1 00:00:00,980 --> 00:00:05,660 In this video, we're going to be looking at polar. 2 00:00:06,930 --> 00:00:13,672 Coordinates. Let's begin by actually 3 00:00:13,672 --> 00:00:20,096 looking at another coordinate system. The Cartesian coordinate 4 00:00:20,096 --> 00:00:23,355 system. Now in that system we 5 00:00:23,355 --> 00:00:28,817 take 2. Axes and X axis which is horizontal. 6 00:00:29,490 --> 00:00:36,094 And Y Axis which is vertical and a fixed .0 called the origin, 7 00:00:36,094 --> 00:00:41,682 which is where these two points cross. These two lines cross. 8 00:00:42,350 --> 00:00:49,420 Now we fix a point P in the plane by saying how far it's 9 00:00:49,420 --> 00:00:54,975 displaced along the X axis to give us the X coordinate. 10 00:00:55,550 --> 00:01:01,724 And how far it's displaced along the Y access to give us the Y 11 00:01:01,724 --> 00:01:03,929 coordinate and so we have. 12 00:01:04,490 --> 00:01:09,198 A point P which is uniquely described by its coordinates XY 13 00:01:09,198 --> 00:01:13,906 and notice I said how far it's displaced because it is 14 00:01:13,906 --> 00:01:17,758 displacement that we're talking about and not distance. That's 15 00:01:17,758 --> 00:01:22,894 what these arrowheads that we put on the axes are all about 16 00:01:22,894 --> 00:01:27,602 their about showings, in which direction we must move so that 17 00:01:27,602 --> 00:01:31,454 if we're moving down this direction, it's a negative 18 00:01:31,454 --> 00:01:34,450 distance and negative displacement that we're making. 19 00:01:34,680 --> 00:01:40,392 Now that is more than one way of describing where a point is in 20 00:01:40,392 --> 00:01:47,651 the plane. And we're going to be having a look at a system 21 00:01:47,651 --> 00:01:49,244 called polar coordinates. 22 00:01:49,250 --> 00:01:57,086 So in this system of polar coordinates, we take a poll. Oh, 23 00:01:57,086 --> 00:02:01,004 and we take a fixed line. 24 00:02:02,740 --> 00:02:08,538 Now, how can we describe a point in the plane using this fixed 25 00:02:08,538 --> 00:02:14,336 .0? The pole and this baseline. Here. One of the ways is to 26 00:02:14,336 --> 00:02:17,904 think of it as. What if we turn? 27 00:02:19,380 --> 00:02:26,753 Around. Centering on oh for the moment we rotate around, 28 00:02:26,753 --> 00:02:33,430 we can pass through a fixed angle. Let's call that theater. 29 00:02:33,990 --> 00:02:40,458 And then along this radius we can go a set distance. 30 00:02:40,970 --> 00:02:44,130 And we'll end up at a point P. 31 00:02:45,170 --> 00:02:51,098 And so the coordinates of that point would be our theater, and 32 00:02:51,098 --> 00:02:54,062 this is our system of polar. 33 00:02:54,600 --> 00:02:57,410 Coordinates. 34 00:02:59,490 --> 00:03:04,476 Now, just as we've got certain conventions with Cartesian 35 00:03:04,476 --> 00:03:08,354 coordinates, we have certain conventions with polar 36 00:03:08,354 --> 00:03:13,340 coordinates, and these are quite strong conventions, so let's 37 00:03:13,340 --> 00:03:19,988 have a look at what these are. First of all, theater is 38 00:03:19,988 --> 00:03:23,265 measured. In 39 00:03:23,265 --> 00:03:29,150 radians. So that's how first convention 40 00:03:29,150 --> 00:03:32,110 theater is measured in radians. 41 00:03:32,750 --> 00:03:35,816 2nd convention 42 00:03:35,816 --> 00:03:42,630 well. Our second convention is this that if this is our 43 00:03:42,630 --> 00:03:47,767 initial line and this is our poll, then we measure theater 44 00:03:47,767 --> 00:03:50,569 positive when we go round in 45 00:03:50,569 --> 00:03:56,296 that direction. Anticlockwise and we measure theater negative. 46 00:03:56,296 --> 00:04:02,470 When we go around in that direction which is 47 00:04:02,470 --> 00:04:09,634 clockwise. So in just the same way as we had an Arrowhead on 48 00:04:09,634 --> 00:04:14,554 our axes X&Y. In a sense, we've got arrowheads here, 49 00:04:14,554 --> 00:04:18,490 distinguishing a positive direction for theater and a 50 00:04:18,490 --> 00:04:20,458 negative direction for measuring 51 00:04:20,458 --> 00:04:24,588 theater. We have 1/3 convention to do with theater 52 00:04:24,588 --> 00:04:29,549 and that is that we never go further round this way. 53 00:04:30,680 --> 00:04:36,536 Number there's our poll. Oh, our fixed point. We never go further 54 