Sine and Cosine come from circles

Now I'm going to clear off the screen here, and we're going to talk about the shape of the sign function. Let's do that. This is a plot of the sign function, where the angle Theta—this is the Theta axis in this plot—where Theta has been plotted out on a straight line instead of wrapped around this circle.

So if we draw a line on here, let's make this, uh, this circle a radius one. If I draw this line up here and it's on a in a circle, the definition of sine of theta—this will be Theta here—is opposite over hypotenuse. So this is the opposite side, and that distance is the opposite leg of that triangle is this value right here. So sine of theta is actually equal to Y over the hypotenuse, and the hypotenuse is one in all cases around this.

So if I plot this on a curve, this is an angle, and I basically go over here and plot it like that. And then as Theta swings around the circle, I'm going to plot the different values of Y. If it comes over this way, down here like this, right? You can see that that plots over there like that. Now when the angle gets back all the way to zero, of course the sine function comes all the way back to zero, and then it repeats again as our vector swings around the other way.

So sine of 2 pi is zero, just like the sine of zero. So every two pi, if I go off the screen, every two pi comes back and repeats to zero. Now I want to do the same thing with the cosine function that we did with sine, where we project the projection of this value onto this time the cosine curve down here. This has the cosine curve with time going down on the page, and our definition of cosine was adjacent over hypotenuse. The hypotenuse is one in our drawing.

So cosine of theta equals adjacent, which is X, the x value, divided by hypotenuse, which is one. So in this diagram, the cosine of theta is actually the x value, which is this x right here.

So let me clean this off for a second, and we'll start at the beginning. Let's start with the radius pointing straight sideways, and we know that cosine of theta equals zero is one. So if I drop that down, if I project that down onto the angle zero, that's this point right here on the curve.

Now as we roll forward, we go to a higher angle. This projection now moves to here on the curve. When the arrow is straight up, we are at this point right here. We go back through the axis. If we continue on, this projects down here, and we're moving this radius vector around in a circle like this. Eventually, this one will be at the same point as before, as the one above, but it'll be on this part of the curve here.

And when we get back to zero again, the projection is to this point here. So that's a way to visualize the cosine curve getting generated by a vector rotating around this circle. The cosine comes out the bottom because it's the projection on the x-axis. When we did the sine, it was the projection on the y-axis that produced the sine wave when we went this way.

So I like to visualize this because this rotating vector is a really simple and powerful idea, and we can see how it actually generates—a way to generate sine and cosine waves. You can see how sort of naturally they come out at different phases, right? The sine starts at zero, and the cosine starts at one. With this way of drawing it, you can see why that happens.

So this relationship between circles and rotating vectors and sines and cosines is a very powerful idea, and we're really going to take advantage of this.