Aside from the obvious knowledge that the roots of $\sin x$ are all integer multiples of $\pi$, is there a formal, algebraic method to calculate the roots of trigonometric functions similar to the quadratic equation?
(e.g. roots of $\sin^2(ax) + \sin(bx) + c$ or some other non-trivial form)
While trigonometric identities exist for multiples of an angle, perhaps the best-known being:
$$ \begin{align} \sin 2x &= 2\sin x \cos x\\ \cos 2x &= \cos^2 x - \sin^2{x} =2\cos^2 x - 1 = 1 - 2 \sin^2x\\ \sin 3x &= 3\sin x - 4 \sin^3 x\\ \cos 3x &= 4\cos^3 x - 3\cos x \end{align} $$
there is no such general formula, and indeed no way to solve such equations, unless the trigonometric arguments are the same and therefore the expression can be factored.