Let's assume $A$ and $B$ are on the same ordinate for the sake of simplicity. Then
$$ b \cdot \sin \alpha = a \cdot \sin \beta = C_y - A_y $$ $$ a \cdot \cos \beta + b \cdot \cos \alpha = c $$
Once you have solved the system for $a$ or $b$ you can calculate $C$ coordinates ($C_x = A_x + a \cdot \cos \beta, C_y = A_y + a \cdot \sin \beta$)