Sine of Conjugate Angle

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Theorem

$\map \sin {2 \pi - \theta} = -\sin \theta$

where $\sin$ denotes sine.


That is, the sine of an angle is the negative of its conjugate.


Proof

\(\ds \map \sin {2 \pi - \theta}\) \(=\) \(\ds \map \sin {2 \pi} \cos \theta - \map \cos {2 \pi} \sin \theta\) Sine of Difference
\(\ds \) \(=\) \(\ds 0 \times \cos \theta - 1 \times \sin \theta\) Sine of Full Angle and Cosine of Full Angle
\(\ds \) \(=\) \(\ds -\sin \theta\)

$\blacksquare$


Also see


Sources