WEBVTT 00:00:20.101 --> 00:00:26.356 The first step to understanding polar coordinates and polar equations is to learn how to plot polar coordinates 00:00:26.358 --> 00:00:35.856 on a polar plane. A polar coordinate will have two values r and theta. Theta will tell us our angle of rotation, 00:00:35.858 --> 00:00:44.356 and r will be a directed distance along this angle. You can see on our polar plane that we have concentric 00:00:44.358 --> 00:00:55.104 circles of radius 1, 2, 3, and 4 units. Looking at our first example, r is 2, which lets me know that this point 00:00:55.106 --> 00:01:03.601 will be lying on the circle with the radius of 2. Our angle of rotation is 2 pi over 3. So let's start 00:01:03.603 --> 00:01:18.854 by drawing the angle 2 pi over 3, and now our point will be on an r - value of 2. 00:01:18.856 --> 00:01:26.356 Looking at the second example, r is 4, and our angle of rotation is negative pi over 2. So this means 00:01:26.358 --> 00:01:34.352 we'll be rotating pi over 2 radians, but in the negative or clockwise direction. So angle of rotation is 00:01:34.354 --> 00:01:47.605 negative pi over 2, and our point lies on an r - value of 4. What might seem a little strange is seeing the next 00:01:47.607 --> 00:01:53.353 example, where we have a negative value for r because in the past our convention has been that r is always 00:01:53.355 --> 00:01:59.603 a non-negative value because it represents the radius of a circle. However, in the polar coordinate system 00:01:59.605 --> 00:02:06.850 r is now a directed distance, which means it might take on a negative value, and that just indicates a change 00:02:06.852 --> 00:02:14.103 in the direction. So let's see what that means, looking at this example. First we have our angle of rotation, 00:02:14.105 --> 00:02:29.606 5 pi over 6. Starting here, I would be at an r - value of 0, and if I moved along this ray, I would be at an r - value 00:02:29.608 --> 00:02:39.352 of 1, 2, 3, or 4. Since r is negative 3, I'm going to move in the opposite direction. So starting here at 0, I would 00:02:39.354 --> 00:02:52.357 move 1, 2, 3 units. So I'm still lying on the circle with a radius of 3 but moving in the opposite direction. 00:02:52.359 --> 00:03:02.605 In our final example, we have an r - value of negative 3, and a theta value of negative 5 pi over 6. So let's start 00:03:02.607 --> 00:03:11.105 with negative 5 pi over 6, which means we're rotating 5 pi over 6 radians in the negative direction. 00:03:11.107 --> 00:03:25.606 Now a value of negative 3 for r means we would move 1, 2, 3 units in the negative or opposite direction. 00:03:25.608 --> 00:03:32.356 Take a minute to think about another way that we might represent this same point but using a positive 00:03:32.358 --> 00:03:50.353 value for r. I might call this point positive 3, and then use an angle of pi over 6. This ordered pair 00:03:50.355 --> 00:03:58.888 (3, pi over 6) would lie on the same spot as this last example. You can see that, unlike a rectangular 00:03:58.890 --> 00:04:06.604 coordinate system, where each point can only be represented in one way, any point on the polar plane 00:04:06.606 --> 00:04:11.850 can be represented in an infinite number of representations. 00:04:13.855 --> 00:04:22.105 As we just saw in our last example, there are different ways of representing a point on the polar plane. 00:04:22.107 --> 00:04:31.105 Let's take for example the point (4, pi over 4). We could represent the same point by using an angle that's 00:04:31.107 --> 00:04:45.354 coterminal to pi over 4. So if I take the angle pi over 4 and add 2 pi, I get an angle 9 pi over 4 that is 00:04:45.356 --> 00:04:56.353 coterminal to pi over 4. Therefore, the ordered pair (4, 9 pi over 4) would lie on the same spot on the 00:04:56.355 --> 00:05:04.855 polar plane. I could find any other coterminal angle by continuing to add or subtract any multiple of 2 pi 00:05:04.857 --> 00:05:15.855 and would still have a point that lies in the same spot on our polar plane. Another way of naming this point 00:05:15.857 --> 00:05:27.856 but using a negative value for r would be to find the angle that forms a line with pi over 4. So if I take pi over 4, 00:05:27.858 --> 00:05:50.602 and add pi, I find that this angle is 5 pi over 4. The ordered pair (negative 4, 5 pi over 4) would lie on the same spot. 00:05:50.604 --> 00:06:01.106 5 pi over 4 would rotate to here, and then a value of negative 4 would move 4 units in the opposite direction. 