In $L_2(\mathbb R^3;\mathbb C^3)$,
we consider a self-adjoint operator $\mathscr L_\varepsilon$,
$\varepsilon >0$,
generated
by the differential expression
$\operatorname{curl}\eta(\mathbf x
/\varepsilon)^{-1}\operatorname{curl}
-\nabla\nu(\mathbf x/\varepsilon)\operatorname{div}$.
Here the matrix function
$\eta(\mathbf x)$
with real entries and the real function
$\nu(\mathbf x)$
are periodic with respect to some lattice,
are positive definite,
and are bounded.
We study the behavior of the operators
$\cos(\tau\mathscr L_\varepsilon^{1/2})$
and
$\mathscr L_\varepsilon^{-1/2}
\sin(\tau\mathscr L_\varepsilon^{1/2})$
for
$\tau\in\mathbb R$
and
small $\varepsilon$.
It is shown that
these operators
converge
to $\cos(\tau(\mathscr L^0)^{1/2})$
and
$(\mathscr L^0)^{-1/2}\sin(\tau(\mathscr L^0)^{1/2})$,
respectively,
in the norm of the operators acting from the Sobolev space $H^s$
(with a suitable $s$)
to $L_2$.
Here
$\mathscr L^0$
is an effective operator
with constant coefficients.
Error estimates are obtained and
the sharpness of the result
with respect to the type of operator norm is studied.
The results
are used for homogenizing the Cauchy problem
for
the model hyperbolic equation
$\partial^2_\tau\mathbf v_\varepsilon
=-\mathscr L_\varepsilon\mathbf v_\varepsilon$,
$\operatorname{div}\mathbf v_\varepsilon=0$,
appearing in electrodynamics.
We study the application
to a nonstationary Maxwell system
for the case
in which the magnetic permeability
is equal to $1$
and the dielectric permittivity
is given by the matrix
$\eta(\mathbf x/\varepsilon)$.
