Finding Angles from Side Length in Pre-Calculus

Question

I had this problem on a recently college exam. I had no idea how to do it, and lost all my points for it. I assumed it would have something to do with finding the correct values for the sides and angles of a triangle given the sine and cosine rules, but the problem didn't seem to give enough space to solve them with that.

On my exam my professor wrote the tangent addition identity:

$$\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \cdot \tan(B)}$$

But I'm not sure what relevance this has.

If C = A + B, find tan(C)

Answer 1

Hint: $$\tan A = \frac{1}{2}$$

$$\tan B = \frac{3}{4}$$

Now, apply $$\tan(A+B) = \frac{\tan A+\tan B}{1-\tan A\tan B}$$

Answer 2

$$\tan(C) =\tan(A+B) = \frac{1/2 + 3/4}{1 - 3/8} = \frac{5/8}{5/8} = 1$$