plus"},{"Start":"01:56.210 ","End":"02:03.365","Text":"the derivative of r with respect to Theta also squared d Theta."},{"Start":"02:03.365 ","End":"02:07.025","Text":"Now we have the expression of r in terms of Theta,"},{"Start":"02:07.025 ","End":"02:11.630","Text":"what we don\u0027t have is our prime or dr by"},{"Start":"02:11.630 ","End":"02:17.810","Text":"d Theta is just a simple differentiation minus sine Theta."},{"Start":"02:17.810 ","End":"02:21.670","Text":"Now plug this and this in here,"},{"Start":"02:21.670 ","End":"02:26.835","Text":"and we get the integral from 0-2 Pi of"},{"Start":"02:26.835 ","End":"02:33.320","Text":"the square root r squared is 1 plus cosine Theta squared."},{"Start":"02:33.320 ","End":"02:35.360","Text":"I\u0027ll just square it already."},{"Start":"02:35.360 ","End":"02:42.645","Text":"It\u0027s 1 plus 2 cosine Theta plus cosine squared Theta,"},{"Start":"02:42.645 ","End":"02:44.850","Text":"that\u0027s the r squared part,"},{"Start":"02:44.850 ","End":"02:46.920","Text":"dr over d Theta,"},{"Start":"02:46.920 ","End":"02:48.510","Text":"when I square it,"},{"Start":"02:48.510 ","End":"02:52.350","Text":"it\u0027s going to be plus sine squared"},{"Start":"02:52.350 ","End":"03:01.300","Text":"Theta plus sine squared Theta, d Theta."},{"Start":"03:01.870 ","End":"03:05.330","Text":"Let\u0027s see if we can evaluate this integral."},{"Start":"03:05.330 ","End":"03:08.200","Text":"Usually, arc length gives"},{"Start":"03:08.200 ","End":"03:13.055","Text":"some nasty integrals and you\u0027ll see that in probably the next exercise."},{"Start":"03:13.055 ","End":"03:14.990","Text":"But here I chose an example,"},{"Start":"03:14.990 ","End":"03:17.855","Text":"we can actually evaluate the integral."},{"Start":"03:17.855 ","End":"03:24.345","Text":"Cosine squared plus sine squared is equal to 1."},{"Start":"03:24.345 ","End":"03:30.440","Text":"We get the integral 0-2 Pi of the square root."},{"Start":"03:30.440 ","End":"03:35.060","Text":"This plus this is 1 together with this 1 is 2, so it\u0027s twice,"},{"Start":"03:35.060 ","End":"03:36.770","Text":"let me take a 2 out the brackets,"},{"Start":"03:36.770 ","End":"03:43.175","Text":"1 plus cosine d Theta."},{"Start":"03:43.175 ","End":"03:48.180","Text":"I\u0027m going to use a trigonometric identity to help us with this."},{"Start":"03:56.690 ","End":"04:01.955","Text":"There is an identity that cosine squared of Alpha"},{"Start":"04:01.955 ","End":"04:08.680","Text":"is 1.5 of 1 plus cosine 2 Alpha."},{"Start":"04:08.680 ","End":"04:10.290","Text":"Now in my case,"},{"Start":"04:10.290 ","End":"04:12.955","Text":"I see I have 1 plus cosine Theta."},{"Start":"04:12.955 ","End":"04:16.280","Text":"There\u0027s also another formula which is less well-known,"},{"Start":"04:16.280 ","End":"04:22.935","Text":"but I just replace Alpha with Theta over 2,"},{"Start":"04:22.935 ","End":"04:28.520","Text":"so I\u0027ve got cosine squared of Theta over 2 is"},{"Start":"04:28.520 ","End":"04:34.410","Text":"equal to 1.5 of 1 plus cosine Theta."},{"Start":"04:34.410 ","End":"04:39.680","Text":"This is the 1 that\u0027s going to help us because we have here 1 plus cosine Theta."},{"Start":"04:39.680 ","End":"04:42.005","Text":"If you read this from right to left,"},{"Start":"04:42.005 ","End":"04:50.465","Text":"what we got here is the integral from 0-2 Pi."},{"Start":"04:50.465 ","End":"04:58.895","Text":"Now, I can put a 2 in the denominator here and make this 2 into a 4."},{"Start":"04:58.895 ","End":"05:05.870","Text":"What I have is the square root of 4 times 1 plus cosine Theta over 2,"},{"Start":"05:05.870 ","End":"05:13.785","Text":"which is cosine squared Theta over 2 d Theta."},{"Start":"05:13.785 ","End":"05:21.940","Text":"Now it gets a little bit tricky because remember that the,"},{"Start":"05:23.150 ","End":"05:32.074","Text":"the square root of a squared in general is not a as you might think,"},{"Start":"05:32.074 ","End":"05:34.700","Text":"but absolute value of a."},{"Start":"05:34.700 ","End":"05:37.430","Text":"Because if a happens to be negative,"},{"Start":"05:37.430 ","End":"05:40.625","Text":"the square root of a squared makes it positive."},{"Start":"05:40.625 ","End":"05:46.190","Text":"This integral is actually equal to,"},{"Start":"05:46.190 ","End":"05:47.780","Text":"and this is not the root I\u0027m going to take,"},{"Start":"05:47.780 ","End":"05:49.850","Text":"but I want to just write it out."},{"Start":"05:49.850 ","End":"05:56.440","Text":"Integral from 0-2 Pi of, well the square root of 4,"},{"Start":"05:56.440 ","End":"05:58.430","Text":"I can take it out in front,"},{"Start":"05:58.430 ","End":"06:06.465","Text":"but I would say absolute value of cosine of Theta over 2 d Theta."},{"Start":"06:06.465 ","End":"06:09.870","Text":"This absolute value is a bit nuisance some"},{"Start":"06:09.870 ","End":"06:15.025","Text":"because I have to separate the positive from the negative."},{"Start":"06:15.025 ","End":"06:17.965","Text":"I get a bit more space here."},{"Start":"06:17.965 ","End":"06:21.530","Text":"If I was going to go this route and it\u0027s possible,"},{"Start":"06:21.530 ","End":"06:24.350","Text":"then I would split this integral up."},{"Start":"06:24.350 ","End":"06:31.624","Text":"I would say that the integral from 0-2 Pi is the integral from 0."},{"Start":"06:31.624 ","End":"06:33.365","Text":"Well, let\u0027s see to what."},{"Start":"06:33.365 ","End":"06:38.910","Text":"When Theta goes from 0 to Pi."},{"Start":"06:39.550 ","End":"06:47.690","Text":"I\u0027m claiming that Pi is the point where it\u0027s switched with the things change sign."},{"Start":"06:47.690 ","End":"06:50.135","Text":"When Theta goes from 0 to Pi,"},{"Start":"06:50.135 ","End":"06:54.870","Text":"Theta over 2 goes from 0 to Pi over 2,"},{"Start":"06:54.870 ","End":"06:56.999","Text":"are in first quadrant."},{"Start":"06:56.999 ","End":"06:59.869","Text":"In first quadrant the cosine is positive."},{"Start":"06:59.869 ","End":"07:02.135","Text":"Then from Pi to 2Pi,"},{"Start":"07:02.135 ","End":"07:05.810","Text":"Theta over 2 goes from Pi over 2 to Pi,"},{"Start":"07:05.810 ","End":"07:08.690","Text":"which means second quadrant where it\u0027s negative."},{"Start":"07:08.690 ","End":"07:11.595","Text":"I could break it up into 2 bits,"},{"Start":"07:11.595 ","End":"07:13.485","Text":"then Pi to 2Pi."},{"Start":"07:13.485 ","End":"07:16.620","Text":"Then here I\u0027d write cosine Theta over 2 here,"},{"Start":"07:16.620 ","End":"07:21.725","Text":"I\u0027d write minus cosine Theta over 2 and throw out the absolute value."},{"Start":"07:21.725 ","End":"07:23.585","Text":"That\u0027s 1 way to go."},{"Start":"07:23.585 ","End":"07:28.270","Text":"But there\u0027s a little trick that saves me having to split this up."},{"Start":"07:28.270 ","End":"07:30.915","Text":"Now I\u0027m going to show you this trick,"},{"Start":"07:30.915 ","End":"07:32.520","Text":"what I\u0027m going to do."},{"Start":"07:32.520 ","End":"07:34.710","Text":"We said that we want"},{"Start":"07:34.710 ","End":"07:40.515","Text":"1 complete revolution or"},{"Start":"07:40.515 ","End":"07:44.980","Text":"to traverse the curve once in counterclockwise."},{"Start":"07:44.980 ","End":"07:47.975","Text":"Now, we didn\u0027t have to start at this point,"},{"Start":"07:47.975 ","End":"07:50.195","Text":"we could have started at another point."},{"Start":"07:50.195 ","End":"07:54.440","Text":"I\u0027m claiming that if we don\u0027t do Theta from 0-2 Pi,"},{"Start":"07:54.440 ","End":"07:58.595","Text":"but if we start here and go around and end here,"},{"Start":"07:58.595 ","End":"08:02.434","Text":"we could avoid this problem."},{"Start":"08:02.434 ","End":"08:03.620","Text":"Let\u0027s say this point,"},{"Start":"08:03.620 ","End":"08:04.880","Text":"instead of calling it Pi,"},{"Start":"08:04.880 ","End":"08:06.920","Text":"we can call it minus Pi,"},{"Start":"08:06.920 ","End":"08:09.140","Text":"every angle and several representations."},{"Start":"08:09.140 ","End":"08:12.290","Text":"If we say that this is minus Pi,"},{"Start":"08:12.290 ","End":"08:18.960","Text":"we start from here and then go all the way up to, then here at 0."},{"Start":"08:19.160 ","End":"08:23.735","Text":"Then we continue, and then we get to plus Pi."},{"Start":"08:23.735 ","End":"08:25.880","Text":"If we go from minus Pi to Pi,"},{"Start":"08:25.880 ","End":"08:27.440","Text":"that should also work."},{"Start":"08:27.440 ","End":"08:31.410","Text":"Now, the advantage is,"},{"Start":"08:33.800 ","End":"08:41.550","Text":"if I take Theta between minus Pi and Pi,"},{"Start":"08:41.950 ","End":"08:51.915","Text":"then Theta over 2 is between Pi over 2 and minus Pi over 2."},{"Start":"08:51.915 ","End":"08:57.950","Text":"From minus 90 degrees to 90 degrees in the fourth and first quadrant,"},{"Start":"08:57.950 ","End":"09:00.275","Text":"the cosine is positive,"},{"Start":"09:00.275 ","End":"09:03.160","Text":"so I won\u0027t need absolute value."},{"Start":"09:03.160 ","End":"09:07.985","Text":"What I\u0027m going to do is change this to,"},{"Start":"09:07.985 ","End":"09:15.634","Text":"instead of this limit will take it from minus Pi to Pi, like I mentioned."},{"Start":"09:15.634 ","End":"09:22.535","Text":"Then this thing, I don\u0027t need it this way."},{"Start":"09:22.535 ","End":"09:29.420","Text":"I\u0027m now all right to take from minus Pi to Pi without the absolute value,"},{"Start":"09:29.420 ","End":"09:31.055","Text":"we just copy what this is,"},{"Start":"09:31.055 ","End":"09:36.480","Text":"cosine of Theta over"},{"Start":"09:36.480 ","End":"09:42.970","Text":"2 d Theta and no need for absolute value."},{"Start":"09:42.970 ","End":"09:54.310","Text":"Now the integral of cosine Theta over 2 is not quite sine Theta over 2."},{"Start":"09:54.310 ","End":"09:57.470","Text":"We need to divide by the inner derivative which is a half,"},{"Start":"09:57.470 ","End":"09:59.315","Text":"I multiply by 2."},{"Start":"09:59.315 ","End":"10:01.670","Text":"The 2 combines with the 2,"},{"Start":"10:01.670 ","End":"10:05.045","Text":"and it comes out to be 4 sine Theta over 2."},{"Start":"10:05.045 ","End":"10:07.895","Text":"If you\u0027re not sure, differentiate this,"},{"Start":"10:07.895 ","End":"10:13.115","Text":"and we get the derivative of sine Theta over 2 is cosine Theta over 2 times a 1/2,"},{"Start":"10:13.115 ","End":"10:15.740","Text":"and that gives us the 2 which is what we had originally."},{"Start":"10:15.740 ","End":"10:21.540","Text":"I need to take this now and just substitute minus Pi and Pi,"},{"Start":"10:21.540 ","End":"10:23.555","Text":"subtract this 1 from this 1,"},{"Start":"10:23.555 ","End":"10:25.490","Text":"let\u0027s see what we get."},{"Start":"10:25.490 ","End":"10:28.805","Text":"If we put Theta equals Pi,"},{"Start":"10:28.805 ","End":"10:36.870","Text":"we have sine of Pi over 2 is at 1, so that\u0027s 4."},{"Start":"10:36.870 ","End":"10:40.035","Text":"If we put in minus Pi,"},{"Start":"10:40.035 ","End":"10:46.020","Text":"we get sine of minus Pi over 2 is minus 1,"},{"Start":"10:46.020 ","End":"10:48.015","Text":"so I\u0027ve got 4 minus,"},{"Start":"10:48.015 ","End":"10:51.210","Text":"minus 4, which is 8,"},{"Start":"10:51.210 ","End":"10:55.980","Text":"and 8 is the answer,"},{"Start":"10:55.980 ","End":"10:58.185","Text":"so I\u0027ll highlight that."},{"Start":"10:58.185 ","End":"11:02.150","Text":"Now we\u0027ve got the circumference of the cardioid."},{"Start":"11:02.150 ","End":"11:06.050","Text":"I\u0027ll leave it to you if you want to pursue it with the other method,"},{"Start":"11:06.050 ","End":"11:08.720","Text":"with the splitting the integral up into 2,"},{"Start":"11:08.720 ","End":"11:10.130","Text":"you should get the same answer,"},{"Start":"11:10.130 ","End":"11:13.560","Text":"of course. Okay, we\u0027re done."