 Let's take a look at some more complicated polar graphs. So many polar graphs involve the trigonometric functions, so it's useful to remember some key values for sine and cosine. So since r equals cosine theta, what I might do is I might pick a bunch of values of theta and find the values for r, which is the same as cosine theta. So I'll pick my key values of theta and then find the value of r or cosine theta. And we'll plot these points. Now at this point, we've only gone part way around the pole, so we do want to continue. So we'll pick a few more values of theta for which we know the sine and cosine of. We'll use these values of theta because these are values of theta which have a reference angle that's equal to one of these. And we'll find our value for r equals cosine theta. Now because our r values are negative, then plotting these points means we're going to turn to face these angles, but then we're going to walk backwards. We're still only halfway around the pole, so we should continue with some additional values of theta, which will give us additional values of r equals cosine theta. And again, our r values are negative, which means that after we've turned to face the direction specified by the angle, we have to walk backwards. Notice that if we walk backwards, however, we're going to land around or possibly on these points we previously plotted. And in fact, if we compare the values of r for the corresponding angles, we see that the distance that we go backwards is going to be the same as the distance we went forwards. As a result, our graph in this interval will just retrace the points we plotted before. And this still doesn't take us all the way around the pole, so let's take up that final lap and find our values for theta, find our values for cosine theta. And this time when we plot the points, because r is positive, we're going to turn to face the direction specified by the angle and then walk out some positive distance, which is going to be the same as the backwards distance we've already traveled to these points. And so again, we just retrace this path. And so that means that our graph is going to look something like this as we connect the dots. All human progress comes from somebody asking, isn't there an easier way of doing this? And the thing to remember is that while we can always plot points, we could use a more sophisticated approach. And this is going to be based on our behavior of sine and cosine. So the thing to remember is that between 0 and pi over 2, sine will increase from 0 to 1, while cosine decreases from 1 down to 0. Between pi over 2 and pi, sine then decreases from 1 down to 0, cosine continues to decrease from 0 down to minus 1. Between pi and 3 pi over 2, sine starts to decrease from 0 to negative 1, and cosine begins to increase from minus 1 to 0. And finally, between 3 pi over 2 and 2 pi, sine increases from minus 1 to 0, and cosine also increases from 0 to 1. So let's take a look at our polar graph again, r equals cosine theta. At theta equals 0, r is equal to 1. As theta goes from 0 to pi over 2, r, or cosine theta, will decrease down to 0. So each time we turn a little farther, we go out a shorter distance. So we might trace a path that looks like this. Now we can keep going. As theta goes from pi over 2 to pi, r, or cosine theta, continues to decrease to negative 1. And since these values are negative, we're actually walking backwards. And so we might trace out a path that looks like this. As theta goes from pi to 3 pi over 2, r, cosine theta, increases from minus 1 to 0. And since these values are negative, we do go backwards. And again, we, in fact, retrace these steps that we've already made. Finally, as theta goes from 3 pi over 2 to 2 pi, r will increase from 0 to 1. And this time we'll be going forward the same distance we went backward before. So again, we'll retrace our steps. And we put this all together. We're going to end up with the graph of a circle. So how about r equals sine of 2 theta? So again, let's consider the behavior of sine and cosine. And it's important to remember that even though we called this theta because we had to call our angle something, what this really is, is it's the angle that we're taking the sine or cosine of. So we can think about this as r equals the sine of 2. And so I know that as long as it is between 0 and pi over 2, r equals the sine of will increase from 0 to 1. Now we do want to say something about theta because that's our direction angle. So we'll put things back where we found them, was 2 theta. So now we can try and solve our inequality for theta, which tells us that r will increase from 0 to 1 over the interval of theta between 0 and pi over 4. And so we could begin to graph our curve. And we can continue. If it is between pi over 2 and pi, then r equals sine of will decrease from 1 to 0. So r is 2 theta. And so we find that r will decrease from 1 to 0 over the interval between pi over 4 and pi over 2. So we might continue our graph this way. If it is between pi and 3 pi over 2, r equals sine of will decrease from 0 to negative 1. So r will decrease from 0 to negative 1 over the interval between pi over 2 and 3 pi over 4. And so we might continue our graph showing this decrease because these are negative numbers. We're actually going backwards, even though we're facing these angles between pi over 2 and 3 pi over 4. From between 3 pi over 2 and 2 pi, r equals sine of will increase from minus 1 to 0. So r will increase from minus 1 to 0 over the interval, which will give us a graph that looks something like this. Now notice that we've only gone halfway around the pole. We've only used angles between 0 and pi. So how do we find the rest of the graph? We might reason as follows. Since sine is periodic, then the cycle of r values will repeat over the interval where is between 2 pi and 4 pi. And so that means this graph will repeat over the interval between pi and 2 pi. Now, what this means is that we'll go out the same distances forward or back as necessary. But this time, we'll be facing the angles in this lower half circle between pi and 2 pi. And so if we continue our graph, we're going to end up with a shape that looks like this.