# Definition:Inverse Sine

## Definition

### Real Numbers

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 $y$.

### Complex Plane

Let $z \in \C$ be a complex number.

The inverse sine of $z$ is the multifunction defined as:

$\sin^{-1} \paren z := \set {w \in \C: \sin \paren w = z}$

where $\sin \paren w$ is the sine of $w$.

## Arcsine

### Real Numbers

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

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

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:

The domain of $\arcsin x$ is $\closedint {-1} 1$
The image of $\arcsin x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

### Complex Plane

The principal branch of the complex inverse sine function is defined as:

$\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.