Sine of Angle plus Integer Multiple of Pi

Theorem

For $n \in \Z, \theta \in \R$:

$\map \sin {\theta + n \pi} = \paren {-1}^n \sin \theta$

Proof

\(\ds \map \sin {\theta + n \pi}\)

\(=\)

\(\ds \sin \theta \cos n \pi + \cos \theta \sin n \pi\)

Sine of Sum

\(\ds \)

\(=\)

\(\ds \sin \theta \cos n \pi\)

Sine of Integer Multiple of Pi

\(\ds \)

\(=\)

\(\ds \paren {-1}^n \sin \theta\)

Cosine of Integer Multiple of Pi

$\blacksquare$

Also see

• Cosine of Angle plus Integer Multiple of Pi