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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 ituses only real arithmetic when $A$ is real. The algorithm exploits an existingalgorithm \texttt{expmv} of Al-Mohy and Higham for $\mathrm{e}^AV$ and itsunderlying backward error analysis. Our experiments show that the new algorithmperforms in a forward stable manner and is generally significantly faster thanalternatives based on multiple invocations of \texttt{expmv} through formulassuch as $\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V +\mathrm{e}^{\mathrm{-i}A}V)/2$.

Computing the Action of Trigonometric and Hyperbolic Matrix Functions