In L ₂(R ³;C ³), we consider a selfadjoint operator L _ε, ε>0, given by the differential expression μ ^−1/2curlη(x/ε) ⁻¹curlμ ^−1/2−μ ^1/2∇ν(x/ε)divμ ^1/2, where μ is a constant positive matrix, a matrix-valued function η(x) and a real-valued function ν(x) are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions cos⁡(τL _ε ^1/2) and L _ε ^−1/2sin⁡(τL _ε ^1/2) for τ∈R and small ε. It is shown that these operators converge to the corresponding operator-valued functions of the operator L ⁰ in the norm of operators acting from the Sobolev space H ^s (with a suitable s) to L ₂. Here L ⁰ is the effective operator with constant coefficients. Also, an approximation with corrector in the (H ^s→H ¹)-norm for the operator L _ε ^−1/2sin⁡(τL _ε ^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 τ. The results are applied to homogenization of the Cauchy problem for the non-stationary Maxwell system in the case where the magnetic permeability is equal to μ, and the dielectric permittivity is given by the matrix η(x/ε).