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{{Use American English|date = March 2019}}
{{Short description|Important functions in solving differential equations}}
The '''trigonometric functions''' (especially [[sine]] and [[cosine]]) for real or complex [[square matrices]] occur in solutions of second-order systems of [[differential equation]]s.<ref>{{cite journal|title=Efficient Algorithms for the Matrix Cosine and Sine|author=Gareth I. Hargreaves |author2=Nicholas J. Higham |journal=Numerical Analysis Report|issue=461|publisher=Manchester Centre for Computational Mathematics|year=2005|volume=40|page=383|doi=10.1007/s11075-005-8141-0|bibcode=2005NuAlg..40..383H|s2cid=1242875|url=http://eprints.maths.manchester.ac.uk/124/1/paper2.pdf}}</ref> They are defined by the same [[Taylor series]] that hold for the trigonometric functions of real and [[complex numbers]]:<ref name="Higham">{{cite book|title=Functions of matrices: theory and computation|author=Nicholas J. Higham|year=2008|pages=287f|isbn={{Format ISBN|9780898717778}}978-0-89871-777-8}}</ref>
