WEBVTT 00:00:00.980 --> 00:00:05.660 In this video, we're going to be looking at polar. 00:00:06.930 --> 00:00:13.672 Coordinates. Let's begin by actually 00:00:13.672 --> 00:00:20.096 looking at another coordinate system. The Cartesian coordinate 00:00:20.096 --> 00:00:23.355 system. Now in that system we 00:00:23.355 --> 00:00:28.817 take 2. Axes and X axis which is horizontal. 00:00:29.490 --> 00:00:36.094 And Y Axis which is vertical and a fixed .0 called the origin, 00:00:36.094 --> 00:00:41.682 which is where these two points cross. These two lines cross. 00:00:42.350 --> 00:00:49.420 Now we fix a point P in the plane by saying how far it's 00:00:49.420 --> 00:00:54.975 displaced along the X axis to give us the X coordinate. 00:00:55.550 --> 00:01:01.724 And how far it's displaced along the Y access to give us the Y 00:01:01.724 --> 00:01:03.929 coordinate and so we have. 00:01:04.490 --> 00:01:09.198 A point P which is uniquely described by its coordinates XY 00:01:09.198 --> 00:01:13.906 and notice I said how far it's displaced because it is 00:01:13.906 --> 00:01:17.758 displacement that we're talking about and not distance. That's 00:01:17.758 --> 00:01:22.894 what these arrowheads that we put on the axes are all about 00:01:22.894 --> 00:01:27.602 their about showings, in which direction we must move so that 00:01:27.602 --> 00:01:31.454 if we're moving down this direction, it's a negative 00:01:31.454 --> 00:01:34.450 distance and negative displacement that we're making. 00:01:34.680 --> 00:01:40.392 Now that is more than one way of describing where a point is in 00:01:40.392 --> 00:01:47.651 the plane. And we're going to be having a look at a system 00:01:47.651 --> 00:01:49.244 called polar coordinates. 00:01:49.250 --> 00:01:57.086 So in this system of polar coordinates, we take a poll. Oh, 00:01:57.086 --> 00:02:01.004 and we take a fixed line. 00:02:02.740 --> 00:02:08.538 Now, how can we describe a point in the plane using this fixed 00:02:08.538 --> 00:02:14.336 .0? The pole and this baseline. Here. One of the ways is to 00:02:14.336 --> 00:02:17.904 think of it as. What if we turn? 00:02:19.380 --> 00:02:26.753 Around. Centering on oh for the moment we rotate around, 00:02:26.753 --> 00:02:33.430 we can pass through a fixed angle. Let's call that theater. 00:02:33.990 --> 00:02:40.458 And then along this radius we can go a set distance. 00:02:40.970 --> 00:02:44.130 And we'll end up at a point P. 00:02:45.170 --> 00:02:51.098 And so the coordinates of that point would be our theater, and 00:02:51.098 --> 00:02:54.062 this is our system of polar. 00:02:54.600 --> 00:02:57.410 Coordinates. 00:02:59.490 --> 00:03:04.476 Now, just as we've got certain conventions with Cartesian 00:03:04.476 --> 00:03:08.354 coordinates, we have certain conventions with polar 00:03:08.354 --> 00:03:13.340 coordinates, and these are quite strong conventions, so let's 00:03:13.340 --> 00:03:19.988 have a look at what these are. First of all, theater is 00:03:19.988 --> 00:03:23.265 measured. In 00:03:23.265 --> 00:03:29.150 radians. So that's how first convention 00:03:29.150 --> 00:03:32.110 theater is measured in radians. 00:03:32.750 --> 00:03:35.816 2nd convention 00:03:35.816 --> 00:03:42.630 well. Our second convention is this that if this is our 00:03:42.630 --> 00:03:47.767 initial line and this is our poll, then we measure theater 00:03:47.767 --> 00:03:50.569 positive when we go round in 00:03:50.569 --> 00:03:56.296 that direction. Anticlockwise and we measure theater negative. 