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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.

enter image description here

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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.

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  • $\begingroup$ Thanks. Turns out I had to use the law of cosines as this particular triangle had a number of possible solutions given the actual values of the variables. But your answer definitely sent me in the right direction, much nicer than circular segmentation. $\endgroup$ – iagreewithjosh May 1 at 6:58

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