Definition:Inverse Sine/Arcsine

Definition


From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$.

From Sine of Multiple of Pi Plus Half, $\sin \left({-\dfrac {\pi} 2}\right) = -1$ and $\sin \dfrac {\pi} 2 = 1$.

Therefore, let $g: \left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right] \to \left[{-1 \,.\,.\, 1}\right]$ be the restriction of $\sin x$ to $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly increasing on $\left[{-1 \,.\,.\, 1}\right]$.

This function is called arcsine of $x$ and is written $\arcsin x$.

Thus:
 * The domain of $\arcsin x$ is $\left[{-1 \,.\,.\, 1}\right]$
 * The image of $\arcsin x$ is $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$.

Caution
There exists the a popular but misleading notation $\sin^{-1} x$, which is supposed to denote the inverse sine function.

However, note that as $\sin x$ is not an injection (even though by restriction of the codomain it can be considered surjective), it does not have an inverse.

The $\arcsin$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\sin^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname{Sin}^{-1}$ (with a capital $\text{S}$) is taken to mean the same as $\arcsin$, although this can also be confusing due to the visual similarity between that and the lowercase $\text{s}$.