$\sin(\alpha) = \frac{\sqrt{n}}{k}$, where $n$ and $k$ are integers and $\alpha$ is a rational multiple of $\pi$

Question

It is well known that the solutions of the equation

$$ \sin\left(\frac\pi x\right)= \frac{\sqrt3}{2} $$

are

$$ x=\frac{3}{6n+2}, n\in\mathbb{Z} $$

and

$$ x=\frac{3}{6n+1}, n\in\mathbb{Z}. $$

Are there any other known values $\alpha$ such that $\sin(\alpha) = \frac{\sqrt{n}}{k}$, where $k$ and $n$ are positive integers and $\alpha$ is a rational multiple of $\pi$?

Answer 1

Since sine is a continuous function, it will take any value $\frac{\sqrt{n}}{k}$ such as $-1\le\frac{\sqrt{n}}{k}\le 1$. (I assumed that you ment $\frac{\sqrt{n}}{k}$ based on your example).