}],"Thumbnail":null,"ID":9953},{"Watched":false,"Name":"Exercise 12","Duration":"4m 11s","ChapterTopicVideoID":10059,"CourseChapterTopicPlaylistID":8898,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.420","Text":"Size, we have to find the arc length of 1 petal of the rose."},{"Start":"00:06.420 ","End":"00:11.085","Text":"The shape is the called a rose and each 1 of these is called the petal."},{"Start":"00:11.085 ","End":"00:15.000","Text":"We have r equals 3 sine 2 Theta."},{"Start":"00:15.000 ","End":"00:17.820","Text":"If we get a complicated integral,"},{"Start":"00:17.820 ","End":"00:19.485","Text":"we don\u0027t have to evaluate it."},{"Start":"00:19.485 ","End":"00:23.265","Text":"These arc length integrals usually come up pretty complicated."},{"Start":"00:23.265 ","End":"00:25.590","Text":"Let me choose the petal,"},{"Start":"00:25.590 ","End":"00:30.070","Text":"this 1 here, just to highlight it."},{"Start":"00:30.170 ","End":"00:35.890","Text":"What we want to know about it is where does Theta go from and to."},{"Start":"00:36.560 ","End":"00:40.920","Text":"If we just use the origin,"},{"Start":"00:40.920 ","End":"00:44.430","Text":"we can say that when Theta is 0,"},{"Start":"00:44.430 ","End":"00:46.650","Text":"certainly we\u0027re at the origin."},{"Start":"00:46.650 ","End":"00:54.015","Text":"Question is, when is the next time that r becomes 0?"},{"Start":"00:54.015 ","End":"00:58.620","Text":"Well, that will happen when sine 2 Theta equals 0."},{"Start":"00:58.620 ","End":"01:03.520","Text":"We can get all the Thetas which return to the origin."},{"Start":"01:03.780 ","End":"01:09.730","Text":"Sine is 0 when the angle is a multiple of Pi,"},{"Start":"01:09.730 ","End":"01:12.465","Text":"so it\u0027s some k times Pi."},{"Start":"01:12.465 ","End":"01:17.220","Text":"Theta is k times Pi over 2."},{"Start":"01:17.220 ","End":"01:20.525","Text":"Every 90 degrees, every Pi over 2,"},{"Start":"01:20.525 ","End":"01:22.215","Text":"we return to the origin."},{"Start":"01:22.215 ","End":"01:27.270","Text":"For the first time 0 we got the angle 0,"},{"Start":"01:27.270 ","End":"01:29.580","Text":"and k is 1 we have Pi over 2."},{"Start":"01:29.580 ","End":"01:31.620","Text":"What I\u0027m saying is that,"},{"Start":"01:31.620 ","End":"01:34.720","Text":"to do this, we can take Theta,"},{"Start":"01:36.710 ","End":"01:42.025","Text":"my 0 looks like a Theta, and that\u0027s better,"},{"Start":"01:42.025 ","End":"01:47.570","Text":"Theta between 0 and Pi over 2 will give us this,"},{"Start":"01:47.570 ","End":"01:51.720","Text":"as it will go in the correct direction."},{"Start":"01:53.060 ","End":"02:02.480","Text":"We just have to apply the integral for the formula for arc length,"},{"Start":"02:02.480 ","End":"02:10.595","Text":"which is that L is equal to the integral from the start Theta,"},{"Start":"02:10.595 ","End":"02:13.130","Text":"which is 0 to the end Theta,"},{"Start":"02:13.130 ","End":"02:15.905","Text":"which is Pi over 2."},{"Start":"02:15.905 ","End":"02:20.405","Text":"Then in general, what we want is the square root"},{"Start":"02:20.405 ","End":"02:27.740","Text":"of r squared plus the derivative of r with respect to Theta,"},{"Start":"02:27.740 ","End":"02:31.400","Text":"also squared, d Theta."},{"Start":"02:31.400 ","End":"02:41.605","Text":"Now this is r and from here I can get the dr over d Theta is,"},{"Start":"02:41.605 ","End":"02:43.895","Text":"derivative of sine is cosine,"},{"Start":"02:43.895 ","End":"02:46.970","Text":"but because of the 2 we have to multiply by the 2,"},{"Start":"02:46.970 ","End":"02:51.635","Text":"so we get 6 cosine of 2 Theta."},{"Start":"02:51.635 ","End":"02:54.810","Text":"Now if we plug that in here,"},{"Start":"02:55.490 ","End":"03:02.580","Text":"we get the integral 0 to Pi over 2 square root."},{"Start":"03:02.580 ","End":"03:07.680","Text":"Now, r squared would be 9"},{"Start":"03:07.680 ","End":"03:16.100","Text":"sine squared 2 Theta plus this thing squared,"},{"Start":"03:16.100 ","End":"03:17.960","Text":"which is going to be"},{"Start":"03:17.960 ","End":"03:23.130","Text":"36 cosine"},{"Start":"03:23.200 ","End":"03:31.560","Text":"squared 2 Theta, d Theta."},{"Start":"03:31.560 ","End":"03:33.835","Text":"We could just leave it at that."},{"Start":"03:33.835 ","End":"03:37.525","Text":"I would like to take the 9 out of the square root,"},{"Start":"03:37.525 ","End":"03:40.390","Text":"I can take 9 out of here and here."},{"Start":"03:40.390 ","End":"03:41.950","Text":"It will come out as 3,"},{"Start":"03:41.950 ","End":"03:44.230","Text":"and I can put that in front of the integral."},{"Start":"03:44.230 ","End":"03:46.580","Text":"Be slightly simpler."},{"Start":"03:46.580 ","End":"03:53.700","Text":"We can write it as this and then the square root of sine squared"},{"Start":"03:53.700 ","End":"04:02.505","Text":"2 Theta plus 4 cosine squared 2 Theta d Theta."},{"Start":"04:02.505 ","End":"04:06.190","Text":"This would be a nasty integral to try to compute,"},{"Start":"04:06.190 ","End":"04:11.570","Text":"so we will leave it at that and we\u0027re done."}],"Thumbnail":null,"ID":9954}],"ID":8898},{"Name":"Area in Polar Coordinates","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Area with Polar Coordinates","Duration":"17m 23s","ChapterTopicVideoID":10257,"CourseChapterTopicPlaylistID":8899,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.760","Text":"In this section we\u0027re going to talk about area with polar coordinates."},{"Start":"00:05.760 ","End":"00:09.850","Text":"I\u0027ll just remind you we already studied area with"},{"Start":"00:09.850 ","End":"00:15.420","Text":"rectangular or Cartesian coordinates where we had y as a function of x,"},{"Start":"00:15.420 ","End":"00:18.120","Text":"so that\u0027s really y equals."},{"Start":"00:18.120 ","End":"00:22.515","Text":"We had a formula for the area which I called S,"},{"Start":"00:22.515 ","End":"00:27.640","Text":"where S is the integral from a to b and make it so"},{"Start":"00:27.640 ","End":"00:33.715","Text":"that it increases from a to b of ydx."},{"Start":"00:33.715 ","End":"00:42.905","Text":"Instead of y we could also put f of x or the integral from a to b of f of x, dx."},{"Start":"00:42.905 ","End":"00:47.725","Text":"In polar coordinates it comes out differently."},{"Start":"00:47.725 ","End":"00:51.260","Text":"When we take an interval for Theta,"},{"Start":"00:51.380 ","End":"01:00.120","Text":"Theta is a constant comes out to be a array or a line through the origin."},{"Start":"01:00.120 ","End":"01:02.979","Text":"If I have Alpha smaller than Beta,"},{"Start":"01:02.979 ","End":"01:08.925","Text":"then when I go from Alpha to Beta I go like this."},{"Start":"01:08.925 ","End":"01:16.500","Text":"This is the area triangular shaped S and the formula is also different."},{"Start":"01:16.500 ","End":"01:22.550","Text":"The formula in this case is that S is equal to,"},{"Start":"01:22.550 ","End":"01:26.755","Text":"instead of the integral of just f of Theta or r,"},{"Start":"01:26.755 ","End":"01:31.520","Text":"it\u0027s actually the integral of 1/2 which you put in front,"},{"Start":"01:31.520 ","End":"01:36.360","Text":"1/2r squared d Theta and of course,"},{"Start":"01:36.360 ","End":"01:37.770","Text":"if you want to put instead of r,"},{"Start":"01:37.770 ","End":"01:41.910","Text":"f of Theta, this is 1/2 of the integral a,"},{"Start":"01:41.910 ","End":"01:44.220","Text":"and it\u0027s from Alpha to Beta."},{"Start":"01:44.220 ","End":"01:51.660","Text":"We make sure that Alpha is less than Beta from Alpha to Beta of f of Theta."},{"Start":"01:51.660 ","End":"01:56.930","Text":"You have to put this in brackets, squared d Theta."},{"Start":"01:56.930 ","End":"02:01.950","Text":"That\u0027s the formula for the area with polar coordinates."},{"Start":"02:02.060 ","End":"02:05.385","Text":"Let\u0027s start with an example."},{"Start":"02:05.385 ","End":"02:07.625","Text":"We don\u0027t need this anymore."},{"Start":"02:07.625 ","End":"02:11.690","Text":"I\u0027d like to take an example that we already did before when"},{"Start":"02:11.690 ","End":"02:15.710","Text":"we studied converting from polar to Cartesian,"},{"Start":"02:15.710 ","End":"02:22.775","Text":"we took the example r equals 3 times 1 minus cosine Theta,"},{"Start":"02:22.775 ","End":"02:25.505","Text":"and we even drew the shape of it,"},{"Start":"02:25.505 ","End":"02:28.380","Text":"we called it a cardioid."},{"Start":"02:28.670 ","End":"02:31.410","Text":"There I\u0027ve brought the picture,"},{"Start":"02:31.410 ","End":"02:35.025","Text":"and this is the pole here,"},{"Start":"02:35.025 ","End":"02:41.485","Text":"and this is the polar axis where Theta equals 0,"},{"Start":"02:41.485 ","End":"02:50.535","Text":"and we even discovered that this distance from here to here was 6, this length."},{"Start":"02:50.535 ","End":"02:57.560","Text":"I\u0027m going to ask the question to find the area of just this little bit"},{"Start":"02:57.560 ","End":"03:05.175","Text":"here called the S. Let\u0027s see how we would do it then."},{"Start":"03:05.175 ","End":"03:09.435","Text":"First of all we need to know what our Alpha and Beta are."},{"Start":"03:09.435 ","End":"03:12.425","Text":"Well, actually it goes from here to here."},{"Start":"03:12.425 ","End":"03:15.470","Text":"Here, we have that Theta is 0,"},{"Start":"03:15.470 ","End":"03:20.070","Text":"so I mean that will become our Alpha."},{"Start":"03:20.070 ","End":"03:22.800","Text":"Here the angle it\u0027s just straight up,"},{"Start":"03:22.800 ","End":"03:29.565","Text":"is 90 degrees or Pi over 2 in Radians."},{"Start":"03:29.565 ","End":"03:34.200","Text":"What we get we should use Radians really."},{"Start":"03:34.200 ","End":"03:37.560","Text":"We would get the integral."},{"Start":"03:37.560 ","End":"03:41.055","Text":"This is what S is going to be in our case,"},{"Start":"03:41.055 ","End":"03:44.640","Text":"from 0, that\u0027s our Alpha."},{"Start":"03:44.640 ","End":"03:45.750","Text":"I\u0027ll just write that down,"},{"Start":"03:45.750 ","End":"03:49.770","Text":"Alpha equals 0, Beta equals Pi over 2,"},{"Start":"03:49.770 ","End":"03:58.695","Text":"and the function, we\u0027ll call this also f of Theta, the function of Theta."},{"Start":"03:58.695 ","End":"04:03.570","Text":"We need from 0 to Pi over 2,"},{"Start":"04:03.570 ","End":"04:06.210","Text":"I didn\u0027t leave room for the 1/2,"},{"Start":"04:06.210 ","End":"04:11.505","Text":"so 1/2 and then r squared which"},{"Start":"04:11.505 ","End":"04:19.710","Text":"is 9 times 1 minus cosine"},{"Start":"04:19.710 ","End":"04:24.120","Text":"Theta squared, d Theta."},{"Start":"04:24.120 ","End":"04:27.995","Text":"Now it\u0027s just a matter of evaluating the integral."},{"Start":"04:27.995 ","End":"04:32.015","Text":"I\u0027m going to take the 9 outside the integral sign,"},{"Start":"04:32.015 ","End":"04:34.445","Text":"and I\u0027m also going to use the formula,"},{"Start":"04:34.445 ","End":"04:36.170","Text":"you know what I mean."},{"Start":"04:36.170 ","End":"04:42.890","Text":"The special products, the a minus b squared is a squared minus 2ab plus b squared."},{"Start":"04:42.890 ","End":"04:44.630","Text":"Sometimes people forget this."},{"Start":"04:44.630 ","End":"04:47.100","Text":"You should know it in your sleep."},{"Start":"04:47.100 ","End":"04:54.330","Text":"Back here we have 9 over 2 times the integral,"},{"Start":"04:54.330 ","End":"05:04.290","Text":"same limits of 1 minus 2 cosine Theta plus cosine squared Theta,"},{"Start":"05:04.290 ","End":"05:08.735","Text":"need to put this in brackets, d Theta."},{"Start":"05:08.735 ","End":"05:11.410","Text":"Each of these is not going to be a problem,"},{"Start":"05:11.410 ","End":"05:12.910","Text":"a constant, the cosine."},{"Start":"05:12.910 ","End":"05:17.360","Text":"The cosine squared, we need another formula for that."},{"Start":"05:17.360 ","End":"05:21.820","Text":"The formula I need is that in general,"},{"Start":"05:21.820 ","End":"05:31.710","Text":"cosine squared Alpha is 1/2 of 1 plus cosine 2 Alpha."