Tangent of Conjugate Angle

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Theorem

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

where $\tan$ denotes tangent.


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


Proof

\(\ds \map \tan {2 \pi - \theta}\) \(=\) \(\ds \frac {\map \sin {2 \pi - \theta} } {\map \cos {2 \pi - \theta} }\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac {-\sin \theta} {\cos \theta}\) Sine of Conjugate Angle and Cosine of Conjugate Angle
\(\ds \) \(=\) \(\ds -\tan \theta\) Tangent is Sine divided by Cosine

$\blacksquare$


Also see


Sources