In the following image, I'm trying to solve for $x$ in terms of $r$, $s$, and $\theta$.
This problem turns up in a non-linear geometry control theory proof I'm writing. However, sadly, I've forgotten much of trigonometry years ago, so any help would be appreciated.
First, you can use the Law of Cosine to compute the distance $d$ between the to centers: $$d^2=r^2+s^2-2rs\cos(\theta_1),$$ where $\theta_1$ is the angle between the radii at the intersection and is equal to $\pi-\theta$.
Next, just note that $x=r+s-d$.