Sine of Angle plus Full Angle/Corollary

Theorem

Let $n \in \Z$ be an integer.

Then:

$\map \sin {x + 2 n \pi} = \sin x$

Proof

From Sine of Angle plus Full Angle:

$\map \sin {x + 2 \pi} = \sin x$

The result follows from the General Periodicity Property:

If:

$\forall x \in X: \map f x = \map f {x + L}$

then:

$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$

$\blacksquare$

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Shifts and periodicity
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Shifts and periodicity