Cotangent of Supplementary Angle

Theorem

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

where $\cot$ denotes tangent.


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


Proof

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

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
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