Cosecant of Conjugate Angle
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Theorem
- $\map \csc {2 \pi - \theta} = -\csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle is the negative of its conjugate.
Proof
\(\ds \map \csc {2 \pi - \theta}\) | \(=\) | \(\ds \frac 1 {\map \sin {2 \pi - \theta} }\) | Cosecant is Reciprocal of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {-\sin \theta}\) | Cosine of Conjugate Angle and Sine of Conjugate Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds -\csc \theta\) | Cosecant is Reciprocal of Sine |
$\blacksquare$
Also see
- Sine of Conjugate Angle
- Cosine of Conjugate Angle
- Tangent of Conjugate Angle
- Cotangent of Conjugate Angle
- Secant 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