},{"Start":"05:31.710 ","End":"05:41.850","Text":"I use that here we get 9 over 2 integral from 0 to Pi over 2."},{"Start":"05:42.740 ","End":"05:46.690","Text":"Let\u0027s see now. First of all,"},{"Start":"05:46.690 ","End":"05:49.625","Text":"I\u0027ll rewrite it as 1,"},{"Start":"05:49.625 ","End":"05:52.730","Text":"and then plus 1/2 from here,"},{"Start":"05:52.730 ","End":"05:55.480","Text":"so that\u0027s 3 over 2,"},{"Start":"05:55.480 ","End":"06:04.180","Text":"minus 2 cosine Theta plus 1/2 cosine 2 Theta,"},{"Start":"06:07.340 ","End":"06:10.920","Text":"d Theta, and this equals."},{"Start":"06:10.920 ","End":"06:12.230","Text":"Let\u0027s do the integration."},{"Start":"06:12.230 ","End":"06:17.020","Text":"The integral of 3 over 2 is 3 over 2 Theta."},{"Start":"06:17.020 ","End":"06:20.150","Text":"This integral of cosine is sine,"},{"Start":"06:20.150 ","End":"06:23.495","Text":"so it\u0027s minus 2 sine Theta."},{"Start":"06:23.495 ","End":"06:26.980","Text":"The integral of cosine 2 Theta,"},{"Start":"06:26.980 ","End":"06:31.395","Text":"we start off by saying sine 2 Theta,"},{"Start":"06:31.395 ","End":"06:37.125","Text":"but if we differentiate this we\u0027ll get cosine 2 Theta times 2."},{"Start":"06:37.125 ","End":"06:39.405","Text":"I need to divide by 2."},{"Start":"06:39.405 ","End":"06:42.480","Text":"I\u0027ve got 1/4 here. Think about that."},{"Start":"06:42.480 ","End":"06:53.525","Text":"Then I have this taken between 0 and Pi over 2 and then 9 over 2 times all of this."},{"Start":"06:53.525 ","End":"06:58.910","Text":"Now, if you look at it when you put 0 in we\u0027re going to get"},{"Start":"06:58.910 ","End":"07:05.104","Text":"0 everywhere because Theta is 0 and sine of 0 is 0,"},{"Start":"07:05.104 ","End":"07:07.385","Text":"and sine of twice 0 is 0,"},{"Start":"07:07.385 ","End":"07:10.300","Text":"so we just need the Pi over 2."},{"Start":"07:10.300 ","End":"07:18.280","Text":"What we get is 9 over 2 and then just plug in Pi over 2."},{"Start":"07:18.830 ","End":"07:25.175","Text":"For the first 1, 9 over 2 times 3 over 2 times Pi over 2,"},{"Start":"07:25.175 ","End":"07:31.140","Text":"sine of Pi over 2 is sine of 90 degrees is 1,"},{"Start":"07:31.140 ","End":"07:37.140","Text":"so we get minus 9 over 2 times 2 times 1."},{"Start":"07:37.140 ","End":"07:42.780","Text":"The sine of 2 Theta is the sine of Pi is 0,"},{"Start":"07:42.780 ","End":"07:45.705","Text":"so this bit will be 0."},{"Start":"07:45.705 ","End":"07:48.360","Text":"Doesn\u0027t matter about the 1/4 and the 9 over 2,"},{"Start":"07:48.360 ","End":"07:50.670","Text":"it\u0027s just going to stay 0."},{"Start":"07:50.670 ","End":"07:58.000","Text":"What we are left with is, let\u0027s see,"},{"Start":"07:58.220 ","End":"08:03.915","Text":"this bit is 9 times 3 is 27 over 8,"},{"Start":"08:03.915 ","End":"08:07.935","Text":"27 Pi over 8,"},{"Start":"08:07.935 ","End":"08:11.580","Text":"and here 2 times 2 times 1,"},{"Start":"08:11.580 ","End":"08:13.290","Text":"2 with a 2 cancels."},{"Start":"08:13.290 ","End":"08:16.455","Text":"It\u0027s just minus 9."},{"Start":"08:16.455 ","End":"08:19.605","Text":"I\u0027m not going to do that on the calculator."},{"Start":"08:19.605 ","End":"08:22.065","Text":"I\u0027m just going to leave that as the answer."},{"Start":"08:22.065 ","End":"08:24.695","Text":"Now I want to continue with another scenario."},{"Start":"08:24.695 ","End":"08:29.905","Text":"Instead of 1 curve let\u0027s take 2 curves."},{"Start":"08:29.905 ","End":"08:33.950","Text":"Clean the board, I kept the cardioid you\u0027ll need it for the next example."},{"Start":"08:33.950 ","End":"08:35.735","Text":"Let me get rid of this,"},{"Start":"08:35.735 ","End":"08:39.920","Text":"and let me bring in a new picture. Here\u0027s the new picture."},{"Start":"08:39.920 ","End":"08:46.125","Text":"This time we\u0027re going to take the area between 2 curves,"},{"Start":"08:46.125 ","End":"08:49.750","Text":"both of them are given in polar."},{"Start":"08:49.910 ","End":"08:52.350","Text":"Let\u0027s call it r_1,"},{"Start":"08:52.350 ","End":"08:55.080","Text":"which is 1 function of Theta f,"},{"Start":"08:55.080 ","End":"08:57.290","Text":"and r_2 is another function of Theta,"},{"Start":"08:57.290 ","End":"08:59.385","Text":"call it g. Again,"},{"Start":"08:59.385 ","End":"09:05.430","Text":"we\u0027re going to take it between Alpha and Beta in a counterclockwise direction and this"},{"Start":"09:05.430 ","End":"09:11.925","Text":"time the formula we get is a variation of the previous formula."},{"Start":"09:11.925 ","End":"09:13.525","Text":"We also had 1/2,"},{"Start":"09:13.525 ","End":"09:16.225","Text":"we also had the integral from Alpha to Beta,"},{"Start":"09:16.225 ","End":"09:18.230","Text":"but instead of r squared,"},{"Start":"09:18.230 ","End":"09:24.320","Text":"we\u0027re going to take the outer 1 squared minus the inner 1 squared."},{"Start":"09:24.320 ","End":"09:26.500","Text":"I call this r_1, r_2,"},{"Start":"09:26.500 ","End":"09:36.510","Text":"d Theta because in practice instead of r_2 we\u0027ll put g of Theta squared and so on,"},{"Start":"09:36.510 ","End":"09:42.700","Text":"minus f of Theta squared."},{"Start":"09:47.870 ","End":"09:50.540","Text":"Let\u0027s just get to the example."},{"Start":"09:50.540 ","End":"09:53.795","Text":"The cardioid is going to be the inner 1."},{"Start":"09:53.795 ","End":"09:55.400","Text":"I know the color is wrong,"},{"Start":"09:55.400 ","End":"09:56.600","Text":"it should be blue."},{"Start":"09:56.600 ","End":"09:58.370","Text":"That\u0027s the r_1."},{"Start":"09:58.370 ","End":"10:07.394","Text":"Make it r_1 and it\u0027s all ready labeled f. For the outer 1, r_2,"},{"Start":"10:07.394 ","End":"10:12.185","Text":"well I\u0027m going to take a circle of radius 2 around"},{"Start":"10:12.185 ","End":"10:17.370","Text":"the pole and the equation is just radius equals 2."},{"Start":"10:17.370 ","End":"10:22.910","Text":"If Theta doesn\u0027t appear we learned about the equation of circle."},{"Start":"10:22.910 ","End":"10:27.205","Text":"That\u0027s going to be the outer 1 that\u0027s g of Theta,"},{"Start":"10:27.205 ","End":"10:32.255","Text":"and just let me draw it in now."},{"Start":"10:32.255 ","End":"10:35.850","Text":"Here it is. I\u0027ve drawn it in a circle of radius 2,"},{"Start":"10:35.850 ","End":"10:40.960","Text":"I marked that as 2 units away from the pole and this I remember was 6."},{"Start":"10:41.670 ","End":"10:48.850","Text":"What I shaded in yellow is what was here green."},{"Start":"10:48.850 ","End":"10:56.185","Text":"This will be my S. According to the formula,"},{"Start":"10:56.185 ","End":"10:59.380","Text":"S equals the integral,"},{"Start":"10:59.380 ","End":"11:01.315","Text":"there was 1/2 in front."},{"Start":"11:01.315 ","End":"11:03.655","Text":"But what is Alpha and Beta?"},{"Start":"11:03.655 ","End":"11:06.040","Text":"Well, I need to find them."},{"Start":"11:06.040 ","End":"11:09.760","Text":"There is a point here where these intersect and there is a point"},{"Start":"11:09.760 ","End":"11:15.070","Text":"here and I have to find the angle."},{"Start":"11:15.070 ","End":"11:22.660","Text":"Let\u0027s see I join this here and I can join this here and each of these has an angle,"},{"Start":"11:22.660 ","End":"11:28.239","Text":"this is going to be a positive angle because I need to go in this direction."},{"Start":"11:28.239 ","End":"11:29.800","Text":"This is going to be Alpha,"},{"Start":"11:29.800 ","End":"11:30.970","Text":"we\u0027ll figure it out,"},{"Start":"11:30.970 ","End":"11:32.860","Text":"and this will be Beta,"},{"Start":"11:32.860 ","End":"11:35.035","Text":"and then we\u0027ll take the interval from Alpha to Beta."},{"Start":"11:35.035 ","End":"11:37.734","Text":"Let\u0027s do a side exercise for that."},{"Start":"11:37.734 ","End":"11:44.095","Text":"What we need is for f of Theta to equal g of Theta or R_1 to equal R_2."},{"Start":"11:44.095 ","End":"11:48.145","Text":"We get an equation that"},{"Start":"11:48.145 ","End":"11:54.580","Text":"3 times 1 minus cosine Theta equals 2."},{"Start":"11:54.580 ","End":"11:56.710","Text":"If you expand this,"},{"Start":"11:56.710 ","End":"12:00.655","Text":"you get that 3 minus 3 cosine Theta equals 2,"},{"Start":"12:00.655 ","End":"12:01.870","Text":"bring the 2 over,"},{"Start":"12:01.870 ","End":"12:04.405","Text":"3 cosine Theta is 1 in short,"},{"Start":"12:04.405 ","End":"12:09.680","Text":"if you think about it you\u0027ll get cosine Theta equals 1/3."},{"Start":"12:09.870 ","End":"12:12.655","Text":"There\u0027s an infinite number of solutions,"},{"Start":"12:12.655 ","End":"12:15.580","Text":"but the main solution is obtained with"},{"Start":"12:15.580 ","End":"12:24.010","Text":"the calculator by doing arc cosine of 1/3."},{"Start":"12:24.010 ","End":"12:26.620","Text":"It depends on the calculator sometimes you just do inverse"},{"Start":"12:26.620 ","End":"12:29.965","Text":"cosine or cosine to the minus 1."},{"Start":"12:29.965 ","End":"12:37.150","Text":"Anyway, it comes out of my calculator 1.23 and a lot of other stuff,"},{"Start":"12:37.150 ","End":"12:38.770","Text":"just save it in the memory,"},{"Start":"12:38.770 ","End":"12:43.030","Text":"I\u0027ll write 1.23 but we\u0027ll use the value from the memory if we need to."},{"Start":"12:43.030 ","End":"12:45.595","Text":"Now that\u0027s not the only solution."},{"Start":"12:45.595 ","End":"12:51.715","Text":"Cosine is an even function and also minus 1.23 will work."},{"Start":"12:51.715 ","End":"12:55.810","Text":"Now we could also add multiples of 2 Pi but which I\u0027m not going to do."},{"Start":"12:55.810 ","End":"12:58.480","Text":"Alpha is minus 1.23,"},{"Start":"12:58.480 ","End":"13:03.515","Text":"etc, and Beta is plus 1.23, etc."},{"Start":"13:03.515 ","End":"13:05.500","Text":"I\u0027m talking about in radians."},{"Start":"13:05.500 ","End":"13:09.370","Text":"Make sure that the calculator is set"},{"Start":"13:09.370 ","End":"13:15.880","Text":"to radians because when we do integrals and derivatives,"},{"Start":"13:15.880 ","End":"13:20.440","Text":"all the formulas only work when we\u0027re dealing with radians."},{"Start":"13:20.440 ","End":"13:24.130","Text":"For just looking up a value of an angle sine or cosine you can use"},{"Start":"13:24.130 ","End":"13:28.510","Text":"degrees but okay. Back to here."},{"Start":"13:28.510 ","End":"13:37.255","Text":"Now we know that this is from minus 1.23 to 1.23 of R_2 squared that\u0027s the outer squared,"},{"Start":"13:37.255 ","End":"13:38.410","Text":"which is 2 squared,"},{"Start":"13:38.410 ","End":"13:41.725","Text":"which is 4 minus this thing squared,"},{"Start":"13:41.725 ","End":"13:44.200","Text":"which is 9 times,"},{"Start":"13:44.200 ","End":"13:48.055","Text":"and now we\u0027ve already used the formula for a minus b squared it\u0027s"},{"Start":"13:48.055 ","End":"13:57.110","Text":"a squared minus 2ab plus b squared, d Theta."},{"Start":"13:58.680 ","End":"14:06.205","Text":"We already had another shortcut via the trigonometric identity for cosine squared Theta,"},{"Start":"14:06.205 ","End":"14:09.100","Text":"let me go and get it, where is it?"},{"Start":"14:09.100 ","End":"14:18.250","Text":"The side there. What we get is equal to 1/2 times the integral."},{"Start":"14:18.250 ","End":"14:20.170","Text":"Now to compute this;"},{"Start":"14:20.170 ","End":"14:24.410","Text":"4 minus 9 is minus 5,"},{"Start":"14:26.250 ","End":"14:31.510","Text":"and then plus 9 times 2 is 18,"},{"Start":"14:31.510 ","End":"14:40.490","Text":"plus 18 cosine Theta and then we have minus 9 of these,"},{"Start":"14:40.860 ","End":"14:47.200","Text":"I have minus 9 over 2,"},{"Start":"14:47.200 ","End":"14:51.295","Text":"which is minus 4.5,"},{"Start":"14:51.295 ","End":"14:59.980","Text":"and then minus 9 over 2 cosine"},{"Start":"14:59.980 ","End":"15:06.610","Text":"of 2 Theta minus 4.5,"},{"Start":"15:06.610 ","End":"15:10.720","Text":"9 over 2, I\u0027m not sure which way to write it leave it as 4.5,"},{"Start":"15:10.720 ","End":"15:16.760","Text":"times cosine of 2 Theta."