  In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator
${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential
expression $\mu_0^{-1/2}\operatorname{curl} \eta(\mathbf{x}/\varepsilon)^{-1}
\operatorname{curl} \mu_0^{-1/2} - \mu_0^{1/2}\nabla
\nu(\mathbf{x}/\varepsilon) \operatorname{div} \mu_0^{1/2}$, where $\mu_0$ is a
constant positive matrix, a matrix-valued function $\eta(\mathbf{x})$ and a
real-valued function $\nu(\mathbf{x})$ are periodic with respect to some
lattice, positive definite and bounded. We study the behavior of the
operator-valued functions $\cos (\tau {\mathcal L}_\varepsilon^{1/2})$ and
${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$
for $\tau \in {\mathbb R}$ and small $\varepsilon$. It is shown that these
operators converge to the corresponding operator-valued functions of the
operator ${\mathcal L}^0$ in the norm of operators acting from the Sobolev
space $H^s$ (with a suitable $s$) to $L_2$. Here ${\mathcal L}^0$ is the
effective operator with constant coefficients. Also, an approximation with
corrector in the $(H^s \to H^1)$-norm for the operator ${\mathcal
L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ is obtained.
We prove error estimates and study the sharpness of the results regarding the
type of the operator norm and regarding the dependence of the estimates on
$\tau$. The results are applied to homogenization of the Cauchy problem for the
nonstationary Maxwell system in the case where the magnetic permeability is
equal to $\mu_0$, and the dielectric permittivity is given by the matrix
$\eta(\mathbf{x}/\varepsilon)$.