00:03:56.296 --> 00:04:02.470 When we go around in that direction which is 00:04:02.470 --> 00:04:09.634 clockwise. So in just the same way as we had an Arrowhead on 00:04:09.634 --> 00:04:14.554 our axes X&Y. In a sense, we've got arrowheads here, 00:04:14.554 --> 00:04:18.490 distinguishing a positive direction for theater and a 00:04:18.490 --> 00:04:20.458 negative direction for measuring 00:04:20.458 --> 00:04:24.588 theater. We have 1/3 convention to do with theater 00:04:24.588 --> 00:04:29.549 and that is that we never go further round this way. 00:04:30.680 --> 00:04:36.536 Number there's our poll. Oh, our fixed point. We never go further 00:04:36.536 --> 00:04:42.392 around this way then there, so theater is always less than or 00:04:42.392 --> 00:04:48.248 equal to pie and we never go round further that way than 00:04:48.248 --> 00:04:52.640 there again. So theater is always strictly greater than 00:04:52.640 --> 00:04:59.472 minus Π - Π ramped there plus Π round to there. And notice that 00:04:59.472 --> 00:05:06.208 we include. This bit of the line if you like this extended bit of 00:05:06.208 --> 00:05:12.144 the line by going route to their having the less than or equal to 00:05:12.144 --> 00:05:15.960 and having strictly greater than Theta strictly greater than 00:05:15.960 --> 00:05:17.232 minus pie there. 00:05:17.960 --> 00:05:24.002 1/4 Convention 1/4 convention is that 00:05:24.002 --> 00:05:27.023 our is always 00:05:27.023 --> 00:05:32.866 positive. One of the things that is quite important is that we be 00:05:32.866 --> 00:05:37.702 able to move from one system of coordinates to another. So the 00:05:37.702 --> 00:05:39.717 question is if we have. 00:05:40.230 --> 00:05:44.184 A point. In our 00:05:44.184 --> 00:05:50.756 XY plane. Who's coordinates are 00:05:50.756 --> 00:05:56.501 X&Y? How can we change from cartesians into 00:05:56.501 --> 00:06:02.144 pohlers? And how can we change back again, but one obvious 00:06:02.144 --> 00:06:08.813 thing to do is to associate the pole with the origin, and then 00:06:08.813 --> 00:06:13.430 to associate the initial line with the X axis. 00:06:13.950 --> 00:06:18.870 And then if we draw the radius out to pee. 00:06:19.920 --> 00:06:21.699 And that's our. 00:06:22.750 --> 00:06:25.708 And that is the angle theater. 00:06:26.350 --> 00:06:30.150 So we can see that in Cartesians, we're 00:06:30.150 --> 00:06:33.950 describing it as XY, and in Pohlers, where 00:06:33.950 --> 00:06:37.750 describing it as our theater. So what's the 00:06:37.750 --> 00:06:39.175 relationship between them? 00:06:40.270 --> 00:06:45.010 Let's drop that perpendicular down and we can see that this is 00:06:45.010 --> 00:06:50.145 a height. Why? Because of the Y coordinate the point and this is 00:06:50.145 --> 00:06:54.885 at a distance X because of the X coordinate of the point. 00:06:55.680 --> 00:07:02.960 And looking at that, we can see that Y is equal to R sign 00:07:02.960 --> 00:07:09.720 theater and X is equal to our cause theater. So given R and 00:07:09.720 --> 00:07:15.440 Theta, we can calculate X&Y. What about moving the other way 00:07:15.440 --> 00:07:22.200 will from Pythagoras? We can see that X squared plus Y squared is 00:07:22.200 --> 00:07:25.320 equal to R-squared. So give now 00:07:25.320 --> 00:07:30.415 X. I'm now why we can calculate all and we can also 00:07:30.415 --> 00:07:34.980 see that if we take the opposite over the adjacent, we 00:07:34.980 --> 00:07:41.205 have Y over X is equal to 10 theater. So given AY in an X, 00:07:41.205 --> 00:07:44.110 we can find out what theater is. 00:07:45.370 --> 00:07:45.970 Now. 00:07:46.990 --> 00:07:52.160 Always when doing these, it's best to draw sketches. If we're 00:07:52.160 --> 00:07:56.860 converting from one sort of point in Cartesians to its 00:07:56.860 --> 00:08:01.560 equivalent in Pohlers, or if we're moving back from Pohlers 00:08:01.560 --> 00:08:06.730 to cartesians, draw a picture, see where that point actually is 00:08:06.730 --> 00:08:12.840 now. Want to have a look at some examples. First of all, we're 00:08:12.840 --> 00:08:16.600 going to have a look at how to 00:08:16.600 --> 00:08:22.504 plot points. Then we're going to have a look at how to convert 00:08:22.504 --> 00:08:27.832 from one system into the other and vice versa. So let's begin 00:08:27.832 --> 00:08:30.230 with plotting. Plot. 00:08:31.490 --> 00:08:37.535 And what I'm going to do is I'm going to plot the following 00:08:37.535 --> 00:08:40.325 points and they're all in polar 00:08:40.325 --> 00:08:47.973 coordinates. I'm going to put them all on the same 00:08:47.973 --> 00:08:55.443 picture so we can get a feel for whereabouts things 00:08:55.443 --> 00:09:02.913 are in the polar play or the plane for the 00:09:02.913 --> 00:09:08.638 polar coordinates. So we put our poll, oh. 00:09:09.180 --> 00:09:11.790 And we have our initial line. 00:09:12.710 --> 00:09:15.314 First one that we've got to plot 00:09:15.314 --> 00:09:22.738 is 2. Pie so we know that Pi is the angle all the 00:09:22.738 --> 00:09:29.472 way around here, so there's pie to there and we want to go 00:09:29.472 --> 00:09:31.026 out to units. 