# When does cosecant=2?

Mar 19, 2018

$x \in \left\{\frac{\pi}{6} , \frac{5 \pi}{6}\right\} + 2 \pi n$ where $n \in m a t h \boldsymbol{Z}$.

#### Explanation:

We want to solve when
$\csc \left(x\right) = 2$

but cosecant is an ugly function. Let's make it into sines and cosines. That's not hard at all:
$\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$
so therefore we want to find when
$\sin \left(x\right) = \frac{1}{2}$

This happens in a 30-60-90 triangle! So we can write down a few times that this happens, based on our knowledge of the unit circle:
$x = \frac{\pi}{6} , \frac{5 \pi}{6} , \frac{13 \pi}{6} , \frac{17 \pi}{6} , \ldots$
i.e.
$x = \left\{\frac{\pi}{6} , \frac{5 \pi}{6}\right\} + 2 \pi n$ where $n \in m a t h \boldsymbol{Z}$.