Mapping from Unit Circle defines Periodic Function

From ProofWiki
Jump to navigation Jump to search


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 $2 \pi$.


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.


Also see