00:09:31.630 --> 00:09:37.377 So it's there. This is the .2. 00:09:37.940 --> 00:09:39.490 Pie. 00:09:41.490 --> 00:09:46.950 Next to .1 N wealthy to is 0 so we're on the initial line. 00:09:48.110 --> 00:09:55.250 An one will be about there, so there is the .1 note. 00:09:56.770 --> 00:10:02.672 2 - Π by 3 - π by 3 means come around this 00:10:02.672 --> 00:10:07.666 way, and so minus π by three is about there, and 00:10:07.666 --> 00:10:12.206 we're coming around there minus π by three, and we 00:10:12.206 --> 00:10:17.654 want to come out a distance to, so that's roughly 2 out 00:10:17.654 --> 00:10:22.648 there, so this would be the .2 - π by 3. 00:10:23.790 --> 00:10:29.458 And finally, we've got the point. One 2/3 of Π. So we take 00:10:29.458 --> 00:10:32.074 the 2/3. That's going all the 00:10:32.074 --> 00:10:34.640 way around. To there. 00:10:35.190 --> 00:10:40.676 And we draw out through there, and we want a distance of one 00:10:40.676 --> 00:10:44.896 along there, which roughly called the scale we're using is 00:10:44.896 --> 00:10:47.428 about there, and so that's the 00:10:47.428 --> 00:10:49.930 .1. 2/3 of Π. 00:10:51.680 --> 00:10:58.100 Notice that we've taken theater first to establish in which 00:10:58.100 --> 00:11:00.668 direction were actually facing. 00:11:02.600 --> 00:11:06.255 OK, let's now have a 00:11:06.255 --> 00:11:11.995 look. Having got used to plotting points, let's now have 00:11:11.995 --> 00:11:16.435 a look in polar coordinates. These points 2. 00:11:17.070 --> 00:11:19.270 Minus Π by 2. 00:11:20.210 --> 00:11:26.210 1. 3/4 of 00:11:26.210 --> 00:11:33.380 Π. And 2 - π by three. Now these 00:11:33.380 --> 00:11:35.840 are all in Pohlers. 00:11:36.400 --> 00:11:41.801 What I want to do is convert them into cartesian coordinates. 00:11:41.801 --> 00:11:47.693 So first a picture whereabouts are they? And I'll do them one 00:11:47.693 --> 00:11:55.058 at a time. So let's take this one 2 - π by two initial point 00:11:55.058 --> 00:11:57.513 poll. Oh, and initial line. 00:11:58.440 --> 00:12:02.157 Minus Π by two? Well, that's coming down here. 00:12:02.730 --> 00:12:10.060 To there. So that's minus π by two, and we've 00:12:10.060 --> 00:12:13.660 come a distance to to 00:12:13.660 --> 00:12:18.820 there. Well, we don't need to do much calculation. I don't think 00:12:18.820 --> 00:12:23.760 to find this. If again we take our origin for our cartesians as 00:12:23.760 --> 00:12:28.700 being the pole, and we align the X axis with our initial line. 00:12:33.360 --> 00:12:40.315 And there's our X. There's RY and we can see straight away the 00:12:40.315 --> 00:12:47.805 point in Pohlers that's 2 - π by two in fact, goes to the 00:12:47.805 --> 00:12:54.225 point. In Cartesians, That's 0 - 2 because it's this point here 00:12:54.225 --> 00:13:01.715 on the Y axis, and it's 2 units below the X axis, so it's 00:13:01.715 --> 00:13:03.320 0 - 2. 00:13:03.370 --> 00:13:07.759 Notice how plotting the point actually saved as having to do 00:13:07.759 --> 00:13:12.148 any of the calculations. So let's take the next point now, 00:13:12.148 --> 00:13:14.542 which was one 3/4 of Π. 00:13:15.280 --> 00:13:23.160 1 3/4 of π. So again, let's plot where it 00:13:23.160 --> 00:13:26.560 is. Take our initial. 00:13:27.220 --> 00:13:34.500 .0 our poll and our initial line 3/4 of π going round. It's 00:13:34.500 --> 00:13:40.100 positive so it drought there be somewhere out along that. 00:13:40.670 --> 00:13:47.347 Direction there's our angle of 3/4 of Π, where somewhere out 00:13:47.347 --> 00:13:54.631 here at a distance one unit. So again, let's take our X&Y 00:13:54.631 --> 00:13:57.059 axes, our X axis. 00:13:58.010 --> 00:14:01.328 To be along the initial line. 00:14:01.850 --> 00:14:07.823 And now why access to be vertical and through the pole? 00:14:08.490 --> 00:14:14.634 Oh So that the polo becomes our origin of, and it's this point. 