# Opposites Theorem

## Theorem

 $\displaystyle \sin \left({- \theta}\right)$ $=$ $\displaystyle - \sin \theta$ $\displaystyle \cos \left({- \theta}\right)$ $=$ $\displaystyle \cos \theta$ $\displaystyle \tan \left({- \theta}\right)$ $=$ $\displaystyle - \tan \theta$

## Proof

 $\displaystyle \sin \left({- \theta}\right)$ $=$ $\displaystyle - \sin \theta$ Sine Function is Odd $\displaystyle \cos \left({- \theta}\right)$ $=$ $\displaystyle \cos \theta$ Cosine Function is Even $\displaystyle \tan \left({- \theta}\right)$ $=$ $\displaystyle - \tan \theta$ Tangent Function is Odd

$\blacksquare$

## Comment

The concept of bagging up these identities into one, and calling it the Opposites Theorem, seems to be a modern innovation designed to ease the learning of trigonometry in schools.