Arc sine function

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Template:Angular function convention

Definition

The arc sine function is denoted $\arcsin$ or $\sin^{-1}$.

For $x$ in the closed interval $[-1,1]$ (i.e., $-1 \le x \le 1$), $\arcsin x$ is defined as the unique value $y$ in the closed interval $[-\pi/2,\pi/2]$ such that $\sin y = x$.

The arc sine function can be considered the inverse function of the restriction of the sine function to the interval $[-\pi/2,\pi/2]$.

Key data

Item	Value
default domain	the closed interval $[-1,1]$
range	the closed interval $[-\pi/2,\pi/2]$
first derivative	$x \mapsto \frac{1}{\sqrt{1 - x^2}}$, defined on the open interval $(-1,1)$. The left-hand derivative at 1 is not defined (as an infinity, it takes the value $+\infty$). The right-hand derivative at -1 is not defined (as an infinity, it takes the value $+\infty$).

Graph