00:14:15.140 --> 00:14:16.229 But where after? 00:14:17.240 --> 00:14:24.170 Now how we going to work this out that remember the 00:14:24.170 --> 00:14:30.470 formula that we had was X equals our cause theater. 00:14:32.010 --> 00:14:39.678 Let's have a look at that. Are is one an we've got 00:14:39.678 --> 00:14:47.346 cause of 3/4 of Π and the cosine of 3/4 of Π 00:14:47.346 --> 00:14:55.014 is minus one over Route 2, so that's minus one over Route 00:14:55.014 --> 00:14:58.848 2. Why is our sign Theta? 00:14:58.880 --> 00:15:06.155 And so this is one times the sign of 3/4 of Π and the sign 00:15:06.155 --> 00:15:13.915 of 3/4 of Π is just one over Route 2, and so we have one over 00:15:13.915 --> 00:15:19.250 Route 2 for RY coordinate. And notice that these answers agree 00:15:19.250 --> 00:15:24.585 with where the point is in this particular quadrant. Negative X 00:15:24.585 --> 00:15:28.950 and positive Y, negative X and positive Y so. 00:15:28.950 --> 00:15:33.617 Even if I've got the calculation wrong in the sense that I, even 00:15:33.617 --> 00:15:37.566 if I've done the arithmetic wrong, have no, I've got the 00:15:37.566 --> 00:15:39.361 point in the right quadrant. 00:15:39.420 --> 00:15:46.130 Let's have a look at the last one of these two. 00:15:46.900 --> 00:15:50.386 Minus Π by 00:15:50.386 --> 00:15:54.350 3. So again, our poll. 00:15:56.020 --> 00:15:59.050 Our initial line. 00:15:59.050 --> 00:16:02.179 Minus Π by three is around here. 00:16:04.760 --> 00:16:11.533 So we've come around there minus π by three, and we're out a 00:16:11.533 --> 00:16:13.617 distance, two along there. 00:16:15.440 --> 00:16:20.550 Take our X axis to coincide with the initial line. 00:16:21.090 --> 00:16:22.500 And now origin. 00:16:23.410 --> 00:16:25.910 Coincide with the pole. 00:16:28.360 --> 00:16:35.170 Let's write down our equations that tell us X is 00:16:35.170 --> 00:16:41.980 our cause theater, which is 2 times the cosine of 00:16:41.980 --> 00:16:48.109 minus π by three, which is equal to 2. 00:16:49.720 --> 00:16:56.980 Times Now we want the cause of minus π by three and 00:16:56.980 --> 00:17:04.240 the cosine of minus π by three is 1/2, and so that 00:17:04.240 --> 00:17:05.450 gives US1. 00:17:06.830 --> 00:17:10.520 Why is equal to our 00:17:10.520 --> 00:17:17.585 sign theater? Which is 2 times the sign of minus π by three, 00:17:17.585 --> 00:17:24.865 which is 2 Times Now we want the sign of my minus Pi π three, and 00:17:24.865 --> 00:17:31.235 that is minus Route 3 over 2. The two is cancelled to give us 00:17:31.235 --> 00:17:32.600 minus Route 3. 00:17:33.130 --> 00:17:37.628 And so again, notice we know that we've got it in the right 00:17:37.628 --> 00:17:41.434 quadrant. 'cause when we drew the diagram, we have positive X 00:17:41.434 --> 00:17:44.894 and negative Y, and that's how we've ended up here. 00:17:46.070 --> 00:17:51.110 What do we do about going back the other way? 00:17:51.650 --> 00:17:59.054 Well, let's have a look at some examples that will do that 00:17:59.054 --> 00:18:05.238 for us. What I'm going to look at as these points, which are 00:18:05.238 --> 00:18:11.230 cartesians. The .22 point minus 3 four. 00:18:12.010 --> 00:18:17.890 The point minus 2 - 2 Route 3. 00:18:18.730 --> 00:18:25.150 And the .1 - 1 now these are all points in Cartesian's. 00:18:25.150 --> 00:18:28.360 So let's begin with this one. 00:18:29.060 --> 00:18:32.228 Show where it is. 00:18:32.740 --> 00:18:38.920 To begin with, on the cartesian axes so it's at 2 for X and two 00:18:38.920 --> 00:18:40.980 for Y. So it's there. 00:18:42.220 --> 00:18:48.110 So again. We'll associate the origin in 00:18:48.110 --> 00:18:53.236 Cartesians with the pole in polar's, and the X axis, with 00:18:53.236 --> 00:18:58.828 the initial line and what we want to calculate is what's that 00:18:58.828 --> 00:19:01.158 angle there an what's that 00:19:01.158 --> 00:19:03.550 radius there? Well. 00:19:04.060 --> 00:19:09.400 All squared is equal to X squared plus Y squared. 