When is Morrie's Law a rational number

Question

I recently stumbled across Morrie's law after noticing that $\sin(20)$ x $\sin(40)$ x $\sin(80)=\frac{1}{8}$. $$\\$$ It's a simple proof to show that, in general: $$\prod_{i=0}^{k}\cos(2^ix)=\frac{\sin(2^{k+1}x)}{2^{k+1}\sin(x)} $$ This led me to think about when the R.H.S is a rational number. For the case where $\sin x, x, \sin(2^{k+1}x)$ are $\in\mathbb{Q}$, I can show that the only solution set is: $k\in\mathbb{N}, x=\frac{\pi}{2},-\frac{\pi}{2}$, but I'm unsure about how to deal with the other cases, I've looked at Niven's Theorem but this doesn't account for the other cases. I've also seen a result that for all rational, $x$, $\sin(x)$ is irrational, is this true (when $x=30, \sin(x)=\frac{1}{2}$) which is rational, see discussion thread: https://www.reddit.com/r/math/comments/4gh1i4/is_sinx_where_x_is_rational_always_an_irrational/? $$\\$$ In particular, I want to know how to find rational solutions when $x$ is rational but $\sin{x}$ isn't rational, and similar conditions on $\sin(2^{k+1}x)$. Any help would be much appreciated.