May be someone can help me to solve the problem. There are circle with radius R1 and circle with radius R2. We also know the distance between A and O and that angle AOB = $\phi$. The aim is to calculate distance between B and C. I've tried to use sine and cosine theorems but didn't get any results.
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As You have mentioned, by cosine theorem, $AB$ is known.
From that, with the given, $\angle A$ can then be found by sine theorem.
In $\triangle AOC$, since $AO, OC$ and $\angle A$ are known, apply cosine theorem again to find $AC$. Result then follows.