What can I do with a polynomial of cosines?

Question

In a mathematical derivation I ended up with a polynomial of the form:

$\cos^3\theta + a_1cos^2\ \theta + a_2cos\ \theta + a_3= 0$

Coefficients $a_n$ are not functions of $\theta$ (but are not known constants either, so they remain symbolic).

1. Is there a name for this polynomial?
2. Is it possible to derive a formula for all roots/ $ \theta $'s that satisfy this equation? Or is this the closest I'll get?

Answer 1

This is a trigonometric polynomial.

For your question, to solve for $\theta$, first solve for the root of $x^3 + a_1 x^2+a_2x + a_3=0$, then solve for $\cos \theta = x$.