Cotangent of Straight Angle

Theorem

$\cot 180^\circ = \cot \pi$ is undefined

where $\cot$ denotes cotangent.


Proof

From Cotangent is Cosine divided by Sine:

$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$

When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.

But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.

When $\theta = \pi$, $\sin \theta = 0$.

Thus $\cot \theta$ is undefined at this value.

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
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