In this paper, we introduce two methods for constructing matrix-valued cosine and sine functions defined on a discrete domain. The first method develops an algorithm to compute the $n\times n$ matrix-valued cosine and sine of a given square matrix $A$ with some restrictions on its eigenvalues. The second method derives a formula based on the Jordan canonical form to compute each term of a Jordan block in the discrete matrix-valued cosine and sine of a given square matrix $A$. To illustrate the utility of our approach, we present some examples.