I have an equation which I am using to describe the squared distance from a polar point $(r_1, \theta_1)$ to a function $g(\theta)$ which is $C_1$ smooth over the period $0-2\pi$. $$r = r_{1}^{2}+g\left(\theta\right)^{2}-2r_{1}g\left(\theta\right)\cos\left(\theta-\theta_{1}\right)$$ To find local minima and maxima we differentiate w.r.t. $\frac{dr}{d\theta}$ and find the roots of this derivative.
I believe this is: $$r = 2g\left(\theta\right)g'\left(\theta\right)-2r_{1}\left(g\left(\theta\right)\cdot-\sin\left(\theta-\theta_{1}\right)+g'\left(\theta\right)\cos\left(\theta-\theta_{1}\right)\right)$$ How do I find roots for trig functions which don't boil down to textbook trig identities? I looked into Chebyshev Polynomials but it looks like those are still only helpful for rootfinding if they are free of trigonometric functions.
Notes:
$g(\theta) = sin(z\theta)+s$, though I'd like to stay general to any function which is $C_1$ smooth and has a period of $2\pi$
I'd also like to only find local minima, so I believe this will require the second derivative, but at least not require root finding on that derivative.