00:06:01.108 --> 00:06:09.856 So in this way we're able to represent any point in an infinite number of ways. Let's look at an example. 00:06:09.858 --> 00:06:17.352 We're asked to name three different representations of the point (2, 2 pi over 3). You may recall this is the 00:06:17.354 --> 00:06:33.863 first point that we plotted in the earlier examples. In our example the first one that we're asked to find, we're 00:06:33.865 --> 00:06:42.880 asked to represent it using a value of r that is positive but a value of theta that lies between negative 2 pi and 0. 00:06:42.882 --> 00:06:56.600 To do that I need to find an angle that's coterminal to 2 pi over 3 but has a negative value. We take 2 pi over 3 00:06:56.602 --> 00:07:10.106 and subtract 2 pi. We get negative 4 pi over 3. So negative 4 pi over 3 is greater than or equal to negative 2 pi, 00:07:10.108 --> 00:07:25.856 and less than 0, and is coterminal to 2 pi over 3. So we can represent this point as (2, - 4 pi over 3). 00:07:25.858 --> 00:07:32.353 Taking a look at the next example notice that we're now asked to represent this point using a negative value 00:07:32.355 --> 00:07:45.602 for r, which means that I need to find the angle that forms a line with 2 pi over 3. To do that, I'm going to 00:07:45.604 --> 00:08:02.854 take 2 pi over 3 and add pi. Notice that I still want my angle to lie between 0 and 2 pi. If I take 2 pi over 3 00:08:02.856 --> 00:08:20.859 and add pi, I get the angle 5 pi over 3. I can represent this point as (negative 2, 5 pi over 3). My angle 5 pi over 3 is here. 00:08:20.861 --> 00:08:32.101 A value of negative 2 would move two units in the opposite direction. Finally we're asked to find a third 00:08:32.103 --> 00:08:38.852 representation, where we have a positive value for r but now a value of theta that is greater than or equal to 00:08:38.854 --> 00:08:48.103 2 pi but less than 4 pi. So I need to find the angle that's coterminal to 2 pi over 3 but that lies between 00:08:48.105 --> 00:09:03.355 2 pi and 4 pi. If I take 2 pi over 3 and add 2 pi, I get an angle of 8 pi over 3. So the third way that we can 00:09:03.357 --> 00:09:17.854 represent this point is (2, 8 pi over 3). Now as we said, there are infinitely many ways that we can represent 00:09:17.856 --> 00:09:28.355 this point, but all four of these representations would result in the same point on our polar plane. 00:09:31.000 --> 00:09:37.354 In order to convert from polar coordinates to rectangular coordinates, we need formulas that help us relate 00:09:37.356 --> 00:09:48.353 r and theta to x and y. Luckily we already know these formulas from previous sections. So look at point P, 00:09:48.355 --> 00:10:03.852 that's r and theta. If we think about this in terms of a right triangle, we have sides x, y, and r. We know that 00:10:03.854 --> 00:10:19.850 sine theta is equal to y over r. If we take this formula and rearrange it to solve for y, we get y equals r sine theta. 00:10:19.852 --> 00:10:30.853 We also know that cosine of theta is equal to x over r. Rearranging this formula to solve for x, we see that 00:10:30.855 --> 00:10:41.102 x is equal to r cosine theta. So these are the two formulas that we will be using to convert from a 00:10:41.104 --> 00:10:48.604 polar coordinate to a rectangular coordinate. You should realize that they are not new formulas to you. 00:10:48.606 --> 00:10:56.105 They're just rearranged into a new form that's more useful for what we're doing here. Looking at our first example, 00:10:56.107 --> 00:11:04.602 we have the polar coordinate (3, 2 pi over 3), which would be in the second quadrant. You can think about it 00:11:04.604 --> 00:11:11.852 as being right here. Picture taking a rectangular coordinate plane and lying it on top of our polar plane. 