Opposites Theorem

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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.