Cosecant of Supplementary Angle
Theorem
- $\map \csc {\pi - \theta} = \csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle equals its supplement.
Proof
| \(\ds \map \csc {\pi - \theta}\) | \(=\) | \(\ds \frac 1 {\map \sin {\pi - \theta} }\) | Cosecant is Reciprocal of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\sin \theta}\) | Sine of Supplementary Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds \csc \theta\) | Cosecant is Reciprocal of Sine |
$\blacksquare$
Examples
Cosecant of $4 \theta - 180 \degrees$
- $\map \csc {4 \theta - 180 \degrees} = -\map \csc {4 \theta}$
Also see
- Sine of Supplementary Angle
- Cosine of Supplementary Angle
- Tangent of Supplementary Angle
- Cotangent of Supplementary Angle
- Secant of Supplementary 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