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[Mighty Midget]

If

angle ABC = a (alpha)

K = sin BAC / sin CAD and
L = 180 - (BAC + CAD)

then

K = sin a / sin (L - a)

the solution will be at

sin a / sin (L - a) - K = 0

(tested and verified* with a graphical calculator)

* verified as in I didn't get any wrong answers with several different, arbitrary values for BAC and CAD.

Now is the problem, I am unable to isolate alpha in sin (L - a)
If you can do that, you have the formula, I believe.