# Trigonometric Functions of Supplementary Angles

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## Theorem

### Sine of Supplementary Angle

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

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

- $\csc \left({\pi - \theta}\right) = \csc \theta$

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