00:11:11.854 --> 00:11:21.603 We want to find to find the corresponding point (x, y) that would be in that same location. To find x, we can 00:11:21.605 --> 00:11:34.104 use the formula x equals r cosine theta. In this case r is 3, and theta is 2 pi over 3. Thinking about cosine of 00:11:34.106 --> 00:11:42.854 2 pi over 3, we see that it has a reference angle of pi over 3. The angle lies in quadrant two where cosine is 00:11:42.856 --> 00:11:55.855 negative, so the cosine of 2 pi over 3 is negative 1/2. Our x-coordinate is negative 3/2. To find the y - value 00:11:55.857 --> 00:12:08.854 for our ordered pair, we use the formula y equals r sine theta. So y is equal to 3 sine of 2 pi over 3. 00:12:08.856 --> 00:12:18.103 Again a reference angle of pi over 3, quadrant two where sine is positive, so the sine of 2 pi over 3 is 00:12:18.105 --> 00:12:32.852 square root of 3 over 2, and so our y - value is 3 square root of 3, over 2. Our rectangular coordinate is 00:12:32.854 --> 00:12:55.102 (- 3/2, 3 square root of 3 over 2). It's always good to check to see that these two points would lie in the same 00:12:55.104 --> 00:13:02.354 quadrant. We already established that this point is in quadrant two. Looking at your rectangular coordinate, 00:13:02.356 --> 00:13:08.603 we have a negative value for x, and a positive value for y so this point in rectangular form 00:13:08.605 --> 00:13:21.855 also lies in quadrant two. Looking at the second example, (7, pi), we could use these formulas in the same way 00:13:21.857 --> 00:13:27.102 that I did in the first example. Feel free to do that if you're more comfortable with that. However, since 00:13:27.104 --> 00:13:34.854 the point lies on the x-axis, I can probably figure out the rectangular coordinate even without using the formulas. 00:13:34.856 --> 00:13:47.605 Let's look at our polar coordinate. Pi means that we would lie on the negative x-axis at an r-value of 7. 00:13:47.607 --> 00:14:01.105 So we would rotate over to pi, and we would move 7 units to the left. So thinking about this in terms of an x - y axis, 00:14:01.107 --> 00:14:11.854 if I move 7 units to the left then x would be negative 7, and since I'm on the x - axis the y - value is 0. 00:14:11.856 --> 00:14:18.854 Now again while you can do this without using the formulas, you could also fill into the formulas just like 00:14:18.856 --> 00:14:28.102 we did in the first example. In our third example, the first thing I see is that I have a negative value for r, 00:14:28.104 --> 00:14:34.104 and a negative value for theta. While I could just take these and substitute them into the formula as they are, 00:14:34.106 --> 00:14:40.854 one way that we might make this easier is to rename this using a different representation as we did in 00:14:40.856 --> 00:14:51.855 previous examples. Let's think about where (-4, -3 pi over 4) would be. Negative 3 pi over 4 00:14:51.857 --> 00:15:01.102 would mean rotate 3 pi over 4 radians in the negative direction which would put me in quadrant three. 00:15:01.104 --> 00:15:10.103 An r - value of negative 4 would mean that I would move 4 units in the opposite direction. 00:15:10.105 --> 00:15:22.852 So I see that this point is in quadrant one. I might rename it by taking negative 3 pi over 4 and adding pi, 00:15:22.854 --> 00:15:34.852 which would give me pi over 4. So I would rename this point as (4, pi over 4) and use that representation to 00:15:34.854 --> 00:15:41.352 convert it to rectangular form. Now again if you did not do this and you used the original representation you 00:15:41.354 --> 00:15:47.852 still should get the same answer, but it's just a more complicated problem to figure out. So filling in to our 00:15:47.854 --> 00:16:04.601 formulas, x is equal to r times the cosine of theta and y is equal to r times the sine of theta. 00:16:04.603 --> 00:16:15.098 The cosine of pi over 4 is 1 over square root of 2, and the sine of pi over 4 is also 1 over square root of 2. 00:16:15.100 --> 00:16:25.855 So our coordinate will be (4 over the square root of 2, 4 over the square root of 2). 