Arcsine of One is Half Pi

Theorem

$\arcsin 1 = \dfrac \pi 2$

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:

$\sin x = 1$

and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 1 = x$.

From Sine of Right Angle, we have that:

$\sin \dfrac \pi 2 = 1$

We therefore have:

$\arcsin 1 = \dfrac \pi 2$

$\blacksquare$