# How do you get the exact value of csc^-1 (2)?

Aug 8, 2015

${\csc}^{- 1} \left(2\right) = \frac{\pi}{6}$ or $\frac{5 \pi}{6}$

#### Explanation:

${\csc}^{- 1} \left(2\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= {\sin}^{- 1} \left(\frac{1}{2}\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= \arcsin \left(\frac{1}{2}\right)$

$\arcsin \left(\frac{1}{2}\right) = \theta$ means $\sin \left(\theta\right) = \frac{1}{2}$

If $\sin \left(\theta\right) = \frac{1}{2}$ (within the range $\theta \in \left[0 , 2 \pi\right]$)
then
$\textcolor{w h i t e}{\text{XXXX}}$$\theta = \frac{\pi}{6}$$\textcolor{w h i t e}{\text{XXXX}}$or$\textcolor{w h i t e}{\text{XXXX}}$$\theta = \frac{5 \pi}{6}$
(This is one of the standard angles)