 So the most natural way of identifying the location of objects is, ironically, one of the methods that we don't teach until very late in the math curriculum. And that's the method of polar coordinates. A common situation we've all been in is trying to point something out to somebody else. Invariably we point, the other person looks in the direction they think we're pointing at, and there's a time of confusion trying to figure out what you're actually pointing at. So every coordinate system is an attempt to solve the problem of expressing the location of a given point relative to a starting point, which is called the origin. So one way to resolve this problem is to imagine that you're standing at the center of a clock and the direction that you're facing is the 12 o'clock position on the clock. You can then describe the direction to an object by reference to the clock time. So if you're talking about the object at 8 o'clock, that doesn't mean the thing that you run into after dinner, but rather the object you would face if you turned to the 8 o'clock position of your clock. And it's the object that's directly in front of us. This leads to the most natural way of describing the position of an object, which is to give the direction to the object and the distance to the object. For example, 7 o'clock, 71 feet. This leads us to the notion of polar coordinates. Given a point, which we'll call the pole, and a reference direction, we can give directions to any other point in the plane by specifying a rotation angle theta, and then specifying the distance r along that angle to the point. And these values give the polar coordinates r theta of the point. Now there are only so many symbols, so we'll have to recycle them, to make sure we can distinguish between coordinates given in polar coordinates and coordinates given in rectangular coordinates. We use the square brackets to indicate polar coordinates. Now there's two important ideas to keep in mind here. The reference direction is always do right, straight to the right, or what you might think about as the direction of the positive axis axis. The other thing is that the reference angle is measured counterclockwise. So for example, let's graph a couple of points in polar coordinates. So remember in polar coordinates, the first coordinate specifies the distance and the second specifies the rotation angle, but it's generally easier to turn along the rotation angle first and then go out the specified distance. So for this first point, 1, 0, we're going to turn an angle of 0, which is to say don't turn at all, but then go out a distance of 1 unit. To get to the second point, we're going to start at our pole, we're going to turn an angle of pi over 2, and then we're going to walk out a distance of 2 units. And for our third point, we're going to start at our pole, we're going to turn an angle of 3 pi counterclockwise. So remember 2 pi takes us once around, and that extra pi is going to take us halfway around. So this is 1 and a half turns around, and we'll walk out a distance of 1 unit. Now it's possible for either the angle or the distance to be negative, and with an angle that just means we're going to measure clockwise, and with a distance rather than walking towards someplace, we're going to walk away, we're going to walk backwards. So for these points, so again we'll set down our pole and reference direction. For this first point, we'll turn an angle of pi over 6, and now we need to go a distance of minus 2. So let's put in a little reverse path, and we'll go 2 units back along this path. For the second point, we'll reset our reference angle, we'll turn through an angle of minus pi over 4, that's pi over 4 clockwise, then go at distance 3 in that direction. So for our final point, we'll turn minus 5 pi over 6, that's a 5 pi over 6 rotation clockwise, we'll need to back up, so we'll put in that backward path, and then we'll go distance 2 along this backwards path. We can also construct the graph of a polar equation. So for example, if I want to graph the polar equation r equals 2. So the problem of graphing this is not essentially different from the problem of graphing an equation given in rectangular coordinates, we need to find values of r and the associated values of theta. And if we might reason as follows, we already know what r is, so let's find some values of theta. So suppose theta is 0, then r is going to be 2. If theta is pi over 4, r is going to be 2. If theta is pi over 2, r is going to be 2. If theta is pi or 3 pi over 2 or 2 pi, then r is going to be, let me think about this for a minute, 2. And we can plot all of these points as before. Now, just like we have graph paper for graphing with rectangular coordinates, we also have graph paper for graphing with polar coordinates. The grid lines for rectangular coordinates are for specific values of x and y. So the grid lines for polar coordinates will be for specific values of r and theta. So remember that the theta values are rotation angles, and convenient rotation angles correspond to the angles for which we know the sine and cosine of. So we might show a rotation angle of 0, both the forward and backward paths. A rotation angle of pi over 6, forward and back. Pi over 4, forward and back. Pi over 3, pi over 2, 2 pi over 3, 3 pi over 4, 5 pi over 6. And because we included the backward component, we've also included all of the other angles for which we know a sine or cosine value. How about r? Remember that r represents the distance from the pole. So if r equals 1, we've gone a distance of 1 from the pole. That describes a circle of radius 1. r equals 2, r equals 3, r equals 4, r equals 5. Remember the pole itself is our reference point. We might not have it as part of our graph. And this gives us some quick and easy polar graph paper. So now we can try and plot our points. At a rotation angle of 0, we go out 2 units. At a rotation angle of pi over 4, we'll go out 2 units. At a rotation angle of pi over 2, we'll go out 2 units. At a rotation angle of pi, or 3 pi over 2, or 2 pi, we'll go out 2 units. And since we're always going out 2 units, we'll always maintain this 2 unit distance from the pole. And so our graph will look like a circle of radius 2. How about the polar graph theta equals pi? And again, this time we might observe that since we know the value of theta, let's pick values for r. So if r is 0, then theta is pi. And if r is 1, then theta is pi. And if r is 2, then theta is cake, I mean pi. And likewise, if r is 3, 4, or 5, we get a whole lot of pi. So again, we can graph these point by point. So we could put down our polar graph paper. At a rotation angle of pi, we go at distance 0. At a rotation angle of pi, we go at distance 1. And if our angle is pi, we'll go at distance of 2, 3, 4, or 5. And in fact, as long as our angle is pi, we'll go at any distance we want. So we end up with a graph that looks like.