Tangent of Supplementary Angle

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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Symmetry
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Symmetry