Definition:Inverse Sine/Real

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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

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:

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

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 $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.