00:04:36,536 --> 00:04:42,392 around this way then there, so theater is always less than or 55 00:04:42,392 --> 00:04:48,248 equal to pie and we never go round further that way than 56 00:04:48,248 --> 00:04:52,640 there again. So theater is always strictly greater than 57 00:04:52,640 --> 00:04:59,472 minus Π - Π ramped there plus Π round to there. And notice that 58 00:04:59,472 --> 00:05:06,208 we include. This bit of the line if you like this extended bit of 59 00:05:06,208 --> 00:05:12,144 the line by going route to their having the less than or equal to 60 00:05:12,144 --> 00:05:15,960 and having strictly greater than Theta strictly greater than 61 00:05:15,960 --> 00:05:17,232 minus pie there. 62 00:05:17,960 --> 00:05:24,002 1/4 Convention 1/4 convention is that 63 00:05:24,002 --> 00:05:27,023 our is always 64 00:05:27,023 --> 00:05:32,866 positive. One of the things that is quite important is that we be 65 00:05:32,866 --> 00:05:37,702 able to move from one system of coordinates to another. So the 66 00:05:37,702 --> 00:05:39,717 question is if we have. 67 00:05:40,230 --> 00:05:44,184 A point. In our 68 00:05:44,184 --> 00:05:50,756 XY plane. Who's coordinates are 69 00:05:50,756 --> 00:05:56,501 X&Y? How can we change from cartesians into 70 00:05:56,501 --> 00:06:02,144 pohlers? And how can we change back again, but one obvious 71 00:06:02,144 --> 00:06:08,813 thing to do is to associate the pole with the origin, and then 72 00:06:08,813 --> 00:06:13,430 to associate the initial line with the X axis. 73 00:06:13,950 --> 00:06:18,870 And then if we draw the radius out to pee. 74 00:06:19,920 --> 00:06:21,699 And that's our. 75 00:06:22,750 --> 00:06:25,708 And that is the angle theater. 76 00:06:26,350 --> 00:06:30,150 So we can see that in Cartesians, we're 77 00:06:30,150 --> 00:06:33,950 describing it as XY, and in Pohlers, where 78 00:06:33,950 --> 00:06:37,750 describing it as our theater. So what's the 79 00:06:37,750 --> 00:06:39,175 relationship between them? 80 00:06:40,270 --> 00:06:45,010 Let's drop that perpendicular down and we can see that this is 81 00:06:45,010 --> 00:06:50,145 a height. Why? Because of the Y coordinate the point and this is 82 00:06:50,145 --> 00:06:54,885 at a distance X because of the X coordinate of the point. 83 00:06:55,680 --> 00:07:02,960 And looking at that, we can see that Y is equal to R sign 84 00:07:02,960 --> 00:07:09,720 theater and X is equal to our cause theater. So given R and 85 00:07:09,720 --> 00:07:15,440 Theta, we can calculate X&Y. What about moving the other way 86 00:07:15,440 --> 00:07:22,200 will from Pythagoras? We can see that X squared plus Y squared is 87 00:07:22,200 --> 00:07:25,320 equal to R-squared. So give now 88 00:07:25,320 --> 00:07:30,415 X. I'm now why we can calculate all and we can also 89 00:07:30,415 --> 00:07:34,980 see that if we take the opposite over the adjacent, we 90 00:07:34,980 --> 00:07:41,205 have Y over X is equal to 10 theater. So given AY in an X, 91 00:07:41,205 --> 00:07:44,110 we can find out what theater is. 92 00:07:45,370 --> 00:07:45,970 Now. 93 00:07:46,990 --> 00:07:52,160 Always when doing these, it's best to draw sketches. If we're 94 00:07:52,160 --> 00:07:56,860 converting from one sort of point in Cartesians to its 95 00:07:56,860 --> 00:08:01,560 equivalent in Pohlers, or if we're moving back from Pohlers 96 00:08:01,560 --> 00:08:06,730 to cartesians, draw a picture, see where that point actually is 97 00:08:06,730 --> 00:08:12,840 now. Want to have a look at some examples. First of all, we're 98 00:08:12,840 --> 00:08:16,600 going to have a look at how to 99 00:08:16,600 --> 00:08:22,504 plot points. Then we're going to have a look at how to convert 100 00:08:22,504 --> 00:08:27,832 from one system into the other and vice versa. So let's begin 101 00:08:27,832 --> 00:08:30,230 with plotting. Plot. 102 00:08:31,490 --> 00:08:37,535 And what I'm going to do is I'm going to plot the following 103 00:08:37,535 --> 00:08:40,325 points and they're all in polar 104 00:08:40,325 --> 00:08:47,973 coordinates. I'm going to put them all on the same 105 00:08:47,973 --> 00:08:55,443 picture so we can get a feel for whereabouts things 106 00:08:55,443 --> 00:09:02,913 are in the polar play or the plane for the 107 00:09:02,913 --> 00:09:08,638 polar coordinates. So we put our poll, oh. 108 00:09:09,180 --> 00:09:11,790 And we have our initial line. 109 00:09:12,710 --> 00:09:15,314 First one that we've got to plot 110 00:09:15,314 --> 00:09:22,738 is 2. Pie so we know that Pi is the angle all the 111 00:09:22,738 --> 00:09:29,472 way around here, so there's pie to there and we want to go 112 00:09:29,472 --> 00:09:31,026 out to units. 