00:16:28.353 --> 00:16:33.353 Now that we've converted from polar to rectangular coordinates, we're going to take a look at converting 00:16:33.355 --> 00:16:42.603 from rectangular to polar. These first two examples are points that lie on the x - or y - axis. When that's the case, 00:16:42.605 --> 00:16:49.853 you don't need a formula to make the conversion. You just need to plot the point and look at where it's located. 00:16:49.855 --> 00:16:57.603 So let's look at our first point, (0, -1). Keep in mind that we're looking at a rectangular coordinate, 00:16:57.605 --> 00:17:11.855 so values x and y. To convert it we're looking for values r and theta. So looking at this as our x - y axis, 00:17:11.857 --> 00:17:22.601 the point (0, negative 1) would be right here on the negative y - axis. Thinking about that in terms of a polar coordinate, 00:17:22.603 --> 00:17:31.353 let's think about the values of r and theta. If I start at 0, and rotate in the positive or counterclockwise 00:17:31.355 --> 00:17:44.351 direction, theta is 3 pi over 2. Now if you think about our polar plane with concentric circles, it would lie on 00:17:44.353 --> 00:17:58.605 a circle with a radius of 1 unit. So our polar coordinate would be (1, 3 pi over 2). Looking at our second example, 00:17:58.607 --> 00:18:19.605 (4, 0) lies on the positive x - axis, so here's our point (4, 0). Now we need to find r and theta. Since 00:18:19.607 --> 00:18:31.852 our point lies on the positive x-axis, theta is 0 radians. Then if you think about our circles, this point 00:18:31.854 --> 00:18:48.102 would lie on a circle with a radius of 4 units. So in polar form, this point would be (4, 0). Now this might 00:18:48.104 --> 00:18:55.102 seem strange because the rectangular form looks the same as the polar form. It's important 00:18:55.104 --> 00:19:04.854 to remember that this (4, 0) representing x and y doesn't have the same meaning that this (4, 0) has where 00:19:04.856 --> 00:19:12.852 it represents r and theta. You also may be thinking about the fact that there are different ways that we could 00:19:12.854 --> 00:19:19.604 represent any of these points. You might have thought that it was (4, 2 pi), which would be a correct 00:19:19.606 --> 00:19:27.354 representation, but the convention is that the directions will specify in most cases that you want an r-value 00:19:27.356 --> 00:19:36.355 that's positive, and a theta value that's greater than or equal to 0 but less than 2 pi. So while there are 00:19:36.357 --> 00:19:45.856 infinitely many representations for these points, you will generally be asked to find this particular representation. 00:19:49.000 --> 00:19:55.104 In our third example of converting a rectangular coordinate to polar coordinate, we now have a point that 00:19:55.106 --> 00:20:04.853 does not lie on the x - or y -axis. So we're going to need some formulas that help us relate x and y to r and theta. 00:20:04.856 --> 00:20:22.103 Looking at our point (x, y), we want to figure out theta our angle of rotation and r the length of the segment. 00:20:22.105 --> 00:20:34.352 Drawing in our right triangle, we know x and y, and we can find r by using the Pythagorean theorem. 00:20:34.354 --> 00:20:43.104 x squared plus y squared equals r squared. Since we're looking for r, we might go ahead and take the square 00:20:43.106 --> 00:20:51.352 root of both sides and write our formula as r equals the square root of x squared plus y squared. This is 00:20:51.354 --> 00:20:59.855 the formula that we will use when converting from rectangular to polar to find r. In this example, 00:20:59.857 --> 00:21:13.352 x is negative 6 square root of 3, and y is negative 6. When squaring negative 6 square root of 3, remember 00:21:13.354 --> 00:21:23.854 that both the negative 6 and the square root of 3 need to be squared. We have negative 6 squared, times 00:21:23.856 --> 00:21:36.602 the square root of 3 squared. Negative 6 squared is 36, the square root of 3 squared is 3, so multiplying 00:21:36.604 --> 00:21:50.854 36 times 3 we get 108 plus negative 6 squared which is 36. We now have that r is equal to the square root 00:21:50.856 --> 00:22:04.855 of 144, and so r is 12. Now we need to find theta. So again recall from earlier sections how can we relate 00:22:04.857 --> 00:22:13.104 theta to x and y. If you think through your trigonometric functions, we would want to use the tangent function. 