},{"Start":"15:17.240 ","End":"15:19.570","Text":"I think I\u0027ll clean this up a bit."},{"Start":"15:19.570 ","End":"15:22.450","Text":"First of all, I changed my mind I\u0027ll go back to 9 over 2,"},{"Start":"15:22.450 ","End":"15:24.895","Text":"and I\u0027ll combine these 2 together,"},{"Start":"15:24.895 ","End":"15:28.450","Text":"5 and 4.5 is 9.5,"},{"Start":"15:28.450 ","End":"15:30.115","Text":"which is 19 over 2,"},{"Start":"15:30.115 ","End":"15:36.225","Text":"so the minus 19 over 2 from combining the numbers here."},{"Start":"15:36.225 ","End":"15:38.820","Text":"Now I can move this a bit closer and I"},{"Start":"15:38.820 ","End":"15:41.955","Text":"changed the 4.5 to 9 over 2 and I still need to put"},{"Start":"15:41.955 ","End":"15:50.890","Text":"brackets d Theta and the limits minus 1.23 to 1.23,"},{"Start":"15:50.890 ","End":"15:53.020","Text":"a bit crowded here."},{"Start":"15:53.020 ","End":"15:56.600","Text":"Now let\u0027s do the integral."},{"Start":"16:00.210 ","End":"16:05.679","Text":"The integral of 1 is just Theta the integral of this is this times Theta,"},{"Start":"16:05.679 ","End":"16:11.185","Text":"minus 19 over 2 but I\u0027m going to combine it with the 1/2,"},{"Start":"16:11.185 ","End":"16:13.510","Text":"it\u0027s over 4 Theta,"},{"Start":"16:13.510 ","End":"16:21.354","Text":"here the 1/2 with the 18 makes it 9 and the integral of cosine Theta is sine Theta,"},{"Start":"16:21.354 ","End":"16:26.780","Text":"this 1/2 with this makes it minus 9 over 4."},{"Start":"16:28.080 ","End":"16:33.715","Text":"The integral of cosine 2 Theta,"},{"Start":"16:33.715 ","End":"16:38.230","Text":"it\u0027s not quite sine 2 Theta because if we differentiate"},{"Start":"16:38.230 ","End":"16:42.730","Text":"this we get 2 sine cosine Theta and it\u0027s divide by an extra 2,"},{"Start":"16:42.730 ","End":"16:45.040","Text":"I\u0027ll change this 4 into an 8."},{"Start":"16:45.040 ","End":"16:49.390","Text":"There we are. All of this has to be evaluated between"},{"Start":"16:49.390 ","End":"16:56.720","Text":"minus 1.23 or the exact value that we stored and 1.23."},{"Start":"16:57.690 ","End":"17:00.040","Text":"I don\u0027t want to waste a lot of time with"},{"Start":"17:00.040 ","End":"17:03.055","Text":"all these computations it\u0027s the idea that\u0027s important,"},{"Start":"17:03.055 ","End":"17:06.160","Text":"I\u0027ll just write down the answer that I got."},{"Start":"17:06.160 ","End":"17:13.765","Text":"I got 3.8622 to 4 decimal places."},{"Start":"17:13.765 ","End":"17:18.145","Text":"I\u0027ll leave you to do the calculations it\u0027s too tedious."},{"Start":"17:18.145 ","End":"17:22.310","Text":"We\u0027ll just call it a day here."}],"Thumbnail":null,"ID":10591},{"Watched":false,"Name":"Exercise 9","Duration":"5m 56s","ChapterTopicVideoID":10056,"CourseChapterTopicPlaylistID":8899,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:04.140","Text":"In this exercise, we\u0027re given a graph and this happens to be"},{"Start":"00:04.140 ","End":"00:08.910","Text":"a closed curve, as theta goes from 0 to 2 pi."},{"Start":"00:08.910 ","End":"00:15.135","Text":"It goes around once, counterclockwise,"},{"Start":"00:15.135 ","End":"00:24.900","Text":"and we want, not the whole area inside this curve, just the bit to the left of the y-axis."},{"Start":"00:24.900 ","End":"00:28.125","Text":"I\u0027ll give you an idea of what this looks like."},{"Start":"00:28.125 ","End":"00:33.135","Text":"I brought in a picture, which we could live without, but just to make it more interesting."},{"Start":"00:33.135 ","End":"00:39.270","Text":"In general, when we have a plus b or rather b plus a cosine theta,"},{"Start":"00:39.270 ","End":"00:42.230","Text":"depending on the values of these numbers,"},{"Start":"00:42.230 ","End":"00:44.405","Text":"we get different types of curves."},{"Start":"00:44.405 ","End":"00:50.690","Text":"But in our case, it would look something like this,"},{"Start":"00:50.690 ","End":"00:53.270","Text":"if you look at the conditions on b and a,"},{"Start":"00:53.270 ","End":"00:58.835","Text":"It\u0027s a Limacon and it\u0027s the dimpled variety,"},{"Start":"00:58.835 ","End":"01:01.040","Text":"but to the left of the y-axis,"},{"Start":"01:01.040 ","End":"01:02.785","Text":"let me shade that,"},{"Start":"01:02.785 ","End":"01:09.585","Text":"that is the bit I just shaded, and as far as theta goes,"},{"Start":"01:09.585 ","End":"01:15.275","Text":"this is where theta is equal to 90 degrees pi over 2,"},{"Start":"01:15.275 ","End":"01:21.230","Text":"and then we continue with theta up to"},{"Start":"01:21.230 ","End":"01:28.100","Text":"the negative y-axis where theta is 3 pi over 2."},{"Start":"01:28.100 ","End":"01:37.670","Text":"This is the limits that we have for the angle, and we just have to use the formula now."},{"Start":"01:37.670 ","End":"01:42.315","Text":"For area, and the area is"},{"Start":"01:42.315 ","End":"01:49.800","Text":"1 half the integral of r squared d theta,"},{"Start":"01:49.800 ","End":"01:56.825","Text":"and in our case, from pi over 2 to 3 pi over 2."},{"Start":"01:56.825 ","End":"02:00.065","Text":"Now, we substitute r as this,"},{"Start":"02:00.065 ","End":"02:08.865","Text":"so we get 1 half the integral of this plus this squared will be 36,"},{"Start":"02:08.865 ","End":"02:14.505","Text":"plus twice 6 times 4 is 48."},{"Start":"02:14.505 ","End":"02:21.375","Text":"Cosine theta, plus cosine squared theta,"},{"Start":"02:21.375 ","End":"02:29.875","Text":"d theta, need brackets, and the limits pi over 2 to 3 pi over 2."},{"Start":"02:29.875 ","End":"02:33.590","Text":"Whoops, I forgot the 4 squared,"},{"Start":"02:33.590 ","End":"02:36.910","Text":"which is 16 in front of this."},{"Start":"02:36.910 ","End":"02:40.260","Text":"Okay, so let\u0027s see what we get."},{"Start":"02:40.260 ","End":"02:42.300","Text":"We get the integral,"},{"Start":"02:42.300 ","End":"02:51.295","Text":"half of 36 is 18 plus 24 cosine theta,"},{"Start":"02:51.295 ","End":"02:55.700","Text":"and here, 8 cosine squared theta,"},{"Start":"02:55.700 ","End":"02:59.795","Text":"but I\u0027m going to use a trigonometric identity,"},{"Start":"02:59.795 ","End":"03:03.749","Text":"that is the cosine squared theta"},{"Start":"03:03.749 ","End":"03:09.420","Text":"is 1 half of 1 plus cosine 2 theta."},{"Start":"03:09.420 ","End":"03:11.459","Text":"If I do that,"},{"Start":"03:11.459 ","End":"03:13.930","Text":"then the half of the 16 is 8,"},{"Start":"03:13.930 ","End":"03:15.655","Text":"but we have another half from here,"},{"Start":"03:15.655 ","End":"03:19.975","Text":"so we\u0027ve got 4 times 1 plus"},{"Start":"03:19.975 ","End":"03:27.770","Text":"cosine 2 theta, and all this d theta."},{"Start":"03:27.770 ","End":"03:37.165","Text":"This will be the integral of 18 plus 4 is 22,"},{"Start":"03:37.165 ","End":"03:41.960","Text":"plus 24 cosine theta,"},{"Start":"03:41.960 ","End":"03:47.320","Text":"plus 4 cosine 2 theta,"},{"Start":"03:47.720 ","End":"03:52.270","Text":"I should have been writing the limits of integration pi over 2,"},{"Start":"03:52.270 ","End":"03:55.969","Text":"pi over 2, 3 pi over 2,"},{"Start":"03:55.969 ","End":"03:58.710","Text":"3 pi over 2."},{"Start":"03:58.710 ","End":"04:05.140","Text":"Okay. Finally, the integral, 22 theta from here,"},{"Start":"04:05.140 ","End":"04:07.770","Text":"integral of cosine is sine,"},{"Start":"04:07.770 ","End":"04:12.020","Text":"so 24 sine theta,"},{"Start":"04:12.020 ","End":"04:19.099","Text":"and here, the integral of cosine 2 theta is sine 2 theta,"},{"Start":"04:19.099 ","End":"04:23.585","Text":"over 2, which with the 4 gives me 2."},{"Start":"04:23.585 ","End":"04:30.405","Text":"So I need this evaluated between pi over 2,"},{"Start":"04:30.405 ","End":"04:34.560","Text":"and 3 pi over 2."},{"Start":"04:34.560 ","End":"04:41.470","Text":"Let\u0027s see, it\u0027s probably easier to apply these limits to each piece, separately."},{"Start":"04:41.470 ","End":"04:48.735","Text":"With the first term, it\u0027s 22 times, just 3 pi over 2 minus pi over 2 is pi,"},{"Start":"04:48.735 ","End":"04:53.040","Text":"so this is 22 pi, from the first bit."},{"Start":"04:53.040 ","End":"04:54.675","Text":"From the second bit,"},{"Start":"04:54.675 ","End":"05:02.930","Text":"the sine of 3 pi over 2 is sine of 270 is minus 1,"},{"Start":"05:02.930 ","End":"05:05.749","Text":"the sine of pi over 2 is 1,"},{"Start":"05:05.749 ","End":"05:15.780","Text":"so this gives us minus 2, if we subtract with 24, it gives us minus 48."},{"Start":"05:15.780 ","End":"05:22.810","Text":"Let\u0027s see, the last bit we\u0027ve got sine of 3 pi,"},{"Start":"05:22.810 ","End":"05:25.910","Text":"minus sine of pi,"},{"Start":"05:25.910 ","End":"05:28.220","Text":"the sine of any multiple of pi is 0,"},{"Start":"05:28.220 ","End":"05:31.110","Text":"so that\u0027s plus nothing,"},{"Start":"05:31.670 ","End":"05:37.140","Text":"so the answer would be 22 pi minus 48."},{"Start":"05:37.140 ","End":"05:40.820","Text":"Pi is more than 3, so this is more than 66,"},{"Start":"05:40.820 ","End":"05:43.030","Text":"so it\u0027s going to be positive,"},{"Start":"05:43.030 ","End":"05:47.670","Text":"don\u0027t have my calculator with me you might want to compute it numerically,"},{"Start":"05:47.670 ","End":"05:56.620","Text":"if not we\u0027ll just leave the answer as 22 pi minus 48, whatever that comes out to be."}],"Thumbnail":null,"ID":9951},{"Watched":false,"Name":"Exercise 10","Duration":"8m 23s","ChapterTopicVideoID":10057,"CourseChapterTopicPlaylistID":8899,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"In this exercise, we want to find the area between 2 circles,"},{"Start":"00:05.085 ","End":"00:08.535","Text":"r equals 1, which is obviously this 1,"},{"Start":"00:08.535 ","End":"00:15.930","Text":"circle centered at the origin and radius 1 and the other 1,"},{"Start":"00:15.930 ","End":"00:22.905","Text":"2 sine Theta, it is a circle and it\u0027s easy to show that."},{"Start":"00:22.905 ","End":"00:28.735","Text":"But it has a center at this point here."},{"Start":"00:28.735 ","End":"00:34.595","Text":"In any event, when you can see what it ranges from when Theta is 0, it\u0027s 0,"},{"Start":"00:34.595 ","End":"00:38.760","Text":"and when Theta is 90 degrees or Pi over 2,"},{"Start":"00:38.760 ","End":"00:42.180","Text":"it\u0027s equal to sine Theta is 1,"},{"Start":"00:42.180 ","End":"00:45.450","Text":"so 2 sine Theta is 2."},{"Start":"00:45.450 ","End":"00:49.590","Text":"Not hard to show that this is what it looks like,"},{"Start":"00:49.720 ","End":"00:51.830","Text":"but that\u0027s not the point here."},{"Start":"00:51.830 ","End":"00:56.480","Text":"Well, at this point we\u0027ll take it on trust that this is a circle as shown in the picture."},{"Start":"00:56.480 ","End":"01:02.394","Text":"Now we want the shaded bit between the 2 circles to find its area."},{"Start":"01:02.394 ","End":"01:04.800","Text":"Now each of these circles,"},{"Start":"01:04.800 ","End":"01:07.875","Text":"Theta is taken from 0-2 Pi,"},{"Start":"01:07.875 ","End":"01:12.690","Text":"I should maybe add that."},{"Start":"01:12.690 ","End":"01:17.390","Text":"The circle\u0027s working or if Theta keeps going around and around and around,"},{"Start":"01:17.390 ","End":"01:23.325","Text":"but all we need is 0-2 Pi to make 1 complete revolution,"},{"Start":"01:23.325 ","End":"01:28.385","Text":"and I want to find a subinterval of that."},{"Start":"01:28.385 ","End":"01:32.105","Text":"In other words, I want to know what this angle is here,"},{"Start":"01:32.105 ","End":"01:39.000","Text":"and also from here to here. Bad drawing."},{"Start":"01:39.000 ","End":"01:42.750","Text":"I need the other Theta also."},{"Start":"01:42.750 ","End":"01:44.495","Text":"Let\u0027s see what they are."},{"Start":"01:44.495 ","End":"01:50.540","Text":"We just simply have to intersect these 2 to get these 2 values of Theta."