A self-adjoint strongly elliptic second-order differential operator Aε on L2(ℝd;ℂn) is considered. It is assumed that the coefficients of Aε are periodic and depend on x/ε, where ε > 0 is a small parameter. Approximations for the operators cos(A 1/2ετ) and A 1/2ε sin(A 1/2ετ) in the norm of operators from the Sobolev space Hs(ℝd;ℂn) to L2(ℝd;ℂn) (for appropriate s) are obtained. Approximation with a corrector for the operator A 1/2ε sin(A 1/2ετ) in the (Hs → H1)-norm is also obtained. The question about the sharpness of the results with respect to the norm type and with respect to the dependence of the estimates on is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation ∂ 2τuε = − Aεuε.