INSTRUCTOR: Today we're going to be talking about how to use the picture of a polar curve sketched on a Cartesian coordinate system to draw a picture of the same polar curve but on polar axes. So in this particular problem, we've been given this curve here, which I've sketched in green. And it's been sketched on a Cartesian coordinate system. And we've been told that instead of x and y axes, we have axes for theta and r. 

But either way, the graph is along a Cartesian coordinate system here. And we need to take the information from this graph and translate it so that we can graph this same curve on a polar coordinate system and see what it looks like. So an exercise like this is really just about identifying points along the curve that you've been given so that you can translate them onto the polar coordinate system. 

So obviously, if we look at the graph we have here, we notice that we have the point here 0, 2. So what that means is the angle theta is 0, and we're out a distance of 2 from the origin of the polar coordinate system. We also have this point here, which tells us that when the angle is pi/2, we're out a distance of 0 from the origin. 

We have this point here, which is supposed to be at pi, negative 2, which means that at an angle pi, we're at a distance of negative 2 from the origin. And we have this point here, same thing-- you get the idea-- and this point here, which is supposed to be at 2 pi 2. So we just need to translate these points onto the polar coordinate system and then connect them. 

So first thing you want to do-- look at this graph. Identify points. Maybe you even want to label them like this, 2 comma 2 pi. In polar coordinates, r is always first and then theta, r, theta. So we have r, value of 2, and theta, value of 2 pi. So maybe you want to label them that way. But either way, we're going to go ahead and translate them onto our polar coordinate system. 

So what this tells us is that this first point here, the angle is 0. Remember that this is our angle 0 here along this axis. So at the angle 0, we're out a distance of 2 from the origin. The angle 0 here, we're out a distance of 2. So let's go ahead and call this here-- let's call this 2. We're out a distance of 2 from the origin, so we're right here. 

OK. Now our graph comes up a little bit and then goes down until it intersects this point here, which is the angle pi/2 out a distance from the origin of 0 because the value of r here is 0, the angle is pi over 2. So on our polar coordinate system, this is the angle 0 here. This is the angle pi/2. This is pi. This is 3 pi over 2. And then this is coming back to 2 pi as well. 

So we go around like that with the angle. 

So at pi/2, the angle pi/2, we're at a distance of 0 from the origin. So that means that we have this next point here, and we want to go ahead and connect them. So what this means is that as we move from this point here, where we're at a distance from the origin of 2, and we move toward the angle pi over 2, toward this axis here, we're going to move out from the origin a little bit. We're going to go farther away from the origin or farther away from this axis. This distance here becomes greater. 

As we move out a little bit from the origin, we're going to curl back and come toward the origin until we hit this point. So we're going to move out from the origin a little bit. And then as we move toward pi/2, we're going to come back and curl in until we hit this point here. Now, from this point, we're moving from the angle pi/2 to pi. That's our next interval. So pi/2 to pi, we're moving from this angle here down toward this angle. 

Well, when we get to the angle pi, we notice that we're out a distance of negative 2 from the origin. So the angle pi is literally along this axis in this direction. If we were at a distance of positive 2, we'd be over here because we'd be going toward this direction of pi here. But because it's negative 2, along this angle, we start from the origin and we move backwards from that angle in this direction until we go a distance of 2, so we end up back at this point again. 

So we know that from the point we're at now right here at the origin, on this interval here, we're going to end up back at this point. We're going to curl back toward this point. And what we can see is that as we do that, we go out a little bit farther than 2. This is 2 right here. And we're going to have this dip down here, where we go out farther than 2 and then curl back toward it. So that's the same symmetrical shape that we had before, where we come out a little bit farther and then go back until we hit that point too, this point right here. 

Now as we move from the angle pi to 3 pi over 2, we're going to come back to a distance out from the origin of 0. So we're moving from the angle pi here to 3 pi over 2. At pi, we were a distance of negative 2, so we were out this way. We're moving to 3 pi over 2, and we're going to come back to the origin. So if we just continue our path, as we move toward 3 pi over 2, we're going to curl back toward the origin. So this section is going to look like this. We're going to come back toward the origin. 

Now we're back at the origin. And here, from negative 3 pi over 2 to 2 pi, we're going to go out again to a distance of 2. So as we move from negative 3 pi over 2 out to 2 pi, we're going to move back out to a distance of 2. At the angle 2 pi, this is the positive distance of 2, where we want to be. So we're going to move back in this direction here out a distance of 2 until we hit that point. And this is the same graph of the polar curve but just translated onto the polar coordinate system. 

So that's how you use the polar curve plotted on a Cartesian coordinate system to visualize and translate the curve onto a polar coordinate system. So I hope you found this video helpful. If you did, like this video down below and subscribe to be notified of future videos.