00:19:10.890 --> 00:19:17.845 So that's 2 squared +2 squared, keeps us 8 and so are is 00:19:17.845 --> 00:19:20.520 equal to 2 Route 2. 00:19:21.260 --> 00:19:23.416 When we take the square root of 00:19:23.416 --> 00:19:31.065 8. What about theater? Well, tan Theta is equal to 00:19:31.065 --> 00:19:33.270 Y over X. 00:19:33.870 --> 00:19:41.010 In this case it's two over 2, which is one, and so 00:19:41.010 --> 00:19:46.960 theater is π by 4, and so therefore the polar 00:19:46.960 --> 00:19:52.910 coordinates of this point are two route 2π over 4. 00:19:55.470 --> 00:19:59.782 Let's have a look at this one now, minus 3 four. 00:20:00.750 --> 00:20:05.310 Let's 00:20:05.310 --> 00:20:13.954 begin. By establishing whereabouts it 00:20:13.954 --> 00:20:17.238 is on our cartesian 00:20:17.238 --> 00:20:22.318 axes. Minus 3 means it's back here somewhere, so there's 00:20:22.318 --> 00:20:28.454 minus three and the four on the Y. It's up there, so I'll 00:20:28.454 --> 00:20:29.870 point is there. 00:20:32.450 --> 00:20:39.665 Join it up to the origin as our point P and we are after. Now 00:20:39.665 --> 00:20:44.956 the polar coordinates for this point. So again we associate the 00:20:44.956 --> 00:20:50.728 pole with the origin and the initial line with the X axis, 00:20:50.728 --> 00:20:56.019 and so there's the value of theater that we're after. And 00:20:56.019 --> 00:21:01.791 this opie is the length are that were after, so R-squared is 00:21:01.791 --> 00:21:03.715 equal to X squared. 00:21:03.740 --> 00:21:10.592 Plus Y squared, which in this case is minus 3 squared, +4 00:21:10.592 --> 00:21:18.586 squared. That's 9 + 16, gives us 25, and so R is the square 00:21:18.586 --> 00:21:22.583 root of 25, which is just five. 00:21:23.210 --> 00:21:27.560 What about finding theater now well? 00:21:28.640 --> 00:21:30.968 Tan Theta is. 00:21:32.100 --> 00:21:34.839 Y over X. 00:21:35.880 --> 00:21:37.488 Which gives us. 00:21:38.480 --> 00:21:41.948 4 over minus three. 00:21:42.510 --> 00:21:47.119 Now when you put that into your Calculator, you will get. 00:21:47.720 --> 00:21:53.110 A slightly odd answers. It will actually give you a negative 00:21:53.110 --> 00:21:56.130 answer. That might be difficult for you to 00:21:56.130 --> 00:21:58.470 interpret. It sits actually telling you this angle out 00:21:58.470 --> 00:21:58.730 here. 00:21:59.890 --> 00:22:06.415 And we want to be all the way around there now the way that I 00:22:06.415 --> 00:22:11.635 think these are best done is actually to look at a right 00:22:11.635 --> 00:22:16.420 angle triangle like this and call that angle Alpha. Now let's 00:22:16.420 --> 00:22:23.380 have a look at what an Alpha is. Tan Alpha is 4 over 3 and when 00:22:23.380 --> 00:22:28.165 you put that into your Calculator it will tell you that 00:22:28.165 --> 00:22:29.905 Alpha is nought .9. 00:22:29.920 --> 00:22:36.300 Three radians. Remember, theater has to be in radians and 00:22:36.300 --> 00:22:43.074 therefore. Theater here is equal to π minus 00:22:43.074 --> 00:22:49.490 Alpha, and so that's π - 4.9. Three, 00:22:49.490 --> 00:22:52.698 which gives us 2.2 00:22:52.698 --> 00:22:58.202 one radians. And that's theater so you can see that 00:22:58.202 --> 00:23:02.292 the calculation of our is always going to be relatively 00:23:02.292 --> 00:23:05.155 straightforward, but the calculation this angle theater 00:23:05.155 --> 00:23:10.063 is going to be quite tricky, and that's one of the reasons 00:23:10.063 --> 00:23:14.562 why it's best to plot these points before you try and 00:23:14.562 --> 00:23:16.198 workout what theater is. 00:23:17.850 --> 00:23:21.765 Now the next example was the point minus 2. 00:23:22.580 --> 00:23:25.980 Minus 2 Route 3. 00:23:26.730 --> 00:23:28.100 So again. 00:23:30.830 --> 00:23:36.147 Let's have a look where it is in the cartesian plane. These are 00:23:36.147 --> 00:23:40.646 its cartesian coordinates, so we've minus two for X. So we 00:23:40.646 --> 00:23:45.554 somewhere back here and minus 2 route 3 four Y. So where 00:23:45.554 --> 00:23:48.826 somewhere down here? So I'll point is here. 00:23:49.690 --> 00:23:52.690 Join it up to our origin. 