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We're now equipped, I think, with all of the tools to convert back and forth between polar and rectangular coordinates. And hopefully you have a reasonable intuition. I mean, they're both just two ways of specifying a point in two dimensions. Cartesian, you kind of say how far left and right you go-- that's the x-coordinate. And then the y-coordinate tells you how far up and down you go when you get to a point. When you do polar coordinates, the angle tells you what direction to point in, and then the r, or you could view it as the radius, tells you how far to walk in that direction. Right? So they're just two ways to specify a point in space. And as we'll see some functions are easier to specify, or they're better described, in polar coordinates, while other functions are better described in Cartesian coordinates. But before we actually do functions, let's do a couple of quick problems using the tools that we've built. Let's say that we're given the polar coordinate 4, 150 degrees, and we wanted to convert this to Cartesian coordinates. So we want to convert it to x, y. Our first thing that we could do is just to get an intuition of where this even is. Right now I think you're already at the point where if I give you an x and y coordinate you already have a visualization of where that shows up. But polar might be relatively new, so let's try to draw it. So for me, when I do polar I like to think about the angle first. So 150 degrees. So if we start at the x-axis. Let's see. That's 90 degrees. All the way around is 180 degrees. So this will be 30 degrees away from 180, so it's going to be in this direction. Right? This angle right here is 150 degrees. And I do find degrees easier to visualize than radians. That's, I think, just because we use that in everyday life. And then we're saying, OK I've pointed you in this direction. How far should you go? It says, well you go 4 units in the 150 degree direction. So you go 4 units. I don't know what that'll be. I don't want to do that. So I want to go 4 units in that direction. Let me do it in a different color. I'm just assuming this is one, two, three, four units. So that's 4, 150 degrees. So we want to convert this to Cartesian coordinates. And so we already have an intuition that our x-coordinate should be some negative number, and our y-coordinate should be some positive number. Right? And let's see if it all bears out. So what are the tools that we came up with in the last video? Well, we came up with x is equal to r cosine theta. And that just fell out of SOH-CAH-TOA. Nothing fancy there. And y is equal to r sine theta. So this coordinate, x is equal to r cosine theta. So this is going to be-- let me write it in a different color. x is r-- so now we're dealing with Cartesian coordinates-- r cosine theta. So r is 4. So it's 4 cosine of 150 degrees. And then the y is r, which is 4, times sine of 150 degrees. And now we can get the calculator out. OK. So let's see. What's the cosine of-- well, I'll just write it out. So 4 times the cosine of-- and make sure your calculator is in degree mode. You don't want to make it think you're saying 150 radians. So it's minus 3.46. So the x-coordinate is minus 3.46. And what is the y-coordinate? It's 4 times sine of 150 degrees, which is equal to 2. And that makes sense, right? I mean if we eyeball it, that looks about right. If we go one, two, three, a little bit more than three, that looks like minus 3.46. And then we go up one, two. That looks like about positive 2. So our intuition was right. You could either kind of orient yourself into the 150 degree direction and then march forward in that direction, or you could say I'm going to go minus 3.46 to the left and 2 up. So we've just done a transition from polar to Cartesian coordinates. Not too bad. So now let's step it up a little bit and actually convert functions from Cartesian to polar and polar to Cartesian, and back and forth. So I just did a random web search for problems and I found these. The first one, they want us to convert this, x squared plus y squared is equal to 4, to polar coordinates. Right? And hopefully you have some intuition already that this is a circle. Right? And it's actually even more obvious when you do it in polar coordinates. So what were the toolkits we had built? We had figured out that x squared plus y squared is equal to r squared. That was just the Pythagorean theorem. We knew that the tangent of theta was equal to the opposite over the adjacent-- y over x. And we also learned that y is equal to r sine of theta, and x is equal to r cosine theta. And you can prove all these just from SOH-CAH-TOA and the Pythagorean theorem. But if you had to memorize one thing going into an exam on polar coordinates it would be these four equations. Not that I'm a big fan. I actually don't have these memorized. I always spent, literally, the first 10 seconds of the exam writing these down or reproving them, and that way you'll never forget it. But anyway, let's convert this to polar coordinates. Well, you should already see we have an x squared plus y squared. And we already say, well x squared plus y squared is equal to r squared. So we could just rewrite this. This r squared is equal to 4. And then you say, oh maybe r is equal to plus or minus 2. Right? And it actually doesn't matter. You could just say r is equal to 2. And what does that mean? Actually, I think it'll be instructive to graph this one. I wasn't planning on it, but I think it'll be neat to show you. I mean r equals 2, that's such a simple equation. How can that be a circle? Well what is a circle? A circle is something of constant radius around some point. Right? So this is staying a constant radius. Radius is equal to 2. So if we take a distance of-- so this is one, two. So it says no matter what the angle is, right? This function doesn't even involve the angle. It says the radius is constant. It doesn't matter what angle I have. So if the angle is 0 the radius is 2. If the angle is, I don't know, 30 degrees the radius is still 2. If the angle is 60 degrees it's still 2. So the radius is 2 no matter what. And I should probably use the circle tool. So as we go around it doesn't matter. The radius is always going to be 2. This distance is always going to be 2, no matter what direction we're pointed in. And to some degree, that's the definition of a circle. A circle, in polar coordinates, is actually a lot easier to write down. Radius is equal to 2. That's a circle. Let's do the next one. Actually, I'm going to erase this because I want to reuse this over here. So let me just black that stuff out. OK. And then here I go. All right. The next one they give us is x squared plus y squared is equal to 9 times y/x squared. All righty. If we do some pattern matching, well x squared plus y squared we know that's equal to r squared. So this is equal to r squared is equal to 9 times-- let's see, y over x. That looks suspicious. Tangent of theta is equal to y/x. So this is tangent of theta. So 9 times tangent of theta squared. And maybe we want to take the square root of both sides. Right? So r is equal to-- remember, we have to take the square root of both of these, because we're taking the square root of the whole both sides. So it's r is equal to, we could say plus or minus 2 tangent of theta, but it actually doesn't matter once again. And that happens a lot in polar coordinates. Actually, let me show you what a negative radius means, just so that you get the intuition why in that previous example it didn't matter whether I took the radius as minus 2 or positive 2. So for any given angle-- I just want to give you an intuition of what a negative radius is-- if I give you a positive radius we're going to go in that direction for this angle. But if I give you a negative radius we go in the other direction. So this would be a positive radius, and this would be a minus radius. You just go backwards. You walk backwards in that direction. So in that previous example we could say, OK the radius is equal to 2 no matter what. Right? So when you're angle is in this direction your radius is equal to 2. Or you could say the radius is always equal to negative 2. Right? When you're pointed in this direction the radius is equal to minus 2. So it's always going to be minus 2. But it ends up being the same thing either way. And actually it turns out the same thing for this equation, so you could just say r is equal to 3 tangent of theta. If you're not sure, I guess it doesn't hurt to put the plus or minus there. Next one. Let me erase this one again. Actually, I'm almost out of time. Let me continue this in the next video.