# Mapping/Examples/Rotation through 30 Degrees

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## Example of Mapping

Let $\Gamma$ be the Cartesian plane.

The rotation $R_{30 \degrees}$ of $\Gamma$ anticlockwise through an angle of $30 \degrees$ about the origin $O$ is a mapping from $\Gamma$ to $\Gamma$.

## Proof

For every point $P$ in $\Gamma$ which is not the origin $O$ it is possible to:

- construct a straight line $OP$
- construct a straight line $OP'$ at an angle of $30 \degrees$ to $OP$ measured in an anticlockwise direction
- construct the point $P'$ such that $\len \paren {OP} = \len \paren {OP'}$.

It can be seen that for every $P$ there is a unique $P'$ to which $R_{30 \degrees}$ maps $P$, apart from $O$ which is mapped to $O$.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20 \ (2)$: Introduction