Find the Inverse y=arcsin(x)

$y = \arcsin (x)$

Interchange the variables.

$x = \arcsin (y)$

Solve for $y$ .

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Rewrite the equation as $\arcsin (y) = x$ .

$\arcsin (y) = x$

Take the inverse arcsine of both sides of the equation to extract $y$ from inside the arcsine.

$y = \sin (x)$

Remove parentheses.

$y = \sin (x)$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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Replace the $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \sin (x)$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (\arcsin (x)) = \sin (\arcsin (x))$

The functions sine and arcsine are inverses.

$g (\arcsin (x)) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \sin (x)$ is the inverse of $f (x) = \arcsin (x)$ .

$f^{- 1} (x) = \sin (x)$

Find the Inverse y=arcsin(x)

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