# How do you calculate csc^-1(-5)?

Nov 23, 2015

$\theta \approx {348.46}^{\circ} , {191.54}^{\circ}$

#### Explanation:

${\csc}^{-} 1 \left(- 5\right)$

$\csc \theta = - 5$

$\frac{1}{\sin} \theta = - 5$

$\sin \theta = - \frac{1}{5}$

$\theta = {\sin}^{-} 1 \left(- \frac{1}{5}\right)$

$\theta \approx - {11.54}^{\circ} \left({348.46}^{\circ}\right)$

However, since the CAST rule states that cosecant is also negative in quadrant $3$, there is a second answer:

$\theta = \left({360}^{\circ} - {348.46}^{\circ}\right) + {180}^{\circ}$
$\theta = {11.54}^{\circ} + {180}^{\circ}$
$\theta \approx {191.54}^{\circ}$