# How do you find the domain and range of  arcsin(sin x)?

Feb 12, 2017

Domain and range $\arcsin \left(\sin x\right)$ is $\left(- \infty , \infty\right)$.

#### Explanation:

From the definition of $\arcsin x$, which is also described as ${\sin}^{- 1} x$, if $\sin x = u$, we have

$\arcsin u = x \implies \arcsin \left(\sin x\right) = x$

i.e. $\arcsin \left(\sin x\right) = x$

as domain and range $x$ is $\left(- \infty , \infty\right)$,

domain and range $\arcsin \left(\sin x\right)$ is $\left(- \infty , \infty\right)$.