# What is the value of tan (sin^-1 x)?

Oct 23, 2015

tan(sin^(-1)(x)) = x/(sqrt(1-x^2)

#### Explanation:

If ${\sin}^{- 1} \left(x\right) = \theta$
then
$\textcolor{w h i t e}{\text{XXX}}$ based on a unit circle (hypotenuse $= 1$)
$\textcolor{w h i t e}{\text{XXX}}$ the side opposite $\theta$ will have a length of $x$
$\textcolor{w h i t e}{\text{XXX}}$and
$\textcolor{w h i t e}{\text{XXX}}$the adjacent side will have a length of $\sqrt{1 - {x}^{2}}$ (based on Pythagorean Theorem)

$\tan \left({\sin}^{- 1} \left(x\right)\right) = \tan \left(\theta\right) = \left(\text{opposite")/("adjacent}\right) = \frac{x}{\sqrt{1 - {x}^{2}}}$