# Cosecant is Reciprocal of Sine

## Theorem

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

Then:

$\csc \theta = \dfrac 1 {\sin \theta}$

where $\csc$ and $\sin$ mean cosecant and sine 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:

 $\displaystyle \csc \theta$ $=$ $\displaystyle \frac r y$ Cosecant of Angle in Cartesian Plane $\displaystyle$ $=$ $\displaystyle \frac 1 {y / r}$ $\displaystyle$ $=$ $\displaystyle \frac 1 {\sin \theta}$ Sine of Angle in Cartesian Plane

When $\sin \theta = 0$, $\dfrac 1 {\sin \theta}$ is not defined.

$\blacksquare$