Lots of Decimal Places

$\theta$ is an integer in the range $[0,360).$ How many values of $\sin \theta^\circ$ are transcendental?

$\textbf{Details and Assumptions}$

A $\textit{transcendental number}$ is a number that cannot be the root of a polynomial with integer coefficients. For example, $\pi$ and $e$ are transcendental, but $\sqrt{2}$ is not because it is a solution to the equation $x^2-2=0.$