Product of Sine: $\prod_{i=1}^n\sin x_i=k$

Question

From the article Products of Sines, we have $\sin 15^\circ\sin75^\circ=\sin 18^\circ\sin54^\circ=\frac{1}{4}$. We can rewrite this as $\sin \frac{\pi}{12}\sin\frac{5\pi}{12}=\sin \frac{\pi}{10}\sin\frac{3\pi}{10}=\frac{1}{4}$. Is there any good method to get $x,y$ such that $\sin x\sin y=\frac{1}{4}$ or more generally to get $x_i$ such that $$\prod_{i=1}^n\sin x_i=k$$ where $k$ is a rational number?