00:23:53.600 --> 00:23:55.828 Marking our point P. 00:23:56.380 --> 00:24:02.395 Again, we'll take the origin to be the pole and the X axis to be 00:24:02.395 --> 00:24:07.207 the initial line, and we can see that the theater were looking 00:24:07.207 --> 00:24:08.811 for is around there. 00:24:09.510 --> 00:24:12.825 That's our theater, and here's 00:24:12.825 --> 00:24:18.876 our. So again, let's calculate R-squared that's X 00:24:18.876 --> 00:24:21.508 squared plus Y squared. 00:24:22.200 --> 00:24:29.328 Is equal to. Well, in this case we've got minus two all 00:24:29.328 --> 00:24:36.456 squared plus minus 2 route 3 all squared, which gives us 4 00:24:36.456 --> 00:24:43.536 + 12. 16 and so R is equal to the square root of 16, which 00:24:43.536 --> 00:24:44.892 is just 4. 00:24:45.730 --> 00:24:50.867 Now, what about this? We can see that theater should be 00:24:50.867 --> 00:24:55.070 negative, so let's just calculate this angle as an 00:24:55.070 --> 00:24:59.273 angle in a right angle triangle. So tan Alpha 00:24:59.273 --> 00:25:03.943 equals, well, it's going to be the opposite, which is 00:25:03.943 --> 00:25:05.344 this side here. 00:25:07.300 --> 00:25:13.324 2 Route 3 in length over the adjacent, which is just two 00:25:13.324 --> 00:25:20.352 which gives us Route 3. So Alpha just calculated as an angle is π 00:25:20.352 --> 00:25:27.380 by three. So if that's pie by three this angle in size is 2π 00:25:27.380 --> 00:25:32.902 by three, but of course we must measure theater negatively when 00:25:32.902 --> 00:25:37.922 we come clockwise from the initial line, and so Theta. 00:25:37.950 --> 00:25:44.255 Is minus 2π by three the 2π three giving us the size the 00:25:44.255 --> 00:25:50.075 minus sign giving us the direction so we can see that the 00:25:50.075 --> 00:25:55.895 point we've got described as minus 2 - 2 route 3 in 00:25:55.895 --> 00:26:03.170 Cartesians is the .4 - 2π over 3 or minus 2/3 of Π in Pollas. 00:26:03.970 --> 00:26:08.470 Now we've taken a point in this quadrant. A point in this 00:26:08.470 --> 00:26:12.220 quadrant appointing this quadrant. Let's have a look at a 00:26:12.220 --> 00:26:16.720 point in the fourth quadrant just to finish off this set of 00:26:16.720 --> 00:26:20.470 examples and the point we chose was 1 - 1. 00:26:21.490 --> 00:26:22.650 So again. 00:26:24.320 --> 00:26:29.060 Let's have a look at where it is in our cartesian system. 00:26:30.020 --> 00:26:35.090 So we've a value of one 4X and the value of minus one 00:26:35.090 --> 00:26:39.770 for Y. So there's our point P. Join it to the origin. 00:26:40.790 --> 00:26:45.510 And again will associate the origin in the Cartesian's with 00:26:45.510 --> 00:26:50.230 the pole of the polar coordinates and the initial line 00:26:50.230 --> 00:26:55.894 will be the X axis, so we're looking for this angle theater. 00:26:57.010 --> 00:27:00.290 And this length of OP. 00:27:02.020 --> 00:27:07.487 So all squared is equal to X squared plus Y squared. 00:27:08.130 --> 00:27:14.478 So that's one squared plus minus one squared, and that's one plus 00:27:14.478 --> 00:27:19.768 one is 2 so far is equal to Route 2. 00:27:20.510 --> 00:27:24.713 Let's not worry about the direction here. Let's just 00:27:24.713 --> 00:27:28.916 calculate the magnitude of theater well. The magnitude of 00:27:28.916 --> 00:27:34.053 theater, in fact, to do that, I'd rather actually call it 00:27:34.053 --> 00:27:39.190 Alpha, just want to calculate the magnitude. So tan Alpha is. 00:27:39.860 --> 00:27:45.896 Opposite, which is one over the adjacent, which is one which is 00:27:45.896 --> 00:27:52.938 just one. So Alpha is in fact Π by 4. That means that my 00:27:52.938 --> 00:27:58.471 angle theater for the coordinate coming around this way is minus 00:27:58.471 --> 00:28:04.507 π by 4, and so my polar coordinates for this point, our 00:28:04.507 --> 00:28:08.028 Route 2 and minus π by 4. 00:28:08.760 --> 00:28:15.126 So. We've seen here why it's so important to plot your points 00:28:15.126 --> 00:28:17.406 before you do any calculation. 