Cosecant is Reciprocal of Sine

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

Then:

\(\ds \csc \theta\) \(=\) \(\ds \frac r y\) Cosecant of Angle in Cartesian Plane
\(\ds \) \(=\) \(\ds \frac 1 {y / r}\)
\(\ds \) \(=\) \(\ds \frac 1 {\sin \theta}\) Sine of Angle in Cartesian Plane


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

$\blacksquare$


Sources

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