# Rational value of sine [duplicate]

Given $x$, $\sin(x) \in \mathbb{Q}$, where $x$ is in degrees, we want to find all $x$ in the range $(0,90)$.

One trivial solution is $x=30$.

## marked as duplicate by Saad, Jyrki LahtonenMar 11 '18 at 13:30

If $x$ is a rational multiple of $\pi$ then $e^{ix}$ is a root of unity, and so an algebraic integer. So $2\sin x=-ie^{ix}+ie^{-ix}$ is an algebraic integer. If $\sin x$ is rational too, $2\sin x$ is rational and an algebraic integer, so is an integer. As $|\sin x|\le1$ then $\sin x\in\{-1,-\frac12,0,\frac12,1\}$.
• Rather than x being a rational multiple of $\pi$, I want x to be a rational number itself when expressed in terms of degrees. – kaushal agrawal Aug 21 '17 at 3:31