  In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a matrix elliptic second
order differential operator $B_\varepsilon >0$. Coefficients of the operator
$B_\varepsilon$ are periodic with respect to some lattice in $\mathbb{R}^d$ and
depend on $\mathbf{x}/\varepsilon$. We study the quantitative homogenization
for the solutions of the hyperbolic system $\partial _t^2\mathbf{u}_\varepsilon
=-B_\varepsilon\mathbf{u}_\varepsilon$. In operator terms, we are interested in
approximations of the operators $\cos (tB_\varepsilon ^{1/2})$ and
$B_\varepsilon ^{-1/2}\sin (tB_\varepsilon ^{1/2})$ in suitable operator norms.
Approximations for the resolvent $B_\varepsilon ^{-1}$ have been already
obtained by T.~A.~Suslina. So, we rewrite hyperbolic equation as a system for
the vector with components $\mathbf{u}_\varepsilon $ and $\partial
_t\mathbf{u}_\varepsilon$, and consider the corresponding unitary group. For
this group, we adapt the proof of the Trotter-Kato theorem by introduction of
some correction term and derive hyperbolic results from elliptic ones.
