\begin{definition}[Definition:Inverse Sine/Real/Arcsine]
{{:Graph of Arcsine Function|Graph}}
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:
:The domain of $\arcsin x$ is $\closedint {-1} 1$
:The image of $\arcsin x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
\end{definition}