},{"Start":"01:50.540 ","End":"01:57.410","Text":"We get that 1 is equal to 2 sine Theta,"},{"Start":"01:57.410 ","End":"02:03.120","Text":"which means that sine Theta is equal to a 1/2,"},{"Start":"02:03.120 ","End":"02:13.025","Text":"and sine Theta is a 1/2 when Theta is either equal to 30 degrees."},{"Start":"02:13.025 ","End":"02:17.470","Text":"30 degrees in radians is Pi over 6,"},{"Start":"02:17.470 ","End":"02:20.210","Text":"and maybe I\u0027ll call that Theta_1,"},{"Start":"02:20.210 ","End":"02:23.310","Text":"and we\u0027ll call this Theta_2."},{"Start":"02:23.310 ","End":"02:29.670","Text":"Theta_2 is going to be 180 minus 30_150,"},{"Start":"02:29.670 ","End":"02:32.670","Text":"or Pi minus Pi over 6,"},{"Start":"02:32.670 ","End":"02:36.345","Text":"5 Pi over 6."},{"Start":"02:36.345 ","End":"02:41.025","Text":"Now, the area between 2 curves,"},{"Start":"02:41.025 ","End":"02:50.115","Text":"in fact, say the inner 1 is r_1 and the outer 1 r_2."},{"Start":"02:50.115 ","End":"02:52.980","Text":"Here I have r_1 and r_2,"},{"Start":"02:52.980 ","End":"02:55.140","Text":"and the formula is,"},{"Start":"02:55.140 ","End":"02:57.105","Text":"we want the integral,"},{"Start":"02:57.105 ","End":"02:59.450","Text":"there\u0027s always a 1/2 in front of it."},{"Start":"02:59.450 ","End":"03:02.780","Text":"Sometimes it\u0027s in the inside of the integral."},{"Start":"03:02.780 ","End":"03:05.855","Text":"Here we want the limits as found here,"},{"Start":"03:05.855 ","End":"03:12.890","Text":"from Pi over 6- 5 Pi over"},{"Start":"03:12.890 ","End":"03:18.965","Text":"6 of the r_2 squared minus r_1 squared."},{"Start":"03:18.965 ","End":"03:24.180","Text":"I want this 1 squared,"},{"Start":"03:24.180 ","End":"03:28.760","Text":"which is 4 sine squared Theta,"},{"Start":"03:28.760 ","End":"03:31.909","Text":"just to let you know that was r_2 squared."},{"Start":"03:31.909 ","End":"03:33.830","Text":"Now I need r_ 1 squared,"},{"Start":"03:33.830 ","End":"03:36.849","Text":"which is 1 squared, which is 1,"},{"Start":"03:36.849 ","End":"03:41.425","Text":"and all this d Theta,"},{"Start":"03:41.425 ","End":"03:46.350","Text":"the difference of the squares of the outer minus the inner."},{"Start":"03:46.490 ","End":"03:50.040","Text":"The sine squared Theta is not so convenient,"},{"Start":"03:50.040 ","End":"03:53.780","Text":"so I\u0027ll use some trigonometric identity here."},{"Start":"03:53.780 ","End":"03:59.770","Text":"What I\u0027d like to say is that sine squared Theta"},{"Start":"03:59.770 ","End":"04:07.585","Text":"is 1/2 of 1 minus cosine 2 Theta."},{"Start":"04:07.585 ","End":"04:10.550","Text":"If I plug that in here and you know what,"},{"Start":"04:10.550 ","End":"04:12.830","Text":"let\u0027s also separate the 2 integrals,"},{"Start":"04:12.830 ","End":"04:21.480","Text":"I have got 1 integral from Pi over 6-5Pi over 6."},{"Start":"04:21.480 ","End":"04:25.680","Text":"The first 1 a 1/2 with the 4 gives me 2."},{"Start":"04:25.680 ","End":"04:29.805","Text":"Now the 2 sines squared Theta will knock out this 1/2,"},{"Start":"04:29.805 ","End":"04:38.345","Text":"so I\u0027ll get 1 minus cosine 2 Theta for the first bit."},{"Start":"04:38.345 ","End":"04:44.480","Text":"I won\u0027t split them up just yet because I also have here minus"},{"Start":"04:44.480 ","End":"04:51.040","Text":"a 1/2 and all this d Theta."},{"Start":"04:51.040 ","End":"04:55.320","Text":"Now I\u0027ll split it up into 2 bits."},{"Start":"04:55.320 ","End":"04:59.190","Text":"The first bit is 1/2."},{"Start":"04:59.190 ","End":"05:04.170","Text":"It\u0027s just 1/2 which I can bring out front of the"},{"Start":"05:04.170 ","End":"05:10.380","Text":"integral from Pi over 6-5Pi over 6,"},{"Start":"05:10.380 ","End":"05:16.785","Text":"just d Theta or 1d Theta if you like to have something written here,"},{"Start":"05:16.785 ","End":"05:23.805","Text":"and then minus the integral from"},{"Start":"05:23.805 ","End":"05:33.795","Text":"Pi over 6-5Pi over 6 of cosine 2 Theta d Theta."},{"Start":"05:33.795 ","End":"05:38.220","Text":"Cosine 2 Theta d Theta."},{"Start":"05:38.220 ","End":"05:42.435","Text":"Now I want to actually do the integrals."},{"Start":"05:42.435 ","End":"05:47.970","Text":"This integral gives me 1/2 of"},{"Start":"05:47.970 ","End":"05:57.490","Text":"just Theta from Pi over 6-5Pi over 6 minus,"},{"Start":"05:57.490 ","End":"06:02.180","Text":"the integral of cosine is sine but because of the 2 Theta,"},{"Start":"06:02.180 ","End":"06:04.220","Text":"we need to divide by 2,"},{"Start":"06:04.220 ","End":"06:07.440","Text":"so it\u0027s minus a 1/2."},{"Start":"06:09.820 ","End":"06:20.820","Text":"Just sine 2 Theta itself from Pi over 6-5Pi over 6."},{"Start":"06:20.890 ","End":"06:24.815","Text":"Let\u0027s see. This 1,"},{"Start":"06:24.815 ","End":"06:34.125","Text":"just have to subtract 5Pi over 6 minus Pi over 6 is 4Pi over 6, which is 2/3Pi."},{"Start":"06:34.125 ","End":"06:36.780","Text":"The 2/3Pi with the 1/2,"},{"Start":"06:36.780 ","End":"06:40.050","Text":"this bit gives me 1/3Pi."},{"Start":"06:40.050 ","End":"06:43.325","Text":"Let\u0027s see what the 2 bit gives."},{"Start":"06:43.325 ","End":"06:47.015","Text":"Let me just do this as a side exercise."},{"Start":"06:47.015 ","End":"06:53.075","Text":"This bit here is sine of"},{"Start":"06:53.075 ","End":"07:00.600","Text":"twice 5Pi over 6 is 5Pi over 3 minus,"},{"Start":"07:00.600 ","End":"07:10.065","Text":"and if I put Pi over 6 I get sine of just Pi over 6 times 2 is Pi over 3."},{"Start":"07:10.065 ","End":"07:14.430","Text":"To help myself, I\u0027ll write them in degrees."},{"Start":"07:14.430 ","End":"07:16.920","Text":"Pi over 3 is 60 degrees."},{"Start":"07:16.920 ","End":"07:20.160","Text":"I\u0027m still used to degrees more."},{"Start":"07:20.160 ","End":"07:26.220","Text":"5Pi over 3 is 300 degrees because Pi is a"},{"Start":"07:26.220 ","End":"07:33.225","Text":"180 over 3 is 60 times 5, that\u0027s 300 degrees."},{"Start":"07:33.225 ","End":"07:38.675","Text":"What I get, sine of 300 is like sine of minus 60,"},{"Start":"07:38.675 ","End":"07:43.560","Text":"which is minus root 3 over 2,"},{"Start":"07:43.560 ","End":"07:48.730","Text":"and sine of 60 is also root 3 over 2,"},{"Start":"07:48.740 ","End":"07:54.230","Text":"so what I\u0027ve got from here is minus root 3."},{"Start":"07:54.230 ","End":"08:02.670","Text":"Back here, I\u0027ve got minus a 1/2 of minus root 3."},{"Start":"08:03.170 ","End":"08:10.890","Text":"The answer is going to be Pi over 3 from here,"},{"Start":"08:10.890 ","End":"08:15.554","Text":"plus root 3 over 2,"},{"Start":"08:15.554 ","End":"08:18.979","Text":"and there\u0027s no point in writing it as a decimal,"},{"Start":"08:18.979 ","End":"08:24.030","Text":"I think we\u0027ll just leave it like that and declare that we are done."}],"Thumbnail":null,"ID":9952}],"ID":8899},{"Name":"Surface Area in Polar Coordinates","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Surface Area in Polar Coordinates","Duration":"13m 50s","ChapterTopicVideoID":10256,"CourseChapterTopicPlaylistID":8900,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.595","Text":"We\u0027re continuing with polar coordinates,"},{"Start":"00:02.595 ","End":"00:04.830","Text":"and this time surface area."},{"Start":"00:04.830 ","End":"00:07.150","Text":"But not surface area in general,"},{"Start":"00:07.150 ","End":"00:12.055","Text":"just with rotating a curve about the x or y-axis."},{"Start":"00:12.055 ","End":"00:15.580","Text":"I wrote that; it\u0027s just of a curve rotated about the x or y-axis."},{"Start":"00:15.580 ","End":"00:18.185","Text":"I\u0027m going to borrow a sketch."},{"Start":"00:18.185 ","End":"00:21.220","Text":"I borrowed the diagram from parametric."},{"Start":"00:21.220 ","End":"00:26.185","Text":"I\u0027m going to modify it to work for polar coordinates,"},{"Start":"00:26.185 ","End":"00:29.245","Text":"which is a little bit similar to parametric."},{"Start":"00:29.245 ","End":"00:34.855","Text":"Instead of t, we\u0027re going to have Theta."},{"Start":"00:34.855 ","End":"00:38.470","Text":"We\u0027re not going to have x and y directly in terms of Theta,"},{"Start":"00:38.470 ","End":"00:40.825","Text":"only indirectly, so out that goes."},{"Start":"00:40.825 ","End":"00:42.790","Text":"The setup will be,"},{"Start":"00:42.790 ","End":"00:49.925","Text":"that we\u0027ll be given r as a function of Theta. I\u0027ll just move this up."},{"Start":"00:49.925 ","End":"00:51.990","Text":"Theta will go from Alpha to Beta."},{"Start":"00:51.990 ","End":"00:56.120","Text":"As before, we\u0027ll assume that as it does that the curve is"},{"Start":"00:56.120 ","End":"01:01.355","Text":"traversed only once and not back and forth or round and round."},{"Start":"01:01.355 ","End":"01:09.980","Text":"Indirectly we have x and y in terms of Theta because x is r cosine Theta still,"},{"Start":"01:09.980 ","End":"01:12.800","Text":"and y is r sine Theta."},{"Start":"01:12.800 ","End":"01:13.940","Text":"But instead of r, of course,"},{"Start":"01:13.940 ","End":"01:15.320","Text":"we could put f of theta."},{"Start":"01:15.320 ","End":"01:19.145","Text":"In a way, x and y are functions of Theta."},{"Start":"01:19.145 ","End":"01:21.590","Text":"I won\u0027t show you the intermediate steps,"},{"Start":"01:21.590 ","End":"01:23.840","Text":"I\u0027ll just erase these formulas."},{"Start":"01:23.840 ","End":"01:26.690","Text":"I brought in the formula for arc length,"},{"Start":"01:26.690 ","End":"01:30.605","Text":"because if you remember the formula for arc length,"},{"Start":"01:30.605 ","End":"01:36.410","Text":"it will be very similar to the formula for surface area of rotation."},{"Start":"01:36.410 ","End":"01:39.590","Text":"Let\u0027s see, about the x-axis,"},{"Start":"01:39.590 ","End":"01:42.810","Text":"what we\u0027ll get is instead of this, L,"},{"Start":"01:42.810 ","End":"01:52.280","Text":"we\u0027ll have an S. What we\u0027ll have is the integral also from Alpha to Beta."},{"Start":"01:52.280 ","End":"01:55.940","Text":"We\u0027ll have a 2Pi in front."},{"Start":"01:55.940 ","End":"01:58.400","Text":"In this case will have,"},{"Start":"01:58.400 ","End":"02:03.255","Text":"a y times the same thing as here."},{"Start":"02:03.255 ","End":"02:06.525","Text":"I just copied this from here."},{"Start":"02:06.525 ","End":"02:10.180","Text":"Essentially the difference between this and this is the 2Pi y."},{"Start":"02:10.180 ","End":"02:13.525","Text":"Sometimes the 2Pi is written inside, doesn\u0027t matter."},{"Start":"02:13.525 ","End":"02:19.300","Text":"The idea is that this bit here with d Theta,"},{"Start":"02:19.300 ","End":"02:21.099","Text":"the little piece of the curve,"},{"Start":"02:21.099 ","End":"02:28.075","Text":"when we rotate it around the x-axis,"},{"Start":"02:28.075 ","End":"02:29.890","Text":"we multiply by the circumference,"},{"Start":"02:29.890 ","End":"02:31.615","Text":"which is 2Pi y."},{"Start":"02:31.615 ","End":"02:34.780","Text":"Of course, afterwards we have to substitute y from here,"},{"Start":"02:34.780 ","End":"02:36.370","Text":"and r from here."},{"Start":"02:36.370 ","End":"02:38.470","Text":"We\u0027ll show that in the example."},{"Start":"02:38.470 ","End":"02:43.255","Text":"Let me just get to the equivalent with rotation about the y-axis."},{"Start":"02:43.255 ","End":"02:45.880","Text":"We use the same idea."},{"Start":"02:45.880 ","End":"02:49.365","Text":"We also have the integral from Alpha to Beta,"},{"Start":"02:49.365 ","End":"02:52.970","Text":"but this time we\u0027re rotating around the y-axis."},{"Start":"02:52.970 ","End":"02:55.220","Text":"If you have a little piece of curve,"},{"Start":"02:55.220 ","End":"02:57.470","Text":"and we multiply by the circumference,"},{"Start":"02:57.470 ","End":"03:01.785","Text":"the circumference is going to be 2Pi x."},{"Start":"03:01.785 ","End":"03:06.240","Text":"Then the same thing from here, like so."