00:28:18.110 --> 00:28:21.854 Having looked at what happens with points, let's see if we can 00:28:21.854 --> 00:28:25.910 now have a look at what happens to a collection of points. In 00:28:25.910 --> 00:28:27.158 other words, a curve. 00:28:28.000 --> 00:28:31.210 Let's take a very simple curve 00:28:31.210 --> 00:28:37.398 in Cartesians. X squared plus Y squared equals a squared. 00:28:37.950 --> 00:28:42.430 Now this is a circle, a circle centered on the 00:28:42.430 --> 00:28:44.222 origin of Radius A. 00:28:45.260 --> 00:28:47.096 So if we think about that. 00:28:49.650 --> 00:28:56.982 Circle centered on the origin of radius a, so it will go through. 00:28:57.930 --> 00:29:01.470 These points on the axis. 00:29:09.150 --> 00:29:10.080 Like so. 00:29:11.580 --> 00:29:16.910 If we think about what that tells us, it tells us that no 00:29:16.910 --> 00:29:18.960 matter what the angle is. 00:29:22.220 --> 00:29:24.770 For any one of our points. 00:29:26.640 --> 00:29:32.412 If we were thinking in Pohlers, the radius is always a constant. 00:29:32.412 --> 00:29:39.146 So if we were to guess at the polar equation, it would be our 00:29:39.146 --> 00:29:44.918 equals A and it wouldn't involve theater at all. Well, it just 00:29:44.918 --> 00:29:51.171 check that we know that X is equal to our cause theater, and 00:29:51.171 --> 00:29:55.019 we know that Y is equal to our 00:29:55.019 --> 00:30:01.549 sign Theta. So we can substitute these in R-squared 00:30:01.549 --> 00:30:07.642 cost, Square theater plus R-squared. Sine squared Theta is 00:30:07.642 --> 00:30:14.412 equal to a squared. We can take out the R-squared. 00:30:16.770 --> 00:30:24.100 And that leaves us with this factor of Cos squared 00:30:24.100 --> 00:30:26.299 plus sign squared. 00:30:26.410 --> 00:30:31.162 Now cost squared plus sign squared is a well known identity 00:30:31.162 --> 00:30:36.346 cost squared plus sign squared at the same angle is always one, 00:30:36.346 --> 00:30:41.098 so this just reduces two R-squared equals a squared or R 00:30:41.098 --> 00:30:46.282 equals AR is a constant, which is what we predicted for looking 00:30:46.282 --> 00:30:49.738 at the situation there now. Another very straightforward 00:30:49.738 --> 00:30:54.922 curve is the straight line Y equals MX. Let's just have a 00:30:54.922 --> 00:30:56.218 look at that. 00:30:56.760 --> 00:31:01.140 Y equals MX is a straight line that goes through the origin. 00:31:02.550 --> 00:31:07.518 Think about it, is it has a constant gradient and of course 00:31:07.518 --> 00:31:12.900 M. The gradient is defined to be the tangent of the angle that 00:31:12.900 --> 00:31:17.454 the line makes with the positive direction of the X axis. 00:31:18.060 --> 00:31:22.361 So if the gradient is a constant, the tangent of the 00:31:22.361 --> 00:31:27.444 angle is a constant, and so this angle theater is a constant. So 00:31:27.444 --> 00:31:31.745 let's just have a look at that. Why is we know? 00:31:32.350 --> 00:31:39.630 All. Sign Theta equals M times 00:31:39.630 --> 00:31:43.534 by our cause theater. 00:31:44.490 --> 00:31:51.224 The ask cancel out and so I have sign theater over Cos Theta 00:31:51.224 --> 00:31:58.423 equals M. And so I have tan Theta equals M and 00:31:58.423 --> 00:32:01.558 so theater does equal a 00:32:01.558 --> 00:32:08.256 constant. But and here there is a big bot for Y equals MX. 00:32:08.256 --> 00:32:13.129 That's the picture that we get if we're working in Cartesians. 00:32:13.840 --> 00:32:20.033 But if we're working in Pohlers, there's our poll. There's our 00:32:20.033 --> 00:32:25.663 initial line Theta equals a constant. There is the angle 00:32:25.663 --> 00:32:33.056 Theta. And remember, we do not have negative values of R and so 00:32:33.056 --> 00:32:39.888 we get a half line. In other words, we only get this bit of 00:32:39.888 --> 00:32:45.084 the line. The half line there. That simple example should 00:32:45.084 --> 00:32:49.957 warnors that whenever we are moving between one sort of curve 00:32:49.957 --> 00:32:53.944 in cartesians into its equivalent in polar's, we need 00:32:53.944 --> 00:32:58.374 to be very careful about the results that we get. 00:32:59.140 --> 00:33:05.185 So let's just have a look at a couple more examples. Let's take 00:33:05.185 --> 00:33:11.695 X squared plus. Y squared is equal to 9. We know that X is 00:33:11.695 --> 00:33:15.880 our cause theater. And why is our sign theater? 