Cosine of Supplementary Angle

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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
\(\ds \) \(=\) \(\ds -\cos \theta\)

$\blacksquare$


Also see


Sources