},{"Start":"03:06.240 ","End":"03:09.000","Text":"I don\u0027t need this anymore."},{"Start":"03:09.000 ","End":"03:11.230","Text":"A remark on notation,"},{"Start":"03:11.230 ","End":"03:14.615","Text":"in case you look on the internet,"},{"Start":"03:14.615 ","End":"03:20.575","Text":"you will sometimes see that this bit here is called ds."},{"Start":"03:20.575 ","End":"03:23.040","Text":"If I call this ds,"},{"Start":"03:23.040 ","End":"03:26.430","Text":"and I don\u0027t want to get into all the reasons why ds,"},{"Start":"03:26.430 ","End":"03:30.920","Text":"then these two formulas have a simpler expression."},{"Start":"03:30.920 ","End":"03:37.455","Text":"I also want to bring in the formula for arc length here, that is."},{"Start":"03:37.455 ","End":"03:40.835","Text":"All these formulas can become simplified."},{"Start":"03:40.835 ","End":"03:50.280","Text":"This one becomes, S equals 2Pi times the integral from Alpha to Beta of y, ds."},{"Start":"03:50.280 ","End":"04:00.570","Text":"This one becomes, S equals 2Pi times the integral from Alpha to Beta of x, ds."},{"Start":"04:00.570 ","End":"04:10.390","Text":"This one becomes, L equals the integral from Alpha to Beta of ds."},{"Start":"04:10.880 ","End":"04:17.450","Text":"This is just in case you encounter in other places the concept of ds."},{"Start":"04:17.450 ","End":"04:21.530","Text":"Like I said, in some places the 2Pi is written inside,"},{"Start":"04:21.530 ","End":"04:24.065","Text":"could be outside, it doesn\u0027t really matter."},{"Start":"04:24.065 ","End":"04:25.670","Text":"If you just remember,"},{"Start":"04:25.670 ","End":"04:29.300","Text":"the ds and all the other formulas are easier to remember,"},{"Start":"04:29.300 ","End":"04:33.120","Text":"in case you don\u0027t get a formula sheet."},{"Start":"04:33.860 ","End":"04:36.185","Text":"Now I\u0027ll do one example,"},{"Start":"04:36.185 ","End":"04:42.085","Text":"each of rotation about the x-axis or the y-axis."},{"Start":"04:42.085 ","End":"04:52.790","Text":"For Exercise 1, I\u0027m given the curve r equals 5 minus 4 sine Theta."},{"Start":"04:52.790 ","End":"04:59.100","Text":"I\u0027m given it in the range where Theta goes from 0-Pi."},{"Start":"04:59.590 ","End":"05:06.335","Text":"I wrote out the problem that we have to just set up or express the integral,"},{"Start":"05:06.335 ","End":"05:08.590","Text":"not to evaluate it;"},{"Start":"05:08.590 ","End":"05:11.870","Text":"the integral that gives the surface area of"},{"Start":"05:11.870 ","End":"05:15.410","Text":"this curve when it\u0027s rotated about the x-axis,"},{"Start":"05:15.410 ","End":"05:17.860","Text":"so we need to use this formula."},{"Start":"05:17.860 ","End":"05:21.929","Text":"The reason we don\u0027t usually evaluate is the problems,"},{"Start":"05:21.929 ","End":"05:27.680","Text":"the integrals come out so difficult and we\u0027re not here to teach how to do integrals,"},{"Start":"05:27.680 ","End":"05:29.660","Text":"just how to set it up."},{"Start":"05:29.660 ","End":"05:37.960","Text":"I\u0027m going to use this formula for the rotation about the x-axis, and let\u0027s see."},{"Start":"05:37.960 ","End":"05:39.560","Text":"I forgot to mention,"},{"Start":"05:39.560 ","End":"05:43.730","Text":"we can assume that the curve is traced out exactly once."},{"Start":"05:43.730 ","End":"05:45.290","Text":"Don\u0027t have to prove that."},{"Start":"05:45.290 ","End":"05:49.280","Text":"I need to get all the bits and pieces."},{"Start":"05:49.280 ","End":"05:52.045","Text":"If we use this formula."},{"Start":"05:52.045 ","End":"05:55.470","Text":"Well, we still need r, and we need dr over d Theta."},{"Start":"05:55.470 ","End":"05:58.060","Text":"Let\u0027s just write everything out."},{"Start":"05:58.760 ","End":"06:02.475","Text":"We have that r, that was given,"},{"Start":"06:02.475 ","End":"06:06.200","Text":"is 5 minus 4 sine Theta."},{"Start":"06:06.200 ","End":"06:09.095","Text":"Then we need dr over d Theta."},{"Start":"06:09.095 ","End":"06:12.770","Text":"Dr over d Theta is the derivative of this,"},{"Start":"06:12.770 ","End":"06:19.050","Text":"which is just minus 4 cosine Theta."},{"Start":"06:19.190 ","End":"06:25.830","Text":"Then we\u0027ll need also y. Y,"},{"Start":"06:25.830 ","End":"06:30.435","Text":"which is r sine Theta,"},{"Start":"06:30.435 ","End":"06:32.310","Text":"is actually equal to,"},{"Start":"06:32.310 ","End":"06:37.710","Text":"if I replace r with what it originally is here,"},{"Start":"06:37.710 ","End":"06:45.130","Text":"is 5 minus 4 sine Theta, sine Theta."},{"Start":"06:45.140 ","End":"06:49.200","Text":"Of course this is the Alpha and this is the Beta."},{"Start":"06:49.200 ","End":"06:56.220","Text":"What we get is we get that"},{"Start":"06:56.220 ","End":"07:06.090","Text":"S is 2Pi times the integral from 0-Pi of,"},{"Start":"07:06.090 ","End":"07:09.925","Text":"now I can use this form here;"},{"Start":"07:09.925 ","End":"07:14.510","Text":"y, which is this thing here,"},{"Start":"07:14.510 ","End":"07:23.700","Text":"5 minus 4 sine Theta, sine Theta."},{"Start":"07:23.700 ","End":"07:28.470","Text":"Then the square root of r-squared,"},{"Start":"07:28.470 ","End":"07:36.960","Text":"which is 5 minus 4 sine Theta squared,"},{"Start":"07:36.960 ","End":"07:44.235","Text":"plus dr over d Theta squared."},{"Start":"07:44.235 ","End":"07:46.615","Text":"I don\u0027t need the minus."},{"Start":"07:46.615 ","End":"07:51.860","Text":"I can just take 4 cosine Theta squared."},{"Start":"07:51.860 ","End":"07:55.425","Text":"Then all this d Theta."},{"Start":"07:55.425 ","End":"07:58.110","Text":"Now here, this is an expression."},{"Start":"07:58.110 ","End":"07:59.550","Text":"It does answer the question,"},{"Start":"07:59.550 ","End":"08:01.230","Text":"but we want to simplify it a bit,"},{"Start":"08:01.230 ","End":"08:03.200","Text":"if not evaluate it."},{"Start":"08:03.200 ","End":"08:05.380","Text":"I don\u0027t want to drag everything with me,"},{"Start":"08:05.380 ","End":"08:09.800","Text":"just a bit under the square root sign can be simplified,"},{"Start":"08:09.800 ","End":"08:11.660","Text":"and I\u0027ll do that at the side."},{"Start":"08:11.660 ","End":"08:16.370","Text":"What we get is from here using the formula for a minus b squared,"},{"Start":"08:16.370 ","End":"08:19.865","Text":"it\u0027s a squared minus 2ab,"},{"Start":"08:19.865 ","End":"08:25.875","Text":"2 times 5 times 4 times sine Theta,"},{"Start":"08:25.875 ","End":"08:29.960","Text":"plus 4 squared sine squared Theta,"},{"Start":"08:29.960 ","End":"08:35.195","Text":"which is 16 sine squared Theta."},{"Start":"08:35.195 ","End":"08:39.110","Text":"5 squared is 25,"},{"Start":"08:39.110 ","End":"08:42.950","Text":"and 2 times 5 times 4 is 40."},{"Start":"08:42.950 ","End":"08:45.635","Text":"But I still haven\u0027t taken the last bit,"},{"Start":"08:45.635 ","End":"08:47.165","Text":"which is 4 squared,"},{"Start":"08:47.165 ","End":"08:56.130","Text":"which is 16, and cosine squared Theta."},{"Start":"08:56.130 ","End":"09:00.020","Text":"Now the only thing that really simplifies is that because"},{"Start":"09:00.020 ","End":"09:04.835","Text":"sine squared Theta plus cosine squared Theta is 1,"},{"Start":"09:04.835 ","End":"09:08.390","Text":"this bit just comes out to be 16."},{"Start":"09:08.390 ","End":"09:17.540","Text":"Now I can rewrite this integral as equal to 2Pi integral from 0-Pi."},{"Start":"09:17.540 ","End":"09:19.715","Text":"This part the same,"},{"Start":"09:19.715 ","End":"09:22.685","Text":"5 minus 4 sine Theta,"},{"Start":"09:22.685 ","End":"09:32.715","Text":"sine Theta times the square root of 25 plus 16 is 41,"},{"Start":"09:32.715 ","End":"09:37.600","Text":"minus 40 sine Theta."},{"Start":"09:37.600 ","End":"09:38.950","Text":"That\u0027s under the square root sign,"},{"Start":"09:38.950 ","End":"09:42.295","Text":"this bit here, d Theta."},{"Start":"09:42.295 ","End":"09:45.795","Text":"That\u0027s one heck of an integral."},{"Start":"09:45.795 ","End":"09:48.385","Text":"That\u0027s why you\u0027re not asked to evaluate it,"},{"Start":"09:48.385 ","End":"09:50.320","Text":"just to express it."},{"Start":"09:50.320 ","End":"09:59.175","Text":"That\u0027s an example of one type of rotation about the x-axis."},{"Start":"09:59.175 ","End":"10:03.095","Text":"Now let\u0027s do one about the y-axis."},{"Start":"10:03.095 ","End":"10:05.320","Text":"I\u0027ll erase this."},{"Start":"10:05.320 ","End":"10:07.480","Text":"Let\u0027s see what I can reuse."},{"Start":"10:07.480 ","End":"10:09.190","Text":"Going to change the 1 to a 2,"},{"Start":"10:09.190 ","End":"10:10.960","Text":"I\u0027m going to change the x to a y,"},{"Start":"10:10.960 ","End":"10:12.909","Text":"and I\u0027m going to choose a different curve,"},{"Start":"10:12.909 ","End":"10:15.290","Text":"but the rest can stay."},{"Start":"10:15.290 ","End":"10:21.710","Text":"Problem 2, something is rotated about the y-axis and the curve will be,"},{"Start":"10:21.710 ","End":"10:26.705","Text":"r equals cosine squared Theta."},{"Start":"10:26.705 ","End":"10:32.345","Text":"Theta is taken between minus Pi over 6,"},{"Start":"10:32.345 ","End":"10:36.860","Text":"that\u0027s minus 30 degrees and plus 30 degrees,"},{"Start":"10:36.860 ","End":"10:39.570","Text":"but we need to work in radians."},{"Start":"10:40.000 ","End":"10:42.800","Text":"Very similar to before,"},{"Start":"10:42.800 ","End":"10:52.185","Text":"we have our x which is r cosine Theta."},{"Start":"10:52.185 ","End":"10:54.330","Text":"Let\u0027s just plug it in."},{"Start":"10:54.330 ","End":"10:59.840","Text":"We get that S equals the"},{"Start":"10:59.840 ","End":"11:05.365","Text":"integral from minus Pi over 6-Pi over 6."},{"Start":"11:05.365 ","End":"11:09.090","Text":"I didn\u0027t leave room for the 2pi, doesn\u0027t matter."},{"Start":"11:09.090 ","End":"11:11.265","Text":"You can put it inside, and later pull it out."},{"Start":"11:11.265 ","End":"11:14.430","Text":"2Pi times the integral."},{"Start":"11:14.430 ","End":"11:17.925","Text":"Now x is r cosine Theta."},{"Start":"11:17.925 ","End":"11:22.215","Text":"But r is this."},{"Start":"11:22.215 ","End":"11:25.920","Text":"This is the r cosine squared Theta,"},{"Start":"11:25.920 ","End":"11:31.450","Text":"I\u0027ll just remind us that this was the r, times cosine Theta."},{"Start":"11:32.330 ","End":"11:36.540","Text":"All these was our x."},{"Start":"11:36.540 ","End":"11:41.490","Text":"Now we need the square root of r-squared."},{"Start":"11:41.490 ","End":"11:43.400","Text":"R is cosine squared Theta,"},{"Start":"11:43.400 ","End":"11:48.120","Text":"so r-squared is cosine^4 Theta."},{"Start":"11:48.280 ","End":"11:52.665","Text":"Dr over d Theta is equal to,"},{"Start":"11:52.665 ","End":"11:56.330","Text":"if r is cosine squared Theta, the derivative."},{"Start":"11:56.330 ","End":"11:58.580","Text":"First of all, it\u0027s like something squared,"},{"Start":"11:58.580 ","End":"12:01.715","Text":"so it\u0027s two cosine Theta."},{"Start":"12:01.715 ","End":"12:04.954","Text":"But then we need the inner derivative of cosine Theta,"},{"Start":"12:04.954 ","End":"12:08.135","Text":"which is minus sine Theta."},{"Start":"12:08.135 ","End":"12:12.060","Text":"If I put that in here, we get plus,"},{"Start":"12:12.060 ","End":"12:14.280","Text":"because it\u0027s squared don\u0027t need the minus,"},{"Start":"12:14.280 ","End":"12:17.910","Text":"2 squared is 4."},{"Start":"12:17.910 ","End":"12:21.185","Text":"We get cosine squared sine squared,"},{"Start":"12:21.185 ","End":"12:27.079","Text":"cosine squared Theta sine squared Theta,"},{"Start":"12:27.079 ","End":"12:30.085","Text":"and finally d Theta."},{"Start":"12:30.085 ","End":"12:33.290","Text":"Now just want to simplify this a bit."},{"Start":"12:33.290 ","End":"12:36.695","Text":"I\u0027ll just do a little bit of simplification."},{"Start":"12:36.695 ","End":"12:40.385","Text":"What we can get here is the integral."