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See the drawing for the situation. Given lenghts a, b and c and also L, but k and angle alpha are unknown. How to compute this angle alpha?

I know it is possible to compute if we first compute k in some way, maybe there is also a direct approach.

Edit: I could not post images directly, but here is the link. I hope it is clear.

enter image description here

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up vote 1 down vote accepted

Let $r$ be the radius of the circle. By the Cosine Law, $$b^2=(r-a)^2+(r-c)^2-2(r-a)(r-c)\cos\alpha.\tag{$\ast$}$$ We also have $L=r\alpha$, so $r=L/\alpha$. Substitute in $(\ast)$. We get an equation in $\alpha$ and the known quantities.

Unfortunately, this equation does not have a closed form solution. For specific numerical values of the parameters $a$, $b$, $c$, and $L$, our favourite root-finding tool can be used to solve the problem numerically.

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