{{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|authorsauthor=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=9780898717778978-0-89871-777-8}}</ref>
:<math>\begin{align}
Line 7:
\cos X & = I - \frac{X^2}{2!} + \frac{X^4}{4!} - \frac{X^6}{6!} + \cdots & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}X^{2n}
\end{align}</math>
with {{math|''X<sup>n</sup>''}} being the {{mvar|n}}th [[Matrix multiplication#Powers of matricesa matrix|power]] of the matrix {{mvar|X}}, and {{mvar|I}} being the [[identity matrix]] of appropriate dimensions.
Equivalently, they can be defined using the [[matrix exponential]] along with the matrix equivalent of [[Euler's formula]], {{math|''e<sup>iX</sup>'' {{=}} cos ''X'' + ''i'' sin ''X''}}, yielding