113 00:09:31,630 --> 00:09:37,377 So it's there. This is the .2. 114 00:09:37,940 --> 00:09:39,490 Pie. 115 00:09:41,490 --> 00:09:46,950 Next to .1 N wealthy to is 0 so we're on the initial line. 116 00:09:48,110 --> 00:09:55,250 An one will be about there, so there is the .1 note. 117 00:09:56,770 --> 00:10:02,672 2 - Π by 3 - π by 3 means come around this 118 00:10:02,672 --> 00:10:07,666 way, and so minus π by three is about there, and 119 00:10:07,666 --> 00:10:12,206 we're coming around there minus π by three, and we 120 00:10:12,206 --> 00:10:17,654 want to come out a distance to, so that's roughly 2 out 121 00:10:17,654 --> 00:10:22,648 there, so this would be the .2 - π by 3. 122 00:10:23,790 --> 00:10:29,458 And finally, we've got the point. One 2/3 of Π. So we take 123 00:10:29,458 --> 00:10:32,074 the 2/3. That's going all the 124 00:10:32,074 --> 00:10:34,640 way around. To there. 125 00:10:35,190 --> 00:10:40,676 And we draw out through there, and we want a distance of one 126 00:10:40,676 --> 00:10:44,896 along there, which roughly called the scale we're using is 127 00:10:44,896 --> 00:10:47,428 about there, and so that's the 128 00:10:47,428 --> 00:10:49,930 .1. 2/3 of Π. 129 00:10:51,680 --> 00:10:58,100 Notice that we've taken theater first to establish in which 130 00:10:58,100 --> 00:11:00,668 direction were actually facing. 131 00:11:02,600 --> 00:11:06,255 OK, let's now have a 132 00:11:06,255 --> 00:11:11,995 look. Having got used to plotting points, let's now have 133 00:11:11,995 --> 00:11:16,435 a look in polar coordinates. These points 2. 134 00:11:17,070 --> 00:11:19,270 Minus Π by 2. 135 00:11:20,210 --> 00:11:26,210 1. 3/4 of 136 00:11:26,210 --> 00:11:33,380 Π. And 2 - π by three. Now these 137 00:11:33,380 --> 00:11:35,840 are all in Pohlers. 138 00:11:36,400 --> 00:11:41,801 What I want to do is convert them into cartesian coordinates. 139 00:11:41,801 --> 00:11:47,693 So first a picture whereabouts are they? And I'll do them one 140 00:11:47,693 --> 00:11:55,058 at a time. So let's take this one 2 - π by two initial point 141 00:11:55,058 --> 00:11:57,513 poll. Oh, and initial line. 142 00:11:58,440 --> 00:12:02,157 Minus Π by two? Well, that's coming down here. 143 00:12:02,730 --> 00:12:10,060 To there. So that's minus π by two, and we've 144 00:12:10,060 --> 00:12:13,660 come a distance to to 145 00:12:13,660 --> 00:12:18,820 there. Well, we don't need to do much calculation. I don't think 146 00:12:18,820 --> 00:12:23,760 to find this. If again we take our origin for our cartesians as 147 00:12:23,760 --> 00:12:28,700 being the pole, and we align the X axis with our initial line. 148 00:12:33,360 --> 00:12:40,315 And there's our X. There's RY and we can see straight away the 149 00:12:40,315 --> 00:12:47,805 point in Pohlers that's 2 - π by two in fact, goes to the 150 00:12:47,805 --> 00:12:54,225 point. In Cartesians, That's 0 - 2 because it's this point here 151 00:12:54,225 --> 00:13:01,715 on the Y axis, and it's 2 units below the X axis, so it's 152 00:13:01,715 --> 00:13:03,320 0 - 2. 153 00:13:03,370 --> 00:13:07,759 Notice how plotting the point actually saved as having to do 154 00:13:07,759 --> 00:13:12,148 any of the calculations. So let's take the next point now, 155 00:13:12,148 --> 00:13:14,542 which was one 3/4 of Π. 156 00:13:15,280 --> 00:13:23,160 1 3/4 of π. So again, let's plot where it 157 00:13:23,160 --> 00:13:26,560 is. Take our initial. 158 00:13:27,220 --> 00:13:34,500 .0 our poll and our initial line 3/4 of π going round. It's 159 00:13:34,500 --> 00:13:40,100 positive so it drought there be somewhere out along that. 160 00:13:40,670 --> 00:13:47,347 Direction there's our angle of 3/4 of Π, where somewhere out 161 00:13:47,347 --> 00:13:54,631 here at a distance one unit. So again, let's take our X&Y 162 00:13:54,631 --> 00:13:57,059 axes, our X axis. 163 00:13:58,010 --> 00:14:01,328 To be along the initial line. 164 00:14:01,850 --> 00:14:07,823 And now why access to be vertical and through the pole? 165 00:14:08,490 --> 00:14:14,634 Oh So that the polo becomes our origin of, and it's this point. 