Problem

The first two terms of a geometric progression are $4\sin^2\theta$, $8\sin^3\theta$, where $\theta$ is an angle such that $0 < \theta < \frac{1}{6}\pi$.

Given that the sum to infinity of the progression is $\frac{1}{2}$, find the value of $\theta$. Give your answer in the form $\sin^{-1} k$, where $k$ is a rational number.