Cotangent of Supplementary Angle

Theorem

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

where $\cot$ denotes tangent.

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

Proof

\(\ds \map \cot {\pi - \theta}\)

\(=\)

\(\ds \frac {\map \cos {\pi - \theta} } {\map \sin {\pi - \theta} }\)

Cotangent is Cosine divided by Sine

\(\ds \)

\(=\)

\(\ds \frac {-\cos \theta} {\sin \theta}\)

Cosine of Supplementary Angle and Sine of Supplementary Angle

\(\ds \)

\(=\)

\(\ds -\cot \theta\)

$\blacksquare$

Also see

• Sine of Supplementary Angle
• Cosine of Supplementary Angle
• Tangent 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