# Definition:Inverse Sine/Real

## Definition

Let $x \in \R$ be a real number such that $-1 \le x \le 1$.

The **inverse sine of $x$** is the multifunction defined as:

- $\sin^{-1} \left({x}\right) := \left\{{y \in \R: \sin \left({y}\right) = x}\right\}$

where $\sin \left({y}\right)$ is the sine of $y$.

### Arcsine

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

From Sine of Half-Integer Multiple of Pi:

- $\map \sin {-\dfrac {\pi} 2} = -1$

and:

- $\sin \dfrac {\pi} 2 = 1$

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

Thus from Inverse of Strictly Monotone Function, $g \paren x$ admits an inverse function, which will be continuous and strictly increasing on $\closedint {-1} 1$.

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

Thus: