# How do you find the values of theta given csctheta=(2sqrt3)/3?

Mar 21, 2017

$\theta = n \pi \pm {\left(- 1\right)}^{n} \frac{\pi}{3}$, where $n$ is an integer.

#### Explanation:

As $\csc \theta = \frac{2 \sqrt{3}}{3} = \frac{2}{\sqrt{3}}$

Therefore, $\sin \theta = \frac{\sqrt{3}}{2} = \sin \left(\frac{\pi}{3}\right)$

Hence, within the interval $0 \le \theta \le 2 \pi$, $\theta = \frac{\pi}{3}$ or $\frac{2 \pi}{3}$

but as trigonometric ratios repeat after $2 \pi$

they are more solutions which can be put as

$\theta = n \pi \pm {\left(- 1\right)}^{n} \frac{\pi}{3}$, where $n$ is an integer.