The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

## Rotation Equations

The equations for a rotation counter-clockwise through an angle α can be derived as follows. Express the starting point, (x, y) in polar coordinates as (r cos θ, r sin θ). Now rotating by α simply adds that angle to the polar coordinate θ, while leaving r unchanged. Therefore, the rotated point will be (u, v), given in polar coordinates by (r cos (θ + α), r sin (θ + α)). To relate these to the original variables, expand the sine and cosine terms using trigonometric identities. Then we will have

(u, v) = (r (cos θ cos α − sin θ sin α), r (sin θ cos α + cos θ sin α))
= (r cos θ cosαr sin θ sin α, r sin θ cos α + r cos θ sinα)
= (x cos αy sin α, y cos α + x sin α)

This shows that the equations for u and v are given by

u = (cos α) x − (sin α) y
v = (sin α) x + (cos α) y.

That is what we were required to prove.

These equations are expressed using matrix notation in the following form:

.