Cosine of Angle plus Integer Multiple of Pi

Theorem

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

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

Proof

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

\(=\)

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

Cosine of Sum

\(\ds \)

\(=\)

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

Sine of Integer Multiple of Pi

\(\ds \)

\(=\)

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

Cosine of Integer Multiple of Pi

$\blacksquare$

Also see

• Sine of Angle plus Integer Multiple of Pi