I am interested in a closed-form method for finding all solutions of the following trigonometric equation in $\theta$:
\begin{equation} a\sin(2\theta) + b\cos(2\theta) + c\sin(\theta) + d\cos(\theta) \,=\, 0 \end{equation}
where $a, b, c$ and $d$ are given real numbers.
If $z = \exp(i\theta)$, your left side is $$a(z^2-z^{-2})/(2i) +b (z^2+z^{-2})/2 + c (z - z^{-1})/(2i) + d (z + z^{-1})/2$$ After multiplying by $z^2$, you have a quartic in $z$. Its roots can be expressed in closed form (though rather complicated).