Finding the "roots" of a Fourier series

Question

I am attempting to find the intersection of a line and a finite Fourier series. Ideally, this would involve no approximations or comparisons. Thus far, I have determined that isolating x using traditional trigonometric functions such as cos(x). My question is, is it possible to calculate the points of intersection of a line and multiple added sinusoidal functions? If so, how? If not, are there any alternatives?

ex:

$$ 2\cos(3x+4) + 5\cos(6x+7) = 0, x = ?$$

Answer 1

I've completed the project at this point. The only way to find the roots is via approximation. Closed form root finding for fourier series and other similar functions (multidimensional sinusoidal functions, fourier transforms). I used Newton approximations in multiple dimensions, but there are innumerable approximation methods one could use.

Answer 2

Perhaps one approach is to substitute $t = 3x+4$ and $6x+7=2t-1$ and then use $$ \cos(2t-1) = \cos(2t)\cos (1)-\sin(2t)\sin(1) $$ eventually expressing the RHS in terms of $\cos t$ and $\sin t$