 In this video, we want to piece together the information we've learned about cosines, so that we can actually establish with the graph of cosine. So thinking of the unit circle diagram, when you're at zero radian angle, the x-coordinate on the unit circle is going to be one. So we get the point when x, the angle is zero, the cosine ratio will be one. Then moving up the unit circle to the top of the unit circle at pi halves, the cosine becomes zero at that point. When we go from pi halves to pi, think of the unit circle, cosine ratio will go from zero to negative one. We get that point right there. And then as we go from pi to three pi halves, cosine will return to zero. And as we go from three pi halves to two pi, we get that cosine will equal one. So we get this type of picture right here. We definitely can add more details to this. So for example, when the angle is pi sixth, we know that cosine is going to be the square root of three over two. So we get that point there. When we're at pi fourth, we know that it's going to be the square root of two over two, like so. And then lastly, we know that at pi thirds, we're going to get one half right there. So if you try to connect this picture together, we're going to get this smooth curve looks something like this. Continuing on, it's going to break down down here, wrap around and come back up like so. And so since cosine is a two pi periodic function, two pi, of course, being a little bit bigger than six right here. Since it's a two pi periodic function, if we know one period, which we have that drawn on the board right here, we can actually just repeat that over and over again to get the entire picture. So what we're going to do is now turn this on the computer and we see exactly our picture is pretty good. I'm going to erase this. Now we're going to zoom out and we see that this is the graph of cosine. It looks a lot like the graph of sine, but there's some important differences. Sine, which if we put that on the screen right now, you get y equals sine of x. Sine, it actually starts at an x intercept goes to a maximum returns to an x intercept goes to its minimum, then comes back up to its x intercept. Cosine on the hand, looking at the y axis, actually starts at its maximum, goes to an x intercept at pi halves, then it goes down to its minimum at pi returns to an x intercept at three pi halves, and then it goes up to its maximum at two pi when it repeats itself. And so this is an important difference. Sine, cosine basically have the same graph, but it would appear that there's some type of shift. This is called a phase shift. There's some type of shift here. It's like one is a little bit farther off than the other. And this is some topic we'll talk about more as we study the graphs of trigonometric functions in this unit.