Tangent of Right Angle

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


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
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