Can I get help on how I can calculate this without a calculator? I know it's the inverse of $sin(1/2)$, but I'm still a little confused on how I get $\pi / 2$ from this.
Thanks!
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$\begingroup$Consider an equilateral triangle if side length $2$. An altitude cuts one of the $\pi/3$ angles into two copies of $\pi/6$, in hypotenuse-$2$ right-angled triangles with opposite $1$. Thus $\pi/6=\arcsin 1/2$.
$\endgroup$ $\begingroup$$\sin30^{\circ}=\frac{1}{2}$ and for all $-1\leq x\leq 1$ we have $-90^{\circ}\leq\arcsin x\leq90^{\circ}.$
Thus, $\arcsin\frac{1}{2}=30^{\circ}.$
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