To find the general solution to y we need take the sine function in both sides of the equation given:
[tex]\begin{gathered} \sin y=\sin (\sin ^{-1}-\frac{\sqrt[]{3}}{2}) \\ \sin y=-\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]Now, we have to find the value of y for which the sine function is equal to the right side. From the properties of the sine function and its definition we conclude that y has to be:
[tex]y=\frac{5\pi}{3}[/tex]Therefore the general solution is:
[tex]y=\frac{5\pi}{3}\pm2\pi k[/tex]