# How do you graph to solve the equation on the interval [-2pi,2pi] for cscx=-(2sqrt3)/3?

Feb 1, 2017

$- \frac{\pi}{3} , \frac{- 2 \pi}{3} , \frac{4 \pi}{3} , \frac{5 \pi}{3}$

#### Explanation:

$\csc x = \frac{1}{\sin x} = - \frac{2 \sqrt{3}}{3}$
$\sin x = - \frac{3}{2 \sqrt{3}} = - \frac{\sqrt{3}}{2}$
Use trig table of special arcs and unit circle:

For interval $\left(- 2 \pi , 0\right)$, there are 2 answers:
$\sin x = - \frac{\sqrt{3}}{2}$ --> $x = - \frac{\pi}{3}$ and $x = \frac{- 2 \pi}{3}$
For interval $\left(0 , 2 \pi\right)$, there are 2 answers:
sin x = - sqrt3.2 ---> $x = \frac{4 \pi}{3}$ and $x = \frac{5 \pi}{3}$