166 00:14:15,140 --> 00:14:16,229 But where after? 167 00:14:17,240 --> 00:14:24,170 Now how we going to work this out that remember the 168 00:14:24,170 --> 00:14:30,470 formula that we had was X equals our cause theater. 169 00:14:32,010 --> 00:14:39,678 Let's have a look at that. Are is one an we've got 170 00:14:39,678 --> 00:14:47,346 cause of 3/4 of Π and the cosine of 3/4 of Π 171 00:14:47,346 --> 00:14:55,014 is minus one over Route 2, so that's minus one over Route 172 00:14:55,014 --> 00:14:58,848 2. Why is our sign Theta? 173 00:14:58,880 --> 00:15:06,155 And so this is one times the sign of 3/4 of Π and the sign 174 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 175 00:15:13,915 --> 00:15:19,250 Route 2 for RY coordinate. And notice that these answers agree 176 00:15:19,250 --> 00:15:24,585 with where the point is in this particular quadrant. Negative X 177 00:15:24,585 --> 00:15:28,950 and positive Y, negative X and positive Y so. 178 00:15:28,950 --> 00:15:33,617 Even if I've got the calculation wrong in the sense that I, even 179 00:15:33,617 --> 00:15:37,566 if I've done the arithmetic wrong, have no, I've got the 180 00:15:37,566 --> 00:15:39,361 point in the right quadrant. 181 00:15:39,420 --> 00:15:46,130 Let's have a look at the last one of these two. 182 00:15:46,900 --> 00:15:50,386 Minus Π by 183 00:15:50,386 --> 00:15:54,350 3. So again, our poll. 184 00:15:56,020 --> 00:15:59,050 Our initial line. 185 00:15:59,050 --> 00:16:02,179 Minus Π by three is around here. 186 00:16:04,760 --> 00:16:11,533 So we've come around there minus π by three, and we're out a 187 00:16:11,533 --> 00:16:13,617 distance, two along there. 188 00:16:15,440 --> 00:16:20,550 Take our X axis to coincide with the initial line. 189 00:16:21,090 --> 00:16:22,500 And now origin. 190 00:16:23,410 --> 00:16:25,910 Coincide with the pole. 191 00:16:28,360 --> 00:16:35,170 Let's write down our equations that tell us X is 192 00:16:35,170 --> 00:16:41,980 our cause theater, which is 2 times the cosine of 193 00:16:41,980 --> 00:16:48,109 minus π by three, which is equal to 2. 194 00:16:49,720 --> 00:16:56,980 Times Now we want the cause of minus π by three and 195 00:16:56,980 --> 00:17:04,240 the cosine of minus π by three is 1/2, and so that 196 00:17:04,240 --> 00:17:05,450 gives US1. 197 00:17:06,830 --> 00:17:10,520 Why is equal to our 198 00:17:10,520 --> 00:17:17,585 sign theater? Which is 2 times the sign of minus π by three, 199 00:17:17,585 --> 00:17:24,865 which is 2 Times Now we want the sign of my minus Pi π three, and 200 00:17:24,865 --> 00:17:31,235 that is minus Route 3 over 2. The two is cancelled to give us 201 00:17:31,235 --> 00:17:32,600 minus Route 3. 202 00:17:33,130 --> 00:17:37,628 And so again, notice we know that we've got it in the right 203 00:17:37,628 --> 00:17:41,434 quadrant. 'cause when we drew the diagram, we have positive X 204 00:17:41,434 --> 00:17:44,894 and negative Y, and that's how we've ended up here. 205 00:17:46,070 --> 00:17:51,110 What do we do about going back the other way? 206 00:17:51,650 --> 00:17:59,054 Well, let's have a look at some examples that will do that 207 00:17:59,054 --> 00:18:05,238 for us. What I'm going to look at as these points, which are 208 00:18:05,238 --> 00:18:11,230 cartesians. The .22 point minus 3 four. 209 00:18:12,010 --> 00:18:17,890 The point minus 2 - 2 Route 3. 210 00:18:18,730 --> 00:18:25,150 And the .1 - 1 now these are all points in Cartesian's. 211 00:18:25,150 --> 00:18:28,360 So let's begin with this one. 212 00:18:29,060 --> 00:18:32,228 Show where it is. 213 00:18:32,740 --> 00:18:38,920 To begin with, on the cartesian axes so it's at 2 for X and two 214 00:18:38,920 --> 00:18:40,980 for Y. So it's there. 215 00:18:42,220 --> 00:18:48,110 So again. We'll associate the origin in 216 00:18:48,110 --> 00:18:53,236 Cartesians with the pole in polar's, and the X axis, with 217 00:18:53,236 --> 00:18:58,828 the initial line and what we want to calculate is what's that 218 00:18:58,828 --> 00:19:01,158 angle there an what's that 219 00:19:01,158 --> 00:19:03,550 radius there? Well. 220 00:19:04,060 --> 00:19:09,400 All squared is equal to X squared plus Y squared. 