# How do i calculate the angle $\theta_L$?

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.