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