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
- Sine of Conjugate Angle
- Cosine of Conjugate Angle
- Cotangent 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