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.

$\blacksquare$

Also see

• Periodic Function as Mapping from Unit Circle

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.8$: Quotient spaces