Inverse Trigonometric Functions · Hard problem

On the interval $\left[\dfrac{\pi}{2},\dfrac{3\pi}{2}\right]$ the sine function is one-to-one, so it has an inverse there; call it $g$ (so $g(y)$ is the number $x$ in $\left[\dfrac{\pi}{2},\dfrac{3\pi}{2}\right]$ with $\sin x=y$). Find the exact value of $g\left(\dfrac13\right)+g\left(-\dfrac13\right)+g(1)$.