Learn about Polar Coordinates 3

Khan Academy Presents: Converting between Cartesian and Polar functions

All right, let’s keep converting cartesian functions to polar coordinates. The next one I have here is 3y-7x=10 and I am about to cut in pieces our tool kit here and let’s see what we can do. We want to convert this to function of r in data. So you see the simplest thing, we have a y and x or we substitute—you know that y is equal to r sin of data and x is r cos of data. Well, let’s just substitute. So (3)(rsin) - 7(x)x, well that’s 7(rcosine) data is equal 10 over this and that just came from SOCATOA, nothing fancier than that. Let’s see if we can simplify this or rather explicit this in terms of data. Let’s see, so we could factor an r, so you get (r) (3sine) data - 7 cosine data. All right, just factor out an r is equal to 10 and I divide both sides by this expression in your left with r = 10. 3 sine of data—10/3 sine of data - 7 cosine of data and we could also write this, you know, we could say, “Well, this is r as a function of data.” We could write r of data― r is a function of data. And I just felt like doing that so you could saw that the function notation even works in polar coordinates, let’s do another one. And then we cut in and paste our tool kit. Look at that, all right. Next one, so it says y is equal to 2x - 3. Well, we could do the same thing. We know that y and xr in terms of r and data, y = r sine of data. So let’s write that, r sine of data = (2) (x). Well, x is r cosine of data, r cosine of data -3. And let’s see if we could separate the r into datas. So let’s subtract this from both sides, so we get r sine of data - 2 rcosine of data is equal to - 3. Just like the last problem, we can factor an r out, so we get (r) (sine of data) - 2 cosine of data is equal to -3. Now, divide both sides by this expression so you're left with r is equal to –3/sine of data -2 cosine of data. There you go. All right! Now, they want us to do some problems and we clear everything and I want to some problem. We’re going to convert the other way. Let’s write our tool kit down, it’s proved to be useful. All right! Now they want us to convert from polar to Cartesian coordinates and so they give us the polar function r is equal to 4sine of data. So how do we convert this into a function of x and y? So let’s see here. There’s no obvious—let’s think about it a little bit. So we know we have this equation, y is equal to r sine of data. So can we write sine of data—well, you can just—let me do a little side here. So y is equal r sine of data. If we divide both sides by r, what do we get? We get y over r = sine of data right? Well, that’s seems to helpful. We have a sine of data and then we have something at least it gets through of the data out of the equation, we still left with an r but it makes it a lot easier to look at. So let’s do that. sin of data = y/r. So let’s substitute that there so we get r = (4) (sine of data) which is y/r. So 4y/r, we could multiply both sides of this equation by r and you’re left with r2 is equal to 4y. And we know what r2 is equal to― Just to tell us right there, r2 is equal to x2+y2, so you get x2+y2= 4y and there we have it. We have at least an implicit form, we don’t have an explicit equation but I think that’s good enough for now. So you see, it’s really just a lot of algebra and I guess it’s a little bit of tricky and I’m really a lot of algebra and I'm just kind of figuring out how to use this tool kit. Let’s do a couple of more. So let’s say, we paste the tool kit. Let’s say that we have the polar coordinates. R=sine of data + cosine of data and just to say you don’t lose the big picture what we’re doing, you know this is― If we could graph this, we could put your graphic calculator on the polar coordinates that I’ll do future videos where we do graphic and we’ll produce some graphs as data changes as we go around the circle that radius will change and they’ll produce something. I don’t know that it might look like flower petals or something. I don’t know. I don’t have to—I don’t have the institution of exactly what this look like but what we’re saying is we convert to x and y as if you have some graphic in Cartesian coordinates it would look the exact the same way. That’s all we’re doing. How do you express the same relationship in terms of x and y’s? So how do we do this? When the last one—we can these two—we can rewrite, we can divide both sides of these equations by r and we can—this is the same thing as sine of data is equal to y/r, right? I just divide both sides by r and this is a the same thing as cosine of data = x/r. Just divided both sides of this equation by r and you get this. So now, we can use this to substitute back here, so r = sine of data, well we know that’s equal to y/r, so it’s equal y/r + cosine of data. Well cosine of data = x/r. And now we can multiply both sides of this equation by r and we’re left with r2 = y + x and we know that r2 is now that should be second nature, hopefully. So it’s x2 plus y2, all right that’s r2 is equal y + x, there you go. Let’s do one more. Let’s do it without our tool kit. Now we’ll do the different color. So let’s say we had r = a2, so they’ll leaving abstract but a is some constant, so we hopefully no intuition from a couple of videos, go to this would be a circle right? If A was 3 then we say, oh, r = 9. Right, 32 and then it just be a circle with the constant radius of lines. So how would you write this as a in Cartesian coordinates? Well, we know that x2 plus y2 = r2. So one thing we could do now is just we’re both sides of this equations. You get r2 with (a2)2 that’s a4, right? It’s A to the 2x2, so A to the fourth. And we know that xr2 is it is x2 + y2. So you get x2 + y2 = a4. Not too bad, not too bad. And that’s really; I mean that will give you pretty far in converting between Cartesian and polar coordinates. In the next video we’ll maybe explore of a couple of this graph and hopefully give a little intuition of really how the relationship between r and data really works. See you in the next video.