Cotangent of Zero
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Theorem
- $\cot 0$ 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 $\theta = 0$, $\sin \theta = 0$.
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