# Cosine of Supplementary Angle

## Theorem

$\map \cos {\pi - \theta} = -\cos \theta$

where $\cos$ denotes cosine.

That is, the cosine of an angle is the negative of its supplement.

## Proof

 $\ds \map \cos {\pi - \theta}$ $=$ $\ds \cos \pi \cos \theta + \sin \pi \sin \theta$ Cosine of Difference $\ds$ $=$ $\ds \paren {-1} \times \cos \theta + 0 \times \sin \theta$ Cosine of Straight Angle and Sine of Straight Angle $\ds$ $=$ $\ds -\cos \theta$

$\blacksquare$