221 00:19:10,890 --> 00:19:17,845 So that's 2 squared +2 squared, keeps us 8 and so are is 222 00:19:17,845 --> 00:19:20,520 equal to 2 Route 2. 223 00:19:21,260 --> 00:19:23,416 When we take the square root of 224 00:19:23,416 --> 00:19:31,065 8. What about theater? Well, tan Theta is equal to 225 00:19:31,065 --> 00:19:33,270 Y over X. 226 00:19:33,870 --> 00:19:41,010 In this case it's two over 2, which is one, and so 227 00:19:41,010 --> 00:19:46,960 theater is π by 4, and so therefore the polar 228 00:19:46,960 --> 00:19:52,910 coordinates of this point are two route 2π over 4. 229 00:19:55,470 --> 00:19:59,782 Let's have a look at this one now, minus 3 four. 230 00:20:00,750 --> 00:20:05,310 Let's 231 00:20:05,310 --> 00:20:13,954 begin. By establishing whereabouts it 232 00:20:13,954 --> 00:20:17,238 is on our cartesian 233 00:20:17,238 --> 00:20:22,318 axes. Minus 3 means it's back here somewhere, so there's 234 00:20:22,318 --> 00:20:28,454 minus three and the four on the Y. It's up there, so I'll 235 00:20:28,454 --> 00:20:29,870 point is there. 236 00:20:32,450 --> 00:20:39,665 Join it up to the origin as our point P and we are after. Now 237 00:20:39,665 --> 00:20:44,956 the polar coordinates for this point. So again we associate the 238 00:20:44,956 --> 00:20:50,728 pole with the origin and the initial line with the X axis, 239 00:20:50,728 --> 00:20:56,019 and so there's the value of theater that we're after. And 240 00:20:56,019 --> 00:21:01,791 this opie is the length are that were after, so R-squared is 241 00:21:01,791 --> 00:21:03,715 equal to X squared. 242 00:21:03,740 --> 00:21:10,592 Plus Y squared, which in this case is minus 3 squared, +4 243 00:21:10,592 --> 00:21:18,586 squared. That's 9 + 16, gives us 25, and so R is the square 244 00:21:18,586 --> 00:21:22,583 root of 25, which is just five. 245 00:21:23,210 --> 00:21:27,560 What about finding theater now well? 246 00:21:28,640 --> 00:21:30,968 Tan Theta is. 247 00:21:32,100 --> 00:21:34,839 Y over X. 248 00:21:35,880 --> 00:21:37,488 Which gives us. 249 00:21:38,480 --> 00:21:41,948 4 over minus three. 250 00:21:42,510 --> 00:21:47,119 Now when you put that into your Calculator, you will get. 251 00:21:47,720 --> 00:21:53,110 A slightly odd answers. It will actually give you a negative 252 00:21:53,110 --> 00:21:56,130 answer. That might be difficult for you to 253 00:21:56,130 --> 00:21:58,470 interpret. It sits actually telling you this angle out 254 00:21:58,470 --> 00:21:58,730 here. 255 00:21:59,890 --> 00:22:06,415 And we want to be all the way around there now the way that I 256 00:22:06,415 --> 00:22:11,635 think these are best done is actually to look at a right 257 00:22:11,635 --> 00:22:16,420 angle triangle like this and call that angle Alpha. Now let's 258 00:22:16,420 --> 00:22:23,380 have a look at what an Alpha is. Tan Alpha is 4 over 3 and when 259 00:22:23,380 --> 00:22:28,165 you put that into your Calculator it will tell you that 260 00:22:28,165 --> 00:22:29,905 Alpha is nought .9. 261 00:22:29,920 --> 00:22:36,300 Three radians. Remember, theater has to be in radians and 262 00:22:36,300 --> 00:22:43,074 therefore. Theater here is equal to π minus 263 00:22:43,074 --> 00:22:49,490 Alpha, and so that's π - 4.9. Three, 264 00:22:49,490 --> 00:22:52,698 which gives us 2.2 265 00:22:52,698 --> 00:22:58,202 one radians. And that's theater so you can see that 266 00:22:58,202 --> 00:23:02,292 the calculation of our is always going to be relatively 267 00:23:02,292 --> 00:23:05,155 straightforward, but the calculation this angle theater 268 00:23:05,155 --> 00:23:10,063 is going to be quite tricky, and that's one of the reasons 269 00:23:10,063 --> 00:23:14,562 why it's best to plot these points before you try and 270 00:23:14,562 --> 00:23:16,198 workout what theater is. 271 00:23:17,850 --> 00:23:21,765 Now the next example was the point minus 2. 272 00:23:22,580 --> 00:23:25,980 Minus 2 Route 3. 273 00:23:26,730 --> 00:23:28,100 So again. 274 00:23:30,830 --> 00:23:36,147 Let's have a look where it is in the cartesian plane. These are 275 00:23:36,147 --> 00:23:40,646 its cartesian coordinates, so we've minus two for X. So we 276 00:23:40,646 --> 00:23:45,554 somewhere back here and minus 2 route 3 four Y. So where 277 00:23:45,554 --> 00:23:48,826 somewhere down here? So I'll point is here. 278 00:23:49,690 --> 00:23:52,690 Join it up to our origin. 