  We study the eigenvalues of the unique connected anti-regular graph $A_n$.
Using Chebyshev polynomials of the second kind, we obtain a trigonometric
equation whose roots are the eigenvalues and perform elementary analysis to
obtain an almost complete characterization of the eigenvalues. In particular,
we show that the interval $\Omega=[\tfrac{-1-\sqrt{2}}{2},
\tfrac{-1+\sqrt{2}}{2}]$ contains only the trivial eigenvalues $\lambda = -1$
or $\lambda=0$, and any closed interval strictly larger than $\Omega$ will
contain eigenvalues of $A_n$ for all $n$ sufficiently large. We also obtain
bounds for the maximum and minimum eigenvalues, and for all other eigenvalues
we obtain interval bounds that improve as $n$ increases. Moreover, our approach
reveals a more complete picture of the bipartite character of the eigenvalues
of $A_n$, namely, as $n$ increases the eigenvalues are (approximately)
symmetric about the number $-\tfrac{1}{2}$. We also obtain an asymptotic
distribution of the eigenvalues as $n\rightarrow\infty$. Finally, the
relationship between the eigenvalues of $A_n$ and the eigenvalues of a general
threshold graph is discussed.
