There’s three equations in my post, and the three unknowns are alpha, beta, and C. You know A and B. Of course C is the only one you care about, but my point is there are three equations and three unknowns, so it should be solvable. It is not a linear equation or a quadratic one, so it would have to be solved by substitution. You could use trig identities to eliminate the trig functions, expressing cos in terms of tan or vice versa. I tried putting it into Maxima to solve but it did not come up with a result. That might only be because Maxima isn’t smart enough to do it.

What did you use as gui ? it looks pretty clean and powerfull, can it be extracted and used as a standalone gui ?