I've encountered the following function while working on a project:
$$ f(\theta) = a \sin \theta + b \cos \theta +c \sin \theta \cos \theta + d \sin^2 \theta + e \cos^2 \theta $$
where a through e are all real non-zero numbers and $0 \leq \theta < 2 \pi$.
I know this function has at least two roots. For reasons I won't bore you with, finding the roots numerically isn't ideal for the project I'm working on.
Applying the tangent half-angle substitution described in this answer results in a fourth order polynomial, which is troublesome to find the roots of without numerical methods.
$$ 0 = (e-b)t^4 + (2a-2c) t^3 - 2et^2 + (2a+2c+4d)t + (b+e) $$
Is there a better non-numerical option for finding the roots?