  For $-\pi\leq\beta_1<\beta_2\leq\pi$ denote by $\Phi_{\beta_1,\beta_2}(Q)$
the number of algebraic numbers on the unit circle with arguments in
$[\beta_1,\beta_2]$ of degree $2m$ and with elliptic height at most $Q$. We
show that \[
\Phi_{\beta_1,\beta_2}(Q)=Q^{m+1}\int\limits_{\beta_1}^{\beta_2}{p(t)}\,{\rm
d}t+O\left(Q^m\,\log Q\right),\quad Q\to\infty, \] where $p(t)$ coincides up to
a constant factor with the density of the roots of some random trigonometric
polynomial. This density is calculated explicitly using the Edelman--Kostlan
formula.