},{"Start":"12:40.385 ","End":"12:46.830","Text":"That\u0027s all we can take the 2Pi out, so 2Pi here."},{"Start":"12:46.830 ","End":"12:52.530","Text":"Cosine squared times cosine is cosine cubed."},{"Start":"12:53.750 ","End":"13:00.500","Text":"Now look, here I could take cosine squared outside the brackets,"},{"Start":"13:00.500 ","End":"13:06.410","Text":"but the square root of cosine squared Theta,"},{"Start":"13:06.410 ","End":"13:09.440","Text":"it\u0027s the absolute value of cosine Theta."},{"Start":"13:09.440 ","End":"13:12.515","Text":"But because cosine is positive in this range,"},{"Start":"13:12.515 ","End":"13:15.395","Text":"between minus 30 and 30 degrees,"},{"Start":"13:15.395 ","End":"13:17.765","Text":"I can drop the bars."},{"Start":"13:17.765 ","End":"13:21.890","Text":"What I end up with after taking the cosine squared out,"},{"Start":"13:21.890 ","End":"13:25.820","Text":"would be the square root of"},{"Start":"13:25.820 ","End":"13:32.830","Text":"cosine squared Theta plus 4 sine squared Theta, d Theta."},{"Start":"13:32.830 ","End":"13:36.240","Text":"But I still need this cosine Theta that I took out,"},{"Start":"13:36.240 ","End":"13:38.825","Text":"so instead of putting an extra cosine Theta,"},{"Start":"13:38.825 ","End":"13:41.470","Text":"I\u0027ll raise this power from 3 to 4."},{"Start":"13:41.470 ","End":"13:45.980","Text":"Of course, I have to write the limits of integration, that\u0027s a must."},{"Start":"13:45.980 ","End":"13:49.140","Text":"I think we\u0027ll call it a day."}],"Thumbnail":null,"ID":10593},{"Watched":false,"Name":"Exercise 13","Duration":"6m 56s","ChapterTopicVideoID":10060,"CourseChapterTopicPlaylistID":8900,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this exercise, we\u0027re going to compute"},{"Start":"00:02.370 ","End":"00:08.295","Text":"a surface of revolution in polar coordinates."},{"Start":"00:08.295 ","End":"00:10.890","Text":"The curve is r equals"},{"Start":"00:10.890 ","End":"00:14.550","Text":"cosine Theta from 0-90,"},{"Start":"00:14.550 ","End":"00:17.250","Text":"that is from 0 to Pi over 2."},{"Start":"00:17.250 ","End":"00:21.000","Text":"This curve, which is illustrated here,"},{"Start":"00:21.000 ","End":"00:24.240","Text":"we\u0027re going to rotate it or revolve it"},{"Start":"00:24.240 ","End":"00:28.515","Text":"around the x-axis this way."},{"Start":"00:28.515 ","End":"00:32.415","Text":"Then we\u0027d get a sphere."},{"Start":"00:32.415 ","End":"00:35.340","Text":"Later we\u0027ll check our result"},{"Start":"00:35.340 ","End":"00:39.960","Text":"using solid geometry without calculus."},{"Start":"00:39.960 ","End":"00:43.715","Text":"But first, we\u0027ll do it with the help of an integral."},{"Start":"00:43.715 ","End":"00:47.599","Text":"Note that this makes sense, this range,"},{"Start":"00:47.599 ","End":"00:50.600","Text":"because cosine goes from 1 down to 0"},{"Start":"00:50.600 ","End":"00:52.670","Text":"as we go from 0-90."},{"Start":"00:52.670 ","End":"00:55.080","Text":"Here, the cosine is 1"},{"Start":"00:55.080 ","End":"00:56.840","Text":"and it gets smaller and smaller."},{"Start":"00:56.840 ","End":"00:58.370","Text":"The radius gets smaller and smaller"},{"Start":"00:58.370 ","End":"01:00.590","Text":"when we get to 90 degrees or Pi over 2,"},{"Start":"01:00.590 ","End":"01:02.305","Text":"we\u0027re down to 0."},{"Start":"01:02.305 ","End":"01:04.370","Text":"The diagram makes sense although"},{"Start":"01:04.370 ","End":"01:07.665","Text":"we don\u0027t need it for the computation."},{"Start":"01:07.665 ","End":"01:09.010","Text":"Let\u0027s see now,"},{"Start":"01:09.010 ","End":"01:12.020","Text":"the formula we want is as follows."},{"Start":"01:12.020 ","End":"01:13.550","Text":"We\u0027ll give it in general."},{"Start":"01:13.550 ","End":"01:17.510","Text":"As in general is 2 Pi times the integral"},{"Start":"01:17.510 ","End":"01:19.850","Text":"from the start angle of Theta"},{"Start":"01:19.850 ","End":"01:25.620","Text":"to the end angle of Theta of y"},{"Start":"01:25.620 ","End":"01:32.580","Text":"times the square root of r squared"},{"Start":"01:32.580 ","End":"01:39.480","Text":"plus dr by d Theta squared, d Theta."},{"Start":"01:39.480 ","End":"01:46.050","Text":"Of course y, we have to replace by sine Thetas."},{"Start":"01:46.050 ","End":"01:49.050","Text":"Of course, it\u0027s sine Theta,"},{"Start":"01:49.050 ","End":"01:53.948","Text":"x is our cosine Theta."},{"Start":"01:53.948 ","End":"01:58.500","Text":"We need dr by d Theta,"},{"Start":"01:58.500 ","End":"02:02.655","Text":"which is just a simple differentiation."},{"Start":"02:02.655 ","End":"02:07.690","Text":"This is going to be minus sine Theta."},{"Start":"02:08.030 ","End":"02:13.140","Text":"Just to plug this in here as we said,"},{"Start":"02:13.140 ","End":"02:18.345","Text":"y is r times sine Theta."},{"Start":"02:18.345 ","End":"02:24.730","Text":"We\u0027ll get this is equal to 2Pi integral."},{"Start":"02:24.730 ","End":"02:29.065","Text":"Now, we said from 0 to Pi over 2,"},{"Start":"02:29.065 ","End":"02:32.230","Text":"the y is r."},{"Start":"02:32.230 ","End":"02:35.020","Text":"But r is cosine Theta,"},{"Start":"02:35.020 ","End":"02:40.070","Text":"so it\u0027s cosine Theta sine Theta."},{"Start":"02:40.070 ","End":"02:43.380","Text":"This is the r sine Theta."},{"Start":"02:43.380 ","End":"02:44.893","Text":"All this is y."},{"Start":"02:44.893 ","End":"02:49.990","Text":"Then the square root of r squared,"},{"Start":"02:49.990 ","End":"02:54.185","Text":"which is cosine squared Theta"},{"Start":"02:54.185 ","End":"03:00.500","Text":"plus dr by d Theta squared,"},{"Start":"03:00.500 ","End":"03:06.755","Text":"which is minus sine Theta squared d Theta."},{"Start":"03:06.755 ","End":"03:08.840","Text":"Now, notice that what\u0027s under"},{"Start":"03:08.840 ","End":"03:10.970","Text":"the square root sign is cosine squared"},{"Start":"03:10.970 ","End":"03:13.180","Text":"plus sine squared and that\u0027s 1."},{"Start":"03:13.180 ","End":"03:15.440","Text":"I can basically just cross this"},{"Start":"03:15.440 ","End":"03:17.435","Text":"whole thing out, that\u0027s 1."},{"Start":"03:17.435 ","End":"03:19.295","Text":"The other thing I\u0027m going to do,"},{"Start":"03:19.295 ","End":"03:21.740","Text":"there\u0027s more than 1 way to do this integral,"},{"Start":"03:21.740 ","End":"03:24.050","Text":"you could do it by substitution."},{"Start":"03:24.050 ","End":"03:28.025","Text":"Let u equals sine Thetas long way."},{"Start":"03:28.025 ","End":"03:30.560","Text":"The other way is to use trigonometric identities."},{"Start":"03:30.560 ","End":"03:33.140","Text":"I\u0027m going to use the trigonometric identity,"},{"Start":"03:33.140 ","End":"03:41.070","Text":"that 2 cosine Theta sine Theta is sine of 2 Theta."},{"Start":"03:41.070 ","End":"03:43.520","Text":"I think an identity will work better"},{"Start":"03:43.520 ","End":"03:46.350","Text":"than a substitution a bit easier."},{"Start":"03:47.060 ","End":"03:51.020","Text":"We get Pi because the 2"},{"Start":"03:51.020 ","End":"03:52.220","Text":"is going to go in here,"},{"Start":"03:52.220 ","End":"03:54.980","Text":"and we\u0027re going to use that times the integral."},{"Start":"03:54.980 ","End":"03:57.140","Text":"Now, we have sine 2 Theta,"},{"Start":"03:57.140 ","End":"04:01.345","Text":"because the 2 with this gives me this,"},{"Start":"04:01.345 ","End":"04:03.720","Text":"d Theta and the limits"},{"Start":"04:03.720 ","End":"04:08.010","Text":"are from 0 to Pi over 2."},{"Start":"04:08.010 ","End":"04:11.190","Text":"The integral of sine 2 theta"},{"Start":"04:11.190 ","End":"04:17.670","Text":"would be minus cosine 2 Theta,"},{"Start":"04:17.670 ","End":"04:20.295","Text":"if it was just sine of something."},{"Start":"04:20.295 ","End":"04:21.690","Text":"But because of the 2,"},{"Start":"04:21.690 ","End":"04:26.560","Text":"we need to divide by 2."},{"Start":"04:30.260 ","End":"04:34.600","Text":"Let me see, I\u0027ll put the 1/2 here."},{"Start":"04:38.260 ","End":"04:40.550","Text":"I\u0027ll just write down what I mean,"},{"Start":"04:40.550 ","End":"04:41.990","Text":"and then I\u0027ll show you."},{"Start":"04:41.990 ","End":"04:45.110","Text":"The minus from the cosine,"},{"Start":"04:45.110 ","End":"04:46.520","Text":"I put in front."},{"Start":"04:46.520 ","End":"04:49.605","Text":"The half, I put under the Pi."},{"Start":"04:49.605 ","End":"04:52.820","Text":"This is what the indefinite integral is."},{"Start":"04:52.820 ","End":"04:55.025","Text":"I need to evaluate it between these limits."},{"Start":"04:55.025 ","End":"04:57.155","Text":"I don\u0027t need to do it on the constant."},{"Start":"04:57.155 ","End":"05:03.820","Text":"I can just do it on here from 0 to Pi over 2."},{"Start":"05:04.290 ","End":"05:07.670","Text":"Let\u0027s see what I get."},{"Start":"05:10.040 ","End":"05:13.820","Text":"The minus means I can reverse"},{"Start":"05:13.820 ","End":"05:17.210","Text":"the order of subtraction if I switch these 2,"},{"Start":"05:17.210 ","End":"05:18.935","Text":"and get rid of this minus,"},{"Start":"05:18.935 ","End":"05:22.295","Text":"and I\u0027ve got Pi over 2."},{"Start":"05:22.295 ","End":"05:31.310","Text":"I\u0027ll take the cosine of twice 0, which is 0,"},{"Start":"05:31.310 ","End":"05:38.315","Text":"minus the cosine of twice Pi over 2, which is Pi."},{"Start":"05:38.315 ","End":"05:42.365","Text":"I did the lower limit minus the upper limit,"},{"Start":"05:42.365 ","End":"05:43.910","Text":"because of the minus here,"},{"Start":"05:43.910 ","End":"05:45.410","Text":"because if you reverse a subtraction."},{"Start":"05:45.410 ","End":"05:46.670","Text":"I think you\u0027re following."},{"Start":"05:46.670 ","End":"05:50.395","Text":"Now, cosine of 0 is 1."},{"Start":"05:50.395 ","End":"05:53.490","Text":"Cosine of Pi is minus 1."},{"Start":"05:53.490 ","End":"05:56.240","Text":"1 minus minus 1 is 2,"},{"Start":"05:56.240 ","End":"05:57.842","Text":"which will cancel with this 2,"},{"Start":"05:57.842 ","End":"06:03.450","Text":"and so the answer is just Pi square units."},{"Start":"06:03.950 ","End":"06:08.000","Text":"I said we could have done this with geometry"},{"Start":"06:08.000 ","End":"06:10.460","Text":"if we identified this as a semicircle."},{"Start":"06:10.460 ","End":"06:12.875","Text":"I\u0027ll just do this for a check."},{"Start":"06:12.875 ","End":"06:15.440","Text":"The formula for surface area of a sphere"},{"Start":"06:15.440 ","End":"06:22.960","Text":"is S equals 4Pi times,"},{"Start":"06:22.960 ","End":"06:24.675","Text":"I don\u0027t want to use r,"},{"Start":"06:24.675 ","End":"06:26.250","Text":"because I\u0027ve used that here,"},{"Start":"06:26.250 ","End":"06:32.180","Text":"times say radius squared."},{"Start":"06:32.180 ","End":"06:37.085","Text":"Now, in this case, the radius is 1/2."},{"Start":"06:37.085 ","End":"06:40.400","Text":"What I get is 4 Pi."},{"Start":"06:40.400 ","End":"06:44.985","Text":"The radius is 1/2 squared."},{"Start":"06:44.985 ","End":"06:46.545","Text":"1/2 squared is a 1/4."},{"Start":"06:46.545 ","End":"06:50.080","Text":"The 1/4 cancels with the 4, so we get Pi,"},{"Start":"06:50.080 ","End":"06:53.389","Text":"so we\u0027ve got a verification of this result,"},{"Start":"06:53.389 ","End":"06:55.950","Text":"and we\u0027re done."}],"Thumbnail":null,"ID":9955},{"Watched":false,"Name":"Exercise 14","Duration":"10m 57s","ChapterTopicVideoID":10061,"CourseChapterTopicPlaylistID":8900,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this exercise, we have to find"},{"Start":"00:02.220 ","End":"00:05.340","Text":"the surface area obtained by revolving"},{"Start":"00:05.340 ","End":"00:09.435","Text":"a curve given in polar form as follows,"},{"Start":"00:09.435 ","End":"00:13.780","Text":"and rotating it about the y-axis."},{"Start":"00:14.000 ","End":"00:20.385","Text":"Note that Theta goes from minus 90 degrees to 90 degrees."},{"Start":"00:20.385 ","End":"00:24.000","Text":"That is minus Pi over 2 to Pi over 2."