Arcsine of Zero is Zero

Theorem

 * $\arcsin 0 = 0$

where $\arcsin$ is the arcsine function.

Proof
By definition, $\arcsin$ is the inverse of the restriction of the sine function to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Therefore, if:


 * $\sin x = 0$

and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 0 = x$.

From Sine of Zero is Zero, we have that:


 * $\sin 0 = 0$

We have $-\dfrac \pi 2 < 0 < \dfrac \pi 2$, so:


 * $\arcsin 0 = 0$