$\sin(k\pi/n)$ an integer multiple of $\sin(\pi/n)$

Question

Is there any positive integers $(n,k) \neq (6,3)$ with $n\geq 6$ and $2\leq k\leq n/2$ such that: $$\frac{\sin(k\pi/n)}{\sin(\pi/n)} \in \mathbb{N} ?$$ For $n\leq 57$ the answer is no (by computation), but $(n,k)= (57,22)$ is very close with $\displaystyle \frac{\sin(k\pi/n)}{\sin(\pi/n)} \simeq 17.00035063$.