For starter this is how I look @ trigonometry:
Therefor your problem will look like the following in my opinion:
Now you end up wit 3 triangles.. (yes 3)
Namely: ABC ; ACD & ABD
To solve your issue I NEED (besides the known 3 angles @ A)
Either 2 sides OR 1 additional angle AND 1 side.
Or else there can't be a solution.