Tangent of Right Angle
(Redirected from Tangent of 90 Degrees)
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Theorem
- $\tan 90 \degrees = \tan \dfrac \pi 2$ is undefined
where $\tan$ denotes tangent.
Proof
From Tangent is Sine divided by Cosine:
- $\tan \theta = \dfrac {\sin \theta} {\cos \theta}$
When $\cos \theta = 0$, $\dfrac {\sin \theta} {\cos \theta}$ can be defined only if $\sin \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = \dfrac \pi 2$, $\cos \theta = 0$.
Thus $\tan \theta$ is undefined at this value.
$\blacksquare$
Also defined as
Some sources give that:
- $\tan 90 \degrees = \infty$
but this naïve approach is overly simplistic and cannot be backed up with mathematical rigour.
Also see
- Sine of Right Angle
- Cosine of Right Angle
- Cotangent of Right Angle
- Secant of Right Angle
- Cosecant of Right Angle
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Special angles
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles