# How do you find sin(cos^-1x)?

Sep 24, 2016

$\sqrt{1 - {x}^{2}}$.

#### Explanation:

Let $a = {\cos}^{- 1} x \in \left[0 , \pi\right]$. for the principal value. Then cos a =x, for

both positive and negative x,..

The given expression is

$\sin a = \sqrt{1 - {\cos}^{2} a} = \sqrt{1 - {x}^{2}}$, for $a \in \left[0 , \pi\right]$..

If the principal value convention is relaxed and x is negative, a could

be in ${Q}_{3}$ wherein both sine and cosine are negative. And then,

$\sin a = - \sqrt{1 - {x}^{2}}$ ,

.