Cotangent of Conjugate Angle

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Theorem

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

where $\cot$ denotes cotangent.


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


Proof

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

$\blacksquare$


Also see


Sources