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
- Sine of Conjugate Angle
- Cosine of Conjugate Angle
- Tangent of Conjugate Angle
- Secant of Conjugate Angle
- Cosecant of Conjugate Angle
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