Definition:Inverse Sine/Arcsine

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

From Basic Properties of Sine Function, $$\sin \left({-\frac {\pi} 2}\right) = -1$$ and $$\sin \frac {\pi} 2 = 1$$.

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

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 $$\left[{-\frac \pi 2 \, . \, . \, \frac \pi 2}\right]$$.

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 S) is taken to mean the same as $$\arcsin$$, although this can also be confusing due to the visual similarity of upper and lower case s.