# How do you calculate cos [Sin^-1 (-1/5) ]?

Apr 30, 2015

In this way:

we have to calculate $\cos \alpha$, where

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

$\sin \alpha = - \frac{1}{5}$.

Since the range of the function $y = {\sin}^{-} 1 x$ is an angle in the first and fourth quadrant, and since the value is negative, the angle $\alpha$ is in the fourth quadrant, and there the function cosine is positive, so:

$\cos \alpha = + \sqrt{1 - {\sin}^{2} \alpha} = \sqrt{1 - \frac{1}{25}} = \sqrt{\frac{24}{25}} = \frac{2}{5} \sqrt{6}$.