# Tangent of Supplementary Angle

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## Theorem

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

where $\tan$ denotes tangent.

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

## Proof

\(\ds \map \tan {\pi - \theta}\) | \(=\) | \(\ds \frac {\map \sin {\pi - \theta} } {\map \cos {\pi - \theta} }\) | Tangent is Sine divided by Cosine | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\sin \theta} {-\cos \theta}\) | Sine of Supplementary Angle and Cosine of Supplementary Angle | |||||||||||

\(\ds \) | \(=\) | \(\ds -\tan \theta\) |

$\blacksquare$

## Also see

- Sine of Supplementary Angle
- Cosine of Supplementary Angle
- Cotangent of Supplementary Angle
- Secant of Supplementary Angle
- Cosecant 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