I need to find the angle $\theta_L$ in the attached sketch.
The variables I know are: $R_L$, $R_G$, and $\theta_G$. So i need a formula for $\theta_L$ in terms of these. $R_g$ is also the distance between the centres of the 2 main circles.
From trig, we can find that: $$ C = 2(R_G-d_L) \tan( \theta_G / 2 ) $$
I think for this purpose we can assume $h_G = 0$ if that makes it easier, although I think that's already been taken into account in the above trig equation for $C$.
The issue is the unknown distance between the chord and the radius of the larger $R_G$ . I'm thinking the final equation will involve a ratio between the two, but I'm unsure of how to do this.
If you join the two centers of these circles you can form a triangle. By using Law of sines, you get that $$\frac{R_G}{\sin (\pi-\frac{\theta_L}{2}-\frac{\theta_G}{2})}=\frac{R_L}{\sin \frac{\theta_G}{2}}$$
Maybe you can use this to get $\theta_L$ in terms of the other variables.