1. How do we rotate points?

(steps and bouncing) (switch clicks) - In this tutorial, we're gonna take a closer look at the algebra of rotations. We'll be working with the trigonometric functions known as sine and cosine. Let's start by rotating about the origin by an angle theta. The usual convention is that rotating by a positive angle is a counterclockwise motion, and rotating by a negative angle is a clockwise motion. We'd like to find a formula that tells us where every point x, y goes when rotated. Let's let x prime, y prime be the coordinates of the point x, y after rotation. We wanna find formulas for x prime and y prime, in terms of x, y, and theta. One such point is really easy. What happens to the point zero, zero when rotated? It stays still. So x prime equals zero, and y prime equals zero. What about the point one, zero? It gets rotated to a point x prime, y prime as shown here. To determine formulas for x prime and y prime in this case, drop a perpendicular from x prime, y prime to the x axis. The orange length is x prime, and the magenta length is y prime. Notice that the orange, magenta, and green triangle is a right triangle. Notice that the length of the green line, the hypotenuse of the triangle, is one, because the point one, zero is one unit away from the origin. And the lengths don't change when you rotate. Notice too that the magenta line is the line opposite theta, and it has length y prime, which we don't know just yet. The ratio of the opposite side over the hypotenuse is sine theta. That is, y prime over one equals sine theta, or in other words, y prime equals sine theta. (switch clicks on) (gentle ringing) Similarly, the orange line is adjacent to theta, and has length x prime, so if I form the ratio of adjacent over hypotenuse, I get x prime over one equals cosine theta, meaning that x prime equals cosine theta. This tells me that the point one, zero gets rotated to the point cosine theta, sine theta. Use the next exercise to get some practice with these ideas.