Can this trigonometric relation be solved?

Question

Given $A$, $B$, and the $b$ angle I need to solve for $C$. The red line is perpendicular to side $A$.

I know that via the law of cosines I have:

$$C^2 = A^2 + B^2 - 2AB\,cos(d)$$

but I only have $b$, not $d$. Can this be solved at all?

Answer 1

You know the angle between $A$ and $C$ and can use this in the cosine formula to obtain a quadratic for $C$. This will have two solutions in general, and you should consider their geometric significance.

Answer 2

Hint: $B^2=A^2+C^2-2AC\cos(\pi/2+b)$ will give you a quadratic equation for $C$ (assuming that the red line and side $A$ are perpendicular)