279 00:23:53,600 --> 00:23:55,828 Marking our point P. 280 00:23:56,380 --> 00:24:02,395 Again, we'll take the origin to be the pole and the X axis to be 281 00:24:02,395 --> 00:24:07,207 the initial line, and we can see that the theater were looking 282 00:24:07,207 --> 00:24:08,811 for is around there. 283 00:24:09,510 --> 00:24:12,825 That's our theater, and here's 284 00:24:12,825 --> 00:24:18,876 our. So again, let's calculate R-squared that's X 285 00:24:18,876 --> 00:24:21,508 squared plus Y squared. 286 00:24:22,200 --> 00:24:29,328 Is equal to. Well, in this case we've got minus two all 287 00:24:29,328 --> 00:24:36,456 squared plus minus 2 route 3 all squared, which gives us 4 288 00:24:36,456 --> 00:24:43,536 + 12. 16 and so R is equal to the square root of 16, which 289 00:24:43,536 --> 00:24:44,892 is just 4. 290 00:24:45,730 --> 00:24:50,867 Now, what about this? We can see that theater should be 291 00:24:50,867 --> 00:24:55,070 negative, so let's just calculate this angle as an 292 00:24:55,070 --> 00:24:59,273 angle in a right angle triangle. So tan Alpha 293 00:24:59,273 --> 00:25:03,943 equals, well, it's going to be the opposite, which is 294 00:25:03,943 --> 00:25:05,344 this side here. 295 00:25:07,300 --> 00:25:13,324 2 Route 3 in length over the adjacent, which is just two 296 00:25:13,324 --> 00:25:20,352 which gives us Route 3. So Alpha just calculated as an angle is π 297 00:25:20,352 --> 00:25:27,380 by three. So if that's pie by three this angle in size is 2π 298 00:25:27,380 --> 00:25:32,902 by three, but of course we must measure theater negatively when 299 00:25:32,902 --> 00:25:37,922 we come clockwise from the initial line, and so Theta. 300 00:25:37,950 --> 00:25:44,255 Is minus 2π by three the 2π three giving us the size the 301 00:25:44,255 --> 00:25:50,075 minus sign giving us the direction so we can see that the 302 00:25:50,075 --> 00:25:55,895 point we've got described as minus 2 - 2 route 3 in 303 00:25:55,895 --> 00:26:03,170 Cartesians is the .4 - 2π over 3 or minus 2/3 of Π in Pollas. 304 00:26:03,970 --> 00:26:08,470 Now we've taken a point in this quadrant. A point in this 305 00:26:08,470 --> 00:26:12,220 quadrant appointing this quadrant. Let's have a look at a 306 00:26:12,220 --> 00:26:16,720 point in the fourth quadrant just to finish off this set of 307 00:26:16,720 --> 00:26:20,470 examples and the point we chose was 1 - 1. 308 00:26:21,490 --> 00:26:22,650 So again. 309 00:26:24,320 --> 00:26:29,060 Let's have a look at where it is in our cartesian system. 310 00:26:30,020 --> 00:26:35,090 So we've a value of one 4X and the value of minus one 311 00:26:35,090 --> 00:26:39,770 for Y. So there's our point P. Join it to the origin. 312 00:26:40,790 --> 00:26:45,510 And again will associate the origin in the Cartesian's with 313 00:26:45,510 --> 00:26:50,230 the pole of the polar coordinates and the initial line 314 00:26:50,230 --> 00:26:55,894 will be the X axis, so we're looking for this angle theater. 315 00:26:57,010 --> 00:27:00,290 And this length of OP. 316 00:27:02,020 --> 00:27:07,487 So all squared is equal to X squared plus Y squared. 317 00:27:08,130 --> 00:27:14,478 So that's one squared plus minus one squared, and that's one plus 318 00:27:14,478 --> 00:27:19,768 one is 2 so far is equal to Route 2. 319 00:27:20,510 --> 00:27:24,713 Let's not worry about the direction here. Let's just 320 00:27:24,713 --> 00:27:28,916 calculate the magnitude of theater well. The magnitude of 321 00:27:28,916 --> 00:27:34,053 theater, in fact, to do that, I'd rather actually call it 322 00:27:34,053 --> 00:27:39,190 Alpha, just want to calculate the magnitude. So tan Alpha is. 323 00:27:39,860 --> 00:27:45,896 Opposite, which is one over the adjacent, which is one which is 324 00:27:45,896 --> 00:27:52,938 just one. So Alpha is in fact Π by 4. That means that my 325 00:27:52,938 --> 00:27:58,471 angle theater for the coordinate coming around this way is minus 326 00:27:58,471 --> 00:28:04,507 π by 4, and so my polar coordinates for this point, our 327 00:28:04,507 --> 00:28:08,028 Route 2 and minus π by 4. 328 00:28:08,760 --> 00:28:15,126 So. We've seen here why it's so important to plot your points 329 00:28:15,126 --> 00:28:17,406 before you do any calculation. 