Cotangent is Cosine divided by Sine

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Theorem

Let $\theta$ be an angle such that $\sin \theta \ne 0$.

Then:

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

where $\cot$, $\sin$ and $\cos$ mean cotangent, sine and cosine respectively.


Proof

Let a point $P = \left({x, y}\right)$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.

Then:

\(\displaystyle \frac {\cos \theta} {\sin \theta}\) \(=\) \(\displaystyle \frac {x / r} {y / r}\) Cosine of Angle in Cartesian Plane and Sine of Angle in Cartesian Plane
\(\displaystyle \) \(=\) \(\displaystyle \frac x r \frac r y\)
\(\displaystyle \) \(=\) \(\displaystyle \frac x y\)
\(\displaystyle \) \(=\) \(\displaystyle \cot \theta\) Cotangent of Angle in Cartesian Plane

When $\sin \theta = 0$ the expression $\dfrac {\cos \theta} {\sin \theta}$ is not defined.

$\blacksquare$


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