Use WolframAlpha!
It says There is no convenient form of the exact value of the question. Are you looking for ten significant figures? Nine? Eight? Twenty? Use WolframAlpha! :P
$\displaystyle \sin^{-1}(x) = \sum_{k\ge 0} \dfrac{x^{1+2 k} \left(\tfrac{1}{2}\right)_k}{k!+2 k\cdot k!}$
Plug in $x = \dfrac{2}{3}$
The $\left(\tfrac{1}{2}\right)_k$ is the Pochammer symbol, meaning falling factorial.
I wonder if this is what they are after.
$\arcsin(\frac 2 3) - \arcsin(- \frac 2 3)=2 \arcsin(\frac 2 3)$
$y = 2 \arcsin(\frac 2 3)$
$\sin(y) = \sin\left( 2 \arcsin(\frac 2 3)\right)=$
$2 \sin\left(\arcsin(\frac 2 3)\right)\cos\left(\arcsin(\frac 2 3)\right)=$
$2(\frac 2 3)(\frac {\sqrt{5}} 3)=\frac {4 \sqrt{5}} 9$
$\sin(y)=\dfrac{4 \sqrt 5} 9$
$y = \arcsin\left(\dfrac{4 \sqrt 5} 9\right)$
It is the correct value.
Complex Solution
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