Equations defining Plane Rotation/Cartesian

Theorem
Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$.

Let $P = \tuple {x, y}$ be an arbitrary point in the plane.

Then:
 * $\map {r_\alpha} P = \tuple {x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha}$

Proof

 * Rotation-equations-origin.png

Let $r_\alpha$ rotate $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.

Let $OP$ form an angle $\theta$ with the $x$-axis.

We have:
 * $OP = OP'$

Thus:

Then:

and:

The result follows.