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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 XY after rotation we want to 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 1 0 it gets rotated to a point X prime Y prime is 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 in 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 1 because the point 1 0 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 1 equals sine theta or in other words Y prime equals sine theta 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 1 equals cosine theta meaning that x prime equals cosine theta this tells me that the point 1 0 gets rotated to the point cosine theta sine theta use the next exercise to get some practice with these ideas