00:22:13.106 --> 00:22:27.102 So the formula that we will use to find theta is tan theta equals y over x. Substituting in y and x into our 00:22:27.104 --> 00:22:40.855 formula, we have tan theta equals negative 6 over negative 6 square root of 3, which reduces to positive 00:22:40.857 --> 00:22:50.853 1 over square root of 3. Now we need to solve this trigonometric equation. To do that, we need to find the 00:22:50.855 --> 00:22:58.605 reference angle. To find the reference angle, we're going to need to find the inverse tangent of the absolute 00:22:58.607 --> 00:23:05.105 value of 1 over square root of 3. In this case it happens to be positive already, but recall that in a case where 00:23:05.107 --> 00:23:14.105 this is a negative value, you would find the inverse tangent of the positive value. So to find theta r, we will 00:23:14.107 --> 00:23:27.353 find the inverse tangent of 1 over square root of 3, and that tells me that my reference angle is pi over 6. 00:23:27.356 --> 00:23:36.104 Looking at our coordinate, I see that my angle needs to be an angle that lies in the third quadrant, so I need 00:23:36.107 --> 00:23:54.604 to take my reference angle theta r, which I now know is pi over 6, and add pi to it. Theta will be pi plus 00:23:54.607 --> 00:24:13.602 pi over 6, which is 7 pi over 6. Now we have our polar coordinates. r is 12 and theta is 7 pi over 6. 00:24:16.845 --> 00:24:22.353 As a preparation for graphing equations in polar form, we're going to look at how to convert an equation 00:24:22.356 --> 00:24:29.854 from rectangular form to polar form. To do that, we'll use the same four formulas that we've used earlier in 00:24:29.857 --> 00:24:38.851 this section. In our rectangular equation, the equation will be in terms of x or y or both, and we will need 00:24:38.854 --> 00:24:47.352 to rewrite it to be in terms of r or theta or both. In the first example, our equation is x squared plus y squared 00:24:47.355 --> 00:24:55.597 equals 49. Looking at our formulas, I see that x squared plus y squared is equal to r squared. Therefore 00:24:55.600 --> 00:25:04.101 I can replace x squared plus y squared with r squared, and rewrite the equation as r squared equals 49. 00:25:04.104 --> 00:25:13.601 Taking the square root of both sides, I get two possible equations. r equals 7 or r equals negative 7. 00:25:13.604 --> 00:25:23.851 Either of these is an acceptable polar form for this rectangular equation. In the second example, I again want 00:25:23.854 --> 00:25:31.352 to replace this, the x's and y's so that they're in terms of r or theta or both. To do this I'm first going to take 00:25:31.355 --> 00:25:41.851 the x minus 1 squared and expand it by multiplying x minus 1 times x minus 1. That gives me x squared 00:25:41.854 --> 00:25:53.850 minus 2x plus 1, plus y squared equals 1. Rearranging my terms on the left hand side, I can write x squared 00:25:53.853 --> 00:26:06.353 plus y squared minus 2x, and then if I subtract 1 from both sides, it's now equal to 0. Again x squared plus 00:26:06.356 --> 00:26:17.353 y squared is equal to r squared, so I can replace x squared plus y squared with r squared. I see that x is equal 00:26:17.356 --> 00:26:30.103 to r cosine theta, so I can replace the x with r cosine theta. I now have a polar equation because it's in terms 00:26:30.106 --> 00:26:39.104 of r and theta. However the convention for a polar equation is generally to write it in terms of r equals. So 00:26:39.107 --> 00:26:47.601 I would like to take this equation and solve for r. To do that I'm going to factor out the common factor of r, 00:26:47.604 --> 00:26:59.851 so I have r times r minus 2 cosine theta equals 0. Setting each of these factors equal to 0, I get two possible 00:26:59.854 --> 00:27:16.103 equations. r equals 0 or r minus 2 cosine theta equals 0, which gives me r equals 2 cosine theta. 