 So this is a quick overview of some of the key identities and results from trigonometry. So let's begin with the unit circle. If we're looking at trigonometric functions, it's convenient, very, very, very useful to think about these in terms of a circle of radius r equal to 1. That's the unit circle. So there's my well-drawn circle. And what I'm going to do, remember that when we measure angles, what we do is we start here at the point on the positive x-axis, and we walk around the circle counterclockwise, and the distance that we have walked marks a point on the circle. And then the line joining that point to the center of the circle marks the other side of an angle. So we'll walk around the circle some distance, we'll mark out whatever our angle is, and then the horizontal and vertical distances to that point on the circle. So I've walked some distance, that marks out some angle, I've walked some distance, I've marked out an angle, and there's a horizontal and vertical distance from the origin to the point. We call those the x and y coordinates of the point, and these correspond to our two fundamental trigonometric functions. The horizontal distance, our x coordinate, corresponds to cosine of theta, cosine of the angle. The vertical distance, what we think about as our y coordinate, is going to correspond to the sine of the angle. Now we always measure this angle by going counterclockwise from the positive x-axis. So I always take my walks, starting at the point of the positive x-axis, that's the coordinates 1, 0, and I walk counterclockwise around the circle, and that's always going to be how we measure the angle. However, it's often convenient to use what's known as the reference angle. That's going to be the angle from either side of the x-axis to the point. So I'm going to go from the nearest point on the x-axis, and I'm going to walk to meet you at wherever you've walked. So you've walked over to here, I walk from the close point on the x-axis to meet you right at that point, and that's going to subtend some other angle. And in this case, we'll call that angle phi. And those reference angles might be measured counterclockwise. So they are here. I've measured counterclockwise, or they might be measured clockwise. For example, if I go, if you walk all the way around and then sum over to here, if I'm going to meet you, I'm going to start here at the negative x-axis and walk counterclockwise to catch up with you. And so there's my reference angle. On the other hand, if you keep walking, you end up way over here. I'm going to walk this time clockwise from the positive x-axis to meet you. Now one of the things that emerge from consideration of these reference angles are a couple of important identities. So suppose I go halfway around the circle. So again, my circle has radius r equal to 1. Once around the circle is the circumference that's 2 pi times the radius 2 pi. And if I only go halfway around the circle, that's going to take the distance of pi. So I go halfway around the circle and then I realize, oops, I've gone too far. Let's back up a little bit. So I'll back up some amount. So I've gone a distance of pi, but I've actually gone backwards some amount theta. So this will be pi minus theta is going to be the angle that I've subtended. Now at this point, remember that the vertical distance to the point corresponds to the value of the sign. So the vertical distance there is going to be the sign of pi minus theta. That's how far I've gone. Now suppose I just measure out the reference angle theta. So I'm going to start again at the positive x-axis. I'm going to walk the same distance theta along the circle. So remember, I originally walked halfway around and then went back. Now I'm just going to walk that distance back. But this time I'm going to start at the point of the positive x-axis. And I consider the angle and I look at that vertical distance. That's going to be the sign of theta. And if I look at the geometry here, the vertical distance here is equal to the vertical distance there. The y-coordinate of this point is the same as the y-coordinate of this point. So that tells me that the sign of pi minus theta and the sign of theta are going to be exactly equal. Likewise, if I consider my horizontal distance. So I've gone halfway around the circle and back some. I've gone an angle pi minus theta. I have my horizontal distance here that's going to correspond to my cosine value. And again, if I just walk the reference angle, then my horizontal distance there is going to be the cosine of theta. And here, again, the geometry tells us that these two as lengths are going to be equal, the length of this side here and the length of this side are going to be the same. But because I'm also considering the sign, remember the SIGN sign, this is a x-coordinate that is negative. This is a x-coordinate that is positive. That tells me that these two have opposite SIGN signs. So that tells me the cosine of pi minus theta. This cosine value is the negative of the cosine of theta.