Tangent of Complement equals Cotangent
Theorem
- $\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$
where $\tan$ and $\cot$ are tangent and cotangent respectively.
That is, the cotangent of an angle is the tangent of its complement.
This relation is defined wherever $\sin \theta \ne 0$.
Proof
| \(\ds \map \tan {\frac \pi 2 - \theta}\) | \(=\) | \(\ds \frac {\map \sin {\frac \pi 2 - \theta} } {\map \cos {\frac \pi 2 - \theta} }\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cos \theta} {\sin \theta}\) | Sine and Cosine of Complementary Angles | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cot \theta\) | Cotangent is Cosine divided by Sine |
The above is valid only where $\sin \theta \ne 0$, as otherwise $\dfrac {\cos \theta} {\sin \theta}$ is undefined.
From Sine of Multiple of Pi it follows that this happens when $\theta \ne n \pi$.
$\blacksquare$
Also see
- Sine of Complement equals Cosine
- Cosine of Complement equals Sine
- Cotangent of Complement equals Tangent
- Secant of Complement equals Cosecant
- Cosecant of Complement equals Secant
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(6)$
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Complementary angles
- 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