330 00:28:18,110 --> 00:28:21,854 Having looked at what happens with points, let's see if we can 331 00:28:21,854 --> 00:28:25,910 now have a look at what happens to a collection of points. In 332 00:28:25,910 --> 00:28:27,158 other words, a curve. 333 00:28:28,000 --> 00:28:31,210 Let's take a very simple curve 334 00:28:31,210 --> 00:28:37,398 in Cartesians. X squared plus Y squared equals a squared. 335 00:28:37,950 --> 00:28:42,430 Now this is a circle, a circle centered on the 336 00:28:42,430 --> 00:28:44,222 origin of Radius A. 337 00:28:45,260 --> 00:28:47,096 So if we think about that. 338 00:28:49,650 --> 00:28:56,982 Circle centered on the origin of radius a, so it will go through. 339 00:28:57,930 --> 00:29:01,470 These points on the axis. 340 00:29:09,150 --> 00:29:10,080 Like so. 341 00:29:11,580 --> 00:29:16,910 If we think about what that tells us, it tells us that no 342 00:29:16,910 --> 00:29:18,960 matter what the angle is. 343 00:29:22,220 --> 00:29:24,770 For any one of our points. 344 00:29:26,640 --> 00:29:32,412 If we were thinking in Pohlers, the radius is always a constant. 345 00:29:32,412 --> 00:29:39,146 So if we were to guess at the polar equation, it would be our 346 00:29:39,146 --> 00:29:44,918 equals A and it wouldn't involve theater at all. Well, it just 347 00:29:44,918 --> 00:29:51,171 check that we know that X is equal to our cause theater, and 348 00:29:51,171 --> 00:29:55,019 we know that Y is equal to our 349 00:29:55,019 --> 00:30:01,549 sign Theta. So we can substitute these in R-squared 350 00:30:01,549 --> 00:30:07,642 cost, Square theater plus R-squared. Sine squared Theta is 351 00:30:07,642 --> 00:30:14,412 equal to a squared. We can take out the R-squared. 352 00:30:16,770 --> 00:30:24,100 And that leaves us with this factor of Cos squared 353 00:30:24,100 --> 00:30:26,299 plus sign squared. 354 00:30:26,410 --> 00:30:31,162 Now cost squared plus sign squared is a well known identity 355 00:30:31,162 --> 00:30:36,346 cost squared plus sign squared at the same angle is always one, 356 00:30:36,346 --> 00:30:41,098 so this just reduces two R-squared equals a squared or R 357 00:30:41,098 --> 00:30:46,282 equals AR is a constant, which is what we predicted for looking 358 00:30:46,282 --> 00:30:49,738 at the situation there now. Another very straightforward 359 00:30:49,738 --> 00:30:54,922 curve is the straight line Y equals MX. Let's just have a 360 00:30:54,922 --> 00:30:56,218 look at that. 361 00:30:56,760 --> 00:31:01,140 Y equals MX is a straight line that goes through the origin. 362 00:31:02,550 --> 00:31:07,518 Think about it, is it has a constant gradient and of course 363 00:31:07,518 --> 00:31:12,900 M. The gradient is defined to be the tangent of the angle that 364 00:31:12,900 --> 00:31:17,454 the line makes with the positive direction of the X axis. 365 00:31:18,060 --> 00:31:22,361 So if the gradient is a constant, the tangent of the 366 00:31:22,361 --> 00:31:27,444 angle is a constant, and so this angle theater is a constant. So 367 00:31:27,444 --> 00:31:31,745 let's just have a look at that. Why is we know? 368 00:31:32,350 --> 00:31:39,630 All. Sign Theta equals M times 369 00:31:39,630 --> 00:31:43,534 by our cause theater. 370 00:31:44,490 --> 00:31:51,224 The ask cancel out and so I have sign theater over Cos Theta 371 00:31:51,224 --> 00:31:58,423 equals M. And so I have tan Theta equals M and 372 00:31:58,423 --> 00:32:01,558 so theater does equal a 373 00:32:01,558 --> 00:32:08,256 constant. But and here there is a big bot for Y equals MX. 374 00:32:08,256 --> 00:32:13,129 That's the picture that we get if we're working in Cartesians. 375 00:32:13,840 --> 00:32:20,033 But if we're working in Pohlers, there's our poll. There's our 376 00:32:20,033 --> 00:32:25,663 initial line Theta equals a constant. There is the angle 377 00:32:25,663 --> 00:32:33,056 Theta. And remember, we do not have negative values of R and so 378 00:32:33,056 --> 00:32:39,888 we get a half line. In other words, we only get this bit of 379 00:32:39,888 --> 00:32:45,084 the line. The half line there. That simple example should 380 00:32:45,084 --> 00:32:49,957 warnors that whenever we are moving between one sort of curve 381 00:32:49,957 --> 00:32:53,944 in cartesians into its equivalent in polar's, we need 382 00:32:53,944 --> 00:32:58,374 to be very careful about the results that we get. 383 00:32:59,140 --> 00:33:05,185 So let's just have a look at a couple more examples. Let's take 384 00:33:05,185 --> 00:33:11,695 X squared plus. Y squared is equal to 9. We know that X is 385 00:33:11,695 --> 00:33:15,880 our cause theater. And why is our sign theater? 