inscribed angles on circle 
That's basically the problem.  I keep getting $\theta=90-\phi/2$. But I have a feeling its not right. What I did was draw line segments BD and AC. From there you get four triangles. I labeled the intersection of BD and AC as point P. From exterior angles I got my answer. 

You are correct. What makes you second-guess yourself? I think your method is sound. However, note that you are asked to express $\phi$ as a function of $\theta$, so you want 
$$\theta = 90 -\dfrac{\phi}{2} \quad \iff \quad \phi = 180 - 2\theta$$
