\(\theta\) 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 \(x^2-2=0.\)

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