00:27:16.106 --> 00:27:25.103 The equation r equals 0 is not necessary to include because r equals 2 cosine theta also includes the r - values 00:27:25.106 --> 00:27:35.851 of 0. So the only equation that we need to express this in polar form is r equals 2 cosine theta, and so that is 00:27:35.854 --> 00:27:48.102 the polar form of this rectangular equation. In the final example, we have the rectangular equation x equals 9. 00:27:48.105 --> 00:28:01.600 Again using the fact that x is equal to r cosine theta, I can replace x with r cosine theta. Solving the equation 00:28:01.603 --> 00:28:10.850 for r, by dividing both sides by cosine theta, I get that r is equal to 9 over cosine theta, which can be written 00:28:10.853 --> 00:28:24.601 that way or using one of our reciprocal identities. I could rewrite this as r equals 9 times secant theta. 00:28:24.604 --> 00:28:31.601 So when converting from rectangular to polar form in a equation, you need to use the formulas that we 00:28:31.604 --> 00:28:40.851 know that relate x and y to r and theta to rewrite the equation to be in terms of r or theta or both. 00:28:43.850 --> 00:28:48.860 Now that we've converted equations from rectangular to polar form, we're going to look at how we can use 00:28:48.863 --> 00:28:56.351 those same four formulas to convert from polar to rectangular form. So now we have equations that are in 00:28:56.354 --> 00:29:04.352 terms of r or theta or both, and we want to rewrite them to be in terms of x or y or both. Looking at the 00:29:04.355 --> 00:29:12.601 first example, I have r equals 5. Scanning the formulas I see that if I had r squared, I could use the third 00:29:12.604 --> 00:29:21.352 formula. x squared plus y squared equals r squared. So I want to take my equation and square both sides 00:29:21.355 --> 00:29:32.100 and rewrite it as r squared equals 25. Now I can replace the r squared with x squared plus y squared, and 00:29:32.103 --> 00:29:39.378 my rectangular form of the equation will be x squared plus y squared equals 25. Looking at the second 00:29:39.381 --> 00:29:46.852 example, it might make sense at first glance to maybe square both sides again. However if you look at the 00:29:46.855 --> 00:29:53.600 right - hand side, if we square this side we would then have sine squared theta. If you look at your formulas 00:29:53.603 --> 00:30:00.600 you see that we don't really have a formula that we could immediately use if we had sine squared. Instead 00:30:00.603 --> 00:30:05.850 we're going to try a little bit different tactic. We're going to take this equation and we're going to 00:30:05.853 --> 00:30:19.351 multiply both sides by r. So if I have r equals negative 5 sine theta, and I multiply both sides by r, I see that 00:30:19.354 --> 00:30:30.350 I have r squared equals negative 5 r sine theta. Now I can again replace r squared with x squared plus 00:30:30.353 --> 00:30:42.853 y squared, and then on the right hand side I can replace r sine theta with y. 00:30:42.856 --> 00:30:56.105 I've now written my equation to where each side is in terms of x or y or both. 00:30:56.108 --> 00:31:03.353 In the third example, I see that it looks very similar to example two. So again I might think that multiplying 00:31:03.356 --> 00:31:12.101 both sides by r would be a good tactic. But again if you look at the right hand side, it is now in terms of 00:31:12.104 --> 00:31:21.101 cosecant and we only have formulas that help us relate cosine, sine, or tangent functions to x and y. 00:31:21.104 --> 00:31:29.599 What we're going to do instead is first use the reciprocal identitiy for cosecant, and rewrite cosecant as 00:31:29.602 --> 00:31:42.102 1 over sine theta. Multiplying both sides by sine theta, we have r sine theta equals negative 5. Now I can 00:31:42.105 --> 00:31:52.100 replace r sine theta with y, and the rectangular form of my equation is y equals negative 5. 00:31:52.103 --> 00:32:00.100 So when converting from polar to rectangular form, you'll often have to manipulate the equation so that 00:32:00.103 --> 00:32:08.350 it matches one of your four formulas in order to be able to make a substitution.