# 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

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:

 $\ds x$ $=$ $\ds OP \cos \theta$ $\ds y$ $=$ $\ds OP \sin \theta$

Then:

 $\ds x'$ $=$ $\ds OP \map \cos {\alpha + \theta}$ from the geometry $\ds$ $=$ $\ds OP \paren {\cos \alpha \cos \theta - \sin \alpha \sin \theta}$ Cosine of Sum $\ds$ $=$ $\ds OP \cos \theta \cos \alpha - OP \sin \theta \sin \alpha$ factoring $\ds$ $=$ $\ds x \cos \alpha - y \sin \alpha$ substituting $x$ and $y$

and:

 $\ds y'$ $=$ $\ds OP \map \sin {\alpha + \theta}$ from the geometry $\ds$ $=$ $\ds OP \paren {\sin \alpha \cos \theta + \cos \alpha \sin \theta}$ Sine of Difference $\ds$ $=$ $\ds OP \cos \theta \sin \alpha + OP \sin \theta \cos \alpha$ factoring $\ds$ $=$ $\ds x \sin \alpha + y \cos \alpha$ substituting $x$ and $y$

The result follows.

$\blacksquare$

## Examples

### Right Angle

Let $r_\Box$ be the rotation of the plane about the origin through a right angle.

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

Then:

$\map {r_\Box} P = \tuple {y, -x}$