= Solution
The two eigenvalues are the points at arguments $0$ and $\gamma$. The counterclockwise arc from $1$ to $e^{i\gamma}$ has length $\gamma$, while the complementary clockwise arc has length $2\pi-\gamma$. The shorter of the two contains the spectrum. Therefore
$$
\boxed{
\theta(U_\gamma)=\min\{\gamma,2\pi-\gamma\}
=\begin{cases}
\gamma,&0\leq\gamma<\pi,\\
2\pi-\gamma,&\pi\leq\gamma<2\pi.
\end{cases}}
$$
The two geometric cases are respectively the short counterclockwise arc and the short complementary clockwise arc; at $\gamma=\pi$ both semicircles have length $\pi$.
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