  We derive a new algorithm for computing the action $f(A)V$ of the cosine,
sine, hyperbolic cosine, and hyperbolic sine of a matrix $A$ on a matrix $V$,
without first computing $f(A)$. The algorithm can compute $\cos(A)V$ and
$\sin(A)V$ simultaneously, and likewise for $\cosh(A)V$ and $\sinh(A)V$, and it
uses only real arithmetic when $A$ is real. The algorithm exploits an existing
algorithm \texttt{expmv} of Al-Mohy and Higham for $\mathrm{e}^AV$ and its
underlying backward error analysis. Our experiments show that the new algorithm
performs in a forward stable manner and is generally significantly faster than
alternatives based on multiple invocations of \texttt{expmv} through formulas
such as $\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V +
\mathrm{e}^{\mathrm{-i}A}V)/2$.
