Calculating the direction of magnetic field at a point above a loop of current

1. Dec 14, 2011

CentreShifter

I've uploaded my attempted solution here. The problem I'm having is highlighted at the bottom in red.

The issue I'm having is expressing the direction of H. I realize the cancellation that occurs at point (0,0,z), where the only the z-component of the H-field remains. I also realize that my final expression for $\bar{H}$ will be $\hat{z}Hcos\phi$, where $cos\phi=\frac{r}{\sqrt{r^2+z^2}}$. I'm really just having a hard time resolving the geometry of these angles to where I can actually equate the two red phi's in the image.