# Tangent of Supplementary Angle

## Theorem

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

where $\tan$ denotes tangent.

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

## Proof

 $\ds \map \tan {\pi - \theta}$ $=$ $\ds \frac {\map \sin {\pi - \theta} } {\map \cos {\pi - \theta} }$ Tangent is Sine divided by Cosine $\ds$ $=$ $\ds \frac {\sin \theta} {-\cos \theta}$ Sine of Supplementary Angle and Cosine of Supplementary Angle $\ds$ $=$ $\ds -\tan \theta$

$\blacksquare$