},{"Start":"00:24.000 ","End":"00:28.485","Text":"Just by the way, this curve is called a cardioid,"},{"Start":"00:28.485 ","End":"00:33.870","Text":"or at least if you do the full circle and you\u0027d get another mirror image of this,"},{"Start":"00:33.870 ","End":"00:36.360","Text":"you would get a heart shape called the cardioid."},{"Start":"00:36.360 ","End":"00:41.520","Text":"Anyway. Let\u0027s just do a preliminary check to see everything is okay."},{"Start":"00:41.520 ","End":"00:47.195","Text":"Theta can be minus Pi over 2 here,"},{"Start":"00:47.195 ","End":"00:50.630","Text":"0 here, and pi over 2 here."},{"Start":"00:50.630 ","End":"00:53.420","Text":"Let\u0027s just see that everything\u0027s okay."},{"Start":"00:53.420 ","End":"00:56.435","Text":"When Theta is minus Pi over 2,"},{"Start":"00:56.435 ","End":"01:04.625","Text":"then r is 4 plus 4 times minus 1 because sine of minus Pi over 2 is minus 1, which is 0."},{"Start":"01:04.625 ","End":"01:07.700","Text":"That would give us this point here."},{"Start":"01:07.700 ","End":"01:11.540","Text":"When Theta is 0 sine of Theta 0,"},{"Start":"01:11.540 ","End":"01:13.790","Text":"we get r equals 4,"},{"Start":"01:13.790 ","End":"01:16.810","Text":"so that would be this point."},{"Start":"01:16.810 ","End":"01:20.940","Text":"When Theta is Pi over 2 sine Theta is 1,"},{"Start":"01:20.940 ","End":"01:23.265","Text":"4 plus 4 is 8."},{"Start":"01:23.265 ","End":"01:25.875","Text":"Looks like we\u0027re on track."},{"Start":"01:25.875 ","End":"01:29.960","Text":"Let\u0027s use the formula now for surface area."},{"Start":"01:29.960 ","End":"01:34.325","Text":"Now it\u0027s the integral with respect to Theta."},{"Start":"01:34.325 ","End":"01:40.160","Text":"Theta goes from minus Pi over 2 to Pi over 2,"},{"Start":"01:40.160 ","End":"01:43.645","Text":"so we\u0027re going to have a d Theta at the end."},{"Start":"01:43.645 ","End":"01:46.820","Text":"Because it\u0027s about the y-axis,"},{"Start":"01:46.820 ","End":"01:49.760","Text":"we need this 2 Pi x factor,"},{"Start":"01:49.760 ","End":"01:53.959","Text":"the 2 Pi we put in front and the x here,"},{"Start":"01:53.959 ","End":"01:57.815","Text":"and then the remaining bit is the square root"},{"Start":"01:57.815 ","End":"02:03.490","Text":"of r squared plus the derivative of r squared."},{"Start":"02:03.490 ","End":"02:06.995","Text":"To emphasize its r with respect to Theta,"},{"Start":"02:06.995 ","End":"02:09.320","Text":"dr over d Theta squared,"},{"Start":"02:09.320 ","End":"02:11.605","Text":"r is a function of Theta."},{"Start":"02:11.605 ","End":"02:14.330","Text":"Of course, instead of x,"},{"Start":"02:14.330 ","End":"02:17.290","Text":"we\u0027ll put r cosine Theta,"},{"Start":"02:17.290 ","End":"02:22.350","Text":"and if we had y would be r sine Theta, the usual transformation."},{"Start":"02:22.350 ","End":"02:26.460","Text":"Let\u0027s see what we get."},{"Start":"02:26.460 ","End":"02:33.965","Text":"2 Pi integral from minus Pi over 2 to Pi over 2,"},{"Start":"02:33.965 ","End":"02:37.985","Text":"x we said is r cosine Theta"},{"Start":"02:37.985 ","End":"02:44.465","Text":"the square root of r-squared plus now we need dr by d Theta."},{"Start":"02:44.465 ","End":"02:50.179","Text":"From here, dr by d Theta constant 4 gives nothing."},{"Start":"02:50.179 ","End":"02:52.565","Text":"Derivative of sine is cosine,"},{"Start":"02:52.565 ","End":"02:56.690","Text":"so it\u0027s 4 cosine sine Theta."},{"Start":"02:56.690 ","End":"02:59.135","Text":"That\u0027s what we put here,"},{"Start":"02:59.135 ","End":"03:10.330","Text":"4 cosine sine Theta squared and then d Theta."},{"Start":"03:10.330 ","End":"03:16.130","Text":"What I\u0027d like to do is work on this as a side computation over here,"},{"Start":"03:16.130 ","End":"03:20.600","Text":"going to replace r by what it\u0027s equal to in terms of Theta."},{"Start":"03:20.600 ","End":"03:25.250","Text":"This thing here becomes"},{"Start":"03:25.250 ","End":"03:30.570","Text":"4 plus 4 sine Theta"},{"Start":"03:31.580 ","End":"03:39.270","Text":"squared plus 4 cosine Theta squared."},{"Start":"03:39.270 ","End":"03:42.150","Text":"What this gives us,"},{"Start":"03:42.150 ","End":"03:46.740","Text":"we can obviously take 4 out of here and 4 out of here,"},{"Start":"03:46.740 ","End":"03:47.835","Text":"and since it\u0027s squared,"},{"Start":"03:47.835 ","End":"03:50.295","Text":"we get 4 squared."},{"Start":"03:50.295 ","End":"03:55.480","Text":"Then we have 1 plus sine Theta squared,"},{"Start":"03:58.070 ","End":"04:02.790","Text":"and it matters while expanded already,"},{"Start":"04:02.790 ","End":"04:07.410","Text":"so 1 plus 2"},{"Start":"04:07.410 ","End":"04:13.700","Text":"sine Theta plus sine squared Theta,"},{"Start":"04:13.700 ","End":"04:15.745","Text":"that\u0027s 1 plus sine theta,"},{"Start":"04:15.745 ","End":"04:18.385","Text":"all squared by the binomial expansion."},{"Start":"04:18.385 ","End":"04:24.370","Text":"Here, just cosine squared Theta,"},{"Start":"04:25.010 ","End":"04:30.865","Text":"brackets here and all this is still within 4 squared times this."},{"Start":"04:30.865 ","End":"04:37.400","Text":"Now, notice that sine squared plus cosine squared together is 1,"},{"Start":"04:37.400 ","End":"04:47.509","Text":"with this 1 will give us 16 times 2 plus 2 sine Theta."},{"Start":"04:47.600 ","End":"04:51.135","Text":"Now let\u0027s get back here."},{"Start":"04:51.135 ","End":"04:55.945","Text":"I\u0027ll tell you my strategy for doing this integral."},{"Start":"04:55.945 ","End":"05:04.810","Text":"Eventually, I\u0027m going to want to substitute 2 plus 2 sine Theta and call it,"},{"Start":"05:04.810 ","End":"05:06.640","Text":"I don\u0027t know, u."},{"Start":"05:06.640 ","End":"05:10.450","Text":"The reason this will be good is because the derivative of"},{"Start":"05:10.450 ","End":"05:15.110","Text":"u is 2 cosine Theta and up to a constant,"},{"Start":"05:15.110 ","End":"05:17.850","Text":"that\u0027s what we have."},{"Start":"05:18.190 ","End":"05:26.160","Text":"I can already say now that my plan is to say that u equals 2 plus 2 sine Theta,"},{"Start":"05:26.160 ","End":"05:34.770","Text":"and then du is going to be 2 cosine Theta d Theta."},{"Start":"05:34.770 ","End":"05:39.040","Text":"Now let\u0027s tidy up here with this in mind."},{"Start":"05:39.040 ","End":"05:46.465","Text":"What we get is 2 Pi the integral same limits."},{"Start":"05:46.465 ","End":"05:51.590","Text":"I still have to substitute this r here as 4 plus 4 sine Theta,"},{"Start":"05:51.590 ","End":"05:57.740","Text":"so it\u0027s 4, 1 plus sine Theta."},{"Start":"05:57.740 ","End":"06:01.190","Text":"Then we need a cosine Theta."},{"Start":"06:01.190 ","End":"06:07.355","Text":"Then this square root will be square root of 16 is 4."},{"Start":"06:07.355 ","End":"06:10.910","Text":"Then square root of"},{"Start":"06:10.910 ","End":"06:19.260","Text":"2 plus 2 sine Theta and then d Theta."},{"Start":"06:19.370 ","End":"06:24.795","Text":"Now I want to shape this to use just u and d u."},{"Start":"06:24.795 ","End":"06:29.525","Text":"What I\u0027m going to do is I\u0027ll take this 4 and just put it in front."},{"Start":"06:29.525 ","End":"06:33.445","Text":"This 4, I\u0027ll split into 2 times 2."},{"Start":"06:33.445 ","End":"06:41.079","Text":"I\u0027ll put 2 here to make it 2 plus 2 sine Theta and 2,"},{"Start":"06:41.570 ","End":"06:44.430","Text":"4 is 2 times 2,"},{"Start":"06:44.430 ","End":"06:47.760","Text":"2 here to make the 2 cosine Theta,"},{"Start":"06:47.760 ","End":"06:49.920","Text":"and if I do that,"},{"Start":"06:49.920 ","End":"07:00.605","Text":"then we get 2 Pi with the 4 is 8 Pi integral minus Pi over 2 to Pi over 2."},{"Start":"07:00.605 ","End":"07:10.690","Text":"Then we have 2 plus 2 sine Theta and then 2 cosine Theta,"},{"Start":"07:10.690 ","End":"07:16.560","Text":"and then the square root of 2 plus 2 sine Theta,"},{"Start":"07:17.360 ","End":"07:21.490","Text":"and finally d Theta."},{"Start":"07:22.370 ","End":"07:28.560","Text":"At this point is where we\u0027ll make the substitution,"},{"Start":"07:28.560 ","End":"07:35.775","Text":"so what we will get will be 8 Pi integral."},{"Start":"07:35.775 ","End":"07:38.165","Text":"I\u0027ll leave the limits for a moment."},{"Start":"07:38.165 ","End":"07:41.720","Text":"Then here we have u."},{"Start":"07:41.720 ","End":"07:47.450","Text":"Here we have 2 cosine Theta d Theta,"},{"Start":"07:47.450 ","End":"07:51.500","Text":"this with this, and this is just du,"},{"Start":"07:51.500 ","End":"07:53.870","Text":"so I\u0027ll put that at the end."},{"Start":"07:53.870 ","End":"08:03.160","Text":"Then we have the square root of this thing here, it\u0027s just u."},{"Start":"08:04.640 ","End":"08:11.825","Text":"The integral, we also have to substitute the limits. Let\u0027s see."},{"Start":"08:11.825 ","End":"08:18.814","Text":"When Theta is equal to minus Pi over 2,"},{"Start":"08:18.814 ","End":"08:27.680","Text":"that gives us that u is equal to 2 plus 2 sine Theta,"},{"Start":"08:27.680 ","End":"08:33.205","Text":"comes out 0 because sine Pi over 2 is negative 1."},{"Start":"08:33.205 ","End":"08:37.370","Text":"When Theta is plus Pi over 2,"},{"Start":"08:37.370 ","End":"08:42.065","Text":"that gives us that u equals 2 plus 2 is 4."},{"Start":"08:42.065 ","End":"08:49.335","Text":"The integral du is from 0-4 of this."},{"Start":"08:49.335 ","End":"08:55.395","Text":"This is an easy 1 because we can use exponents,"},{"Start":"08:55.395 ","End":"08:59.330","Text":"square root of u is u to the half,"},{"Start":"08:59.330 ","End":"09:01.370","Text":"and this is u^1."},{"Start":"09:01.370 ","End":"09:07.060","Text":"What we have is 8 Pi integral of u^1.5,"},{"Start":"09:07.060 ","End":"09:14.535","Text":"u^3/2 du."},{"Start":"09:14.535 ","End":"09:20.880","Text":"Then just raise the power by 1 and divide by the new power."},{"Start":"09:20.880 ","End":"09:24.210","Text":"We have 8 Pi,"},{"Start":"09:24.210 ","End":"09:30.115","Text":"and then we\u0027ll have u^5/2."},{"Start":"09:30.115 ","End":"09:32.180","Text":"Then we have to divide by it,"},{"Start":"09:32.180 ","End":"09:42.310","Text":"so it\u0027s 2/5 and all this is going to be evaluated from 0-4."},{"Start":"09:42.310 ","End":"09:51.660","Text":"We can put the constants together and then just substitute u from 0-4."},{"Start":"09:52.060 ","End":"09:57.905","Text":"What we get is 8 Pi times 2 over 5"},{"Start":"09:57.905 ","End":"10:03.530","Text":"is 16 Pi over 5."},{"Start":"10:03.530 ","End":"10:06.140","Text":"Now, when u is 4,"},{"Start":"10:06.140 ","End":"10:12.630","Text":"we need 4^5/2 minus"},{"Start":"10:14.200 ","End":"10:17.210","Text":"0^5/2 plus 0 to this power,"},{"Start":"10:17.210 ","End":"10:21.980","Text":"is just 0, 4^5/2 we can take first"},{"Start":"10:21.980 ","End":"10:27.290","Text":"the square root and then raised to the power of 5 comes out 32."},{"Start":"10:27.290 ","End":"10:35.100","Text":"We have 16 Pi over 5 times 32."},{"Start":"10:35.700 ","End":"10:40.920","Text":"Let\u0027s see 16 times 32 is 512,"},{"Start":"10:40.920 ","End":"10:49.525","Text":"so 512 over 5 times Pi."},{"Start":"10:49.525 ","End":"10:51.620","Text":"I won\u0027t evaluate this,"},{"Start":"10:51.620 ","End":"10:57.840","Text":"just highlight it, and that\u0027s our answer, and we\u0027re done."}],"Thumbnail":null,"ID":9956}],"ID":8900}]

[{"ID":4005,"Videos":[9936,9937]},{"ID":8896,"Videos":[9938,9939,9940,9941,9942,9943,9948]},{"ID":8897,"Videos":[10590,9949,9950]},{"ID":8898,"Videos":[10592,9953,9954]},{"ID":8899,"Videos":[10591,9951,9952]},{"ID":8900,"Videos":[10593,9955,9956]}];

[10593,9955,9956];

1.1

1

Get unlimited access to **1500 subjects** including **personalized modules**

Start your free trial
We couldn't find any results for

Upload your syllabus now and our team will create a customized module especially for you!

Alert

and we will create a personalized module (just for you) in less than **48 hours...**