It is element-wise sin, which is not entirely correct. You can take the sin of, or any analytic function of a matrix, however you'd have to compute the characteristic polynomial for the matrix and apply the function to that. It is a result of the Cayley-Hamilton Theorem. here is a tutorial: http://web.mit.edu/2.151/www/Handouts/CayleyHamilton.pdf
You don't need to know the characteristic polynomial to do it. Just sum as many terms of the power series as you want.
What the characteristic polynomial lets you do is to calculate it faster and more accurately because very large powers can be rewritten in terms of smaller powers that you already computed.
