Trigonometric Functions of Supplementary Angles

Theorem

Sine of Supplementary Angle

$\map \sin {\pi - \theta} = \sin \theta$

where $\sin$ denotes sine.

That is, the sine of an angle equals its supplement.

Cosine of Supplementary Angle

$\map \cos {\pi - \theta} = -\cos \theta$

where $\cos$ denotes cosine.

That is, the cosine of an angle is the negative of its supplement.

Tangent of Supplementary Angle

$\map \tan {\pi - \theta} = -\tan \theta$

where $\tan$ denotes tangent.

That is, the tangent of an angle is the negative of its supplement.

Cotangent of Supplementary Angle

$\map \cot {\pi - \theta} = -\cot \theta$

where $\cot$ denotes tangent.

That is, the cotangent of an angle is the negative of its supplement.

Secant of Supplementary Angle

$\map \sec {\pi - \theta} = -\sec \theta$

where $\sec$ denotes secant.

That is, the secant of an angle is the negative of its supplement.

Cosecant of Supplementary Angle

$\map \csc {\pi - \theta} = \csc \theta$

where $\csc$ denotes cosecant.

That is, the cosecant of an angle equals its supplement.

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