# Mapping from Unit Circle defines Periodic Function

## Theorem

Let $\SS$ denote the unit circle whose center is at the origin of the Cartesian plane $\R^2$.

Let $p: \R \to \SS$ be the mapping defined as:

$\forall x \in \R: \map p x = \tuple {\cos x, \sin x}$

Let $f': \SS \to \R$ be a real-valued function.

Then the composition $f' \circ p$ defines a periodic real function whose period is $2 \pi$.

## Proof

Let $f := f' \circ p$ denote the composition of $f$ with $p$.

We have:

 $\ds \forall x \in \R: \,$ $\ds \map f {x + 2 \pi}$ $=$ $\ds \map {f'} {\map p {x + 2 \pi} }$ Definition of Composition of Mappings $\ds$ $=$ $\ds \map {f'} {\map \cos {x + 2 \pi}, \map \sin {x + 2 \pi} }$ Definition of $p$ $\ds$ $=$ $\ds \map {f'} {\cos x, \sin x}$ Cosine of Angle plus Full Angle, Sine of Angle plus Full Angle $\ds$ $=$ $\ds \map {f'} {\map p x}$ Definition of $p$ $\ds$ $=$ $\ds \map f x$ Definition of Composition of Mappings

Hence the result.

$\blacksquare$