Definition:Inverse Sine/Arcsine

Definition

Real Numbers

Arcsine Function

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:

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

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 $1 - z^2$.