Arcsine of Zero is Zero
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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:
- $\arcsin 0 = 0$
$\blacksquare$