What is the phase of the Fourier Transform of $h(n)={\frac13,\frac13,\frac13}$ (zero position is at 2nd element)
The result on the phase comes from writing $H(\omega)$ as a modulus/phase product: $H(\omega) = |H(\omega)|e^{-\imath\phi(\omega)}$. The value of $\phi(\omega)$ is consistent with the sign of $(1+2\cos \omega)$.



There is a little inconsistency is your notation (maybe from the book!): the phase seems to takes both values $0$ and $\pi$ at $w=\frac{2\pi}{3}$. This is not vital as the modulus vanishes at this point. 

What you might have forgotten in your reasoning is the fact that $t\to \tan t$ is $\pi$-periodic, and that $\tan \pi  =0$ too.