00:33:16.720 --> 00:33:23.170 We can plug those in R-squared, Cos squared Theta plus 00:33:23.170 --> 00:33:25.750 R-squared, sine squared Theta 00:33:25.750 --> 00:33:29.300 equals 9. All squared is a 00:33:29.300 --> 00:33:35.246 common factor. So we can take it out and we've got cost squared 00:33:35.246 --> 00:33:40.238 Theta plus sign squared. Theta is equal to 9 cost squared plus 00:33:40.238 --> 00:33:45.230 sign squared is an identity cost squared plus sign squared of the 00:33:45.230 --> 00:33:50.638 same angle is always one, and so R-squared equals 9. R is equal 00:33:50.638 --> 00:33:53.550 to three IE a circle of radius 00:33:53.550 --> 00:33:56.455 3. Let's 00:33:56.455 --> 00:34:03.265 take. The rectangular hyperbola XY is 00:34:03.265 --> 00:34:05.830 equal to 4. 00:34:06.280 --> 00:34:13.168 And again, we're going to use X equals our cause theater and 00:34:13.168 --> 00:34:18.908 Y equals R sign theater. So we're multiplying X&Y together. 00:34:18.908 --> 00:34:25.222 So when we do that, we're going to have our squared. 00:34:25.290 --> 00:34:32.870 Sign theater Cos Theta equals 4. 00:34:34.350 --> 00:34:41.936 Now. Sign Theta Cos Theta will twice sign tita cost 00:34:41.936 --> 00:34:45.156 theater would be signed to 00:34:45.156 --> 00:34:51.006 theater. But I've taken 2 lots there, so if I've taken 2 lots 00:34:51.006 --> 00:34:54.889 there, it's the equivalent of multiplying that side by two. So 00:34:54.889 --> 00:34:57.007 I've got to multiply that side 00:34:57.007 --> 00:34:59.386 by two. So I end up with that. 00:35:00.580 --> 00:35:01.819 For my equation. 00:35:02.670 --> 00:35:06.856 I still some the other way round now, but one point to notice 00:35:06.856 --> 00:35:10.730 before we do. Knowledge of trig identity's is very 00:35:10.730 --> 00:35:14.105 important. We've used cost squared plus sign. Squared is 00:35:14.105 --> 00:35:18.605 one and we've now used sign to Theta is equal to two 00:35:18.605 --> 00:35:21.980 scientist accosts theater. So knowledge of those is very 00:35:21.980 --> 00:35:26.855 important. So as I said, let's see if we can turn this around 00:35:26.855 --> 00:35:31.355 now and have a look at some examples going the other way. 00:35:32.660 --> 00:35:39.954 First one will take is 2 over, R is equal to 1 plus cause 00:35:39.954 --> 00:35:45.930 theater. I don't like really the way it's written, so let's 00:35:45.930 --> 00:35:51.762 multiply up by R so I get R Plus R cause theater. 00:35:52.540 --> 00:35:57.530 Now, because I've done that, let's just remember that are 00:35:57.530 --> 00:36:02.021 squared is equal to X squared plus Y squared. 00:36:03.080 --> 00:36:09.968 So that means I can replace this are here by the square 00:36:09.968 --> 00:36:13.412 root of X squared plus Y 00:36:13.412 --> 00:36:19.595 squared. Our costs theater. Will, our Cos Theta is equal 00:36:19.595 --> 00:36:25.145 to X so I can replace this bit by X. 00:36:26.420 --> 00:36:31.136 Now it looks untidy's got a square root in it, so naturally 00:36:31.136 --> 00:36:36.638 we would want to get rid of that square root. So let's take X 00:36:36.638 --> 00:36:38.210 away from each side. 00:36:38.370 --> 00:36:44.090 And then let's Square both sides. So that gives us X 00:36:44.090 --> 00:36:50.850 squared plus Y squared there and on this side it's 2 minus X 00:36:50.850 --> 00:36:57.610 all squared, which will give us 4 - 4 X plus X squared. 00:36:58.170 --> 00:37:04.570 So I've got an X squared on each side that will go out and so I'm 00:37:04.570 --> 00:37:08.970 left with Y squared is equal to 4 - 4 X. 00:37:10.410 --> 00:37:15.503 And what you should notice there is that actually a parabola. 00:37:16.180 --> 00:37:22.251 So this would seem to be the way in which we define a 00:37:22.251 --> 00:37:24.119 parabola in polar coordinates.