Algebraic values of sine function

Question

Are there algebraic inputs to the sine function that produce algebraic outputs? Other than zero, that is? This is assuming the sine function in radians.

Answer 1

No, if you take an algebraic $x$ such that $sin(x)$ is algebraic, as $cos^2(x) = 1 - sin^2(x)$, $cos(x)$ would also be algebraic, and therefore, $e^{ix}$ would also be algebraic.

Hence we would have $x$ and $e^{ix}$ algebraic, which implies (Lindemann - Weierstrass theorem) that $x = 0$