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 (A)V and (A)V, and it uses only real arithmetic when A is real. The algorithm exploits an existing algorithm expmv of Al-Mohy and Higham for e^AV and its underlying backward error analysis. Our experiments show that the new algorithm performs

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... a forward stable manner and is generally significantly faster than alternatives based on multiple invocations of expmv through formulas such as cos(A)V = (e^iAV + e^-iAV)/2.