386 00:33:16,720 --> 00:33:23,170 We can plug those in R-squared, Cos squared Theta plus 387 00:33:23,170 --> 00:33:25,750 R-squared, sine squared Theta 388 00:33:25,750 --> 00:33:29,300 equals 9. All squared is a 389 00:33:29,300 --> 00:33:35,246 common factor. So we can take it out and we've got cost squared 390 00:33:35,246 --> 00:33:40,238 Theta plus sign squared. Theta is equal to 9 cost squared plus 391 00:33:40,238 --> 00:33:45,230 sign squared is an identity cost squared plus sign squared of the 392 00:33:45,230 --> 00:33:50,638 same angle is always one, and so R-squared equals 9. R is equal 393 00:33:50,638 --> 00:33:53,550 to three IE a circle of radius 394 00:33:53,550 --> 00:33:56,455 3. Let's 395 00:33:56,455 --> 00:34:03,265 take. The rectangular hyperbola XY is 396 00:34:03,265 --> 00:34:05,830 equal to 4. 397 00:34:06,280 --> 00:34:13,168 And again, we're going to use X equals our cause theater and 398 00:34:13,168 --> 00:34:18,908 Y equals R sign theater. So we're multiplying X&Y together. 399 00:34:18,908 --> 00:34:25,222 So when we do that, we're going to have our squared. 400 00:34:25,290 --> 00:34:32,870 Sign theater Cos Theta equals 4. 401 00:34:34,350 --> 00:34:41,936 Now. Sign Theta Cos Theta will twice sign tita cost 402 00:34:41,936 --> 00:34:45,156 theater would be signed to 403 00:34:45,156 --> 00:34:51,006 theater. But I've taken 2 lots there, so if I've taken 2 lots 404 00:34:51,006 --> 00:34:54,889 there, it's the equivalent of multiplying that side by two. So 405 00:34:54,889 --> 00:34:57,007 I've got to multiply that side 406 00:34:57,007 --> 00:34:59,386 by two. So I end up with that. 407 00:35:00,580 --> 00:35:01,819 For my equation. 408 00:35:02,670 --> 00:35:06,856 I still some the other way round now, but one point to notice 409 00:35:06,856 --> 00:35:10,730 before we do. Knowledge of trig identity's is very 410 00:35:10,730 --> 00:35:14,105 important. We've used cost squared plus sign. Squared is 411 00:35:14,105 --> 00:35:18,605 one and we've now used sign to Theta is equal to two 412 00:35:18,605 --> 00:35:21,980 scientist accosts theater. So knowledge of those is very 413 00:35:21,980 --> 00:35:26,855 important. So as I said, let's see if we can turn this around 414 00:35:26,855 --> 00:35:31,355 now and have a look at some examples going the other way. 415 00:35:32,660 --> 00:35:39,954 First one will take is 2 over, R is equal to 1 plus cause 416 00:35:39,954 --> 00:35:45,930 theater. I don't like really the way it's written, so let's 417 00:35:45,930 --> 00:35:51,762 multiply up by R so I get R Plus R cause theater. 418 00:35:52,540 --> 00:35:57,530 Now, because I've done that, let's just remember that are 419 00:35:57,530 --> 00:36:02,021 squared is equal to X squared plus Y squared. 420 00:36:03,080 --> 00:36:09,968 So that means I can replace this are here by the square 421 00:36:09,968 --> 00:36:13,412 root of X squared plus Y 422 00:36:13,412 --> 00:36:19,595 squared. Our costs theater. Will, our Cos Theta is equal 423 00:36:19,595 --> 00:36:25,145 to X so I can replace this bit by X. 424 00:36:26,420 --> 00:36:31,136 Now it looks untidy's got a square root in it, so naturally 425 00:36:31,136 --> 00:36:36,638 we would want to get rid of that square root. So let's take X 426 00:36:36,638 --> 00:36:38,210 away from each side. 427 00:36:38,370 --> 00:36:44,090 And then let's Square both sides. So that gives us X 428 00:36:44,090 --> 00:36:50,850 squared plus Y squared there and on this side it's 2 minus X 429 00:36:50,850 --> 00:36:57,610 all squared, which will give us 4 - 4 X plus X squared. 430 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 431 00:37:04,570 --> 00:37:08,970 left with Y squared is equal to 4 - 4 X. 432 00:37:10,410 --> 00:37:15,503 And what you should notice there is that actually a parabola. 433 00:37:16,180 --> 00:37:22,251 So this would seem to be the way in which we define a 434 00:37:22,251 --> 00:37:24,119 parabola in polar coordinates.