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{{Use American English|date = March 2019}}
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|authors=Gareth I. Hargreaves, Nicholas J. Higham|journal=Numerical Analysis Report|issue=461|publisher=Manchester Centre for Computational Mathematics|year=2005}}</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=9780898717778}}</ref>▼
{{Short description|Important functions in solving differential equations}}
▲The '''trigonometric functions''' (especially [[sine]] and [[cosine]]) for
:<math>\begin{align}
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\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
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
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:<math>\sin^2 X + \cos^2 X = I</math>
If {{mvar|X}} is a [[diagonal matrix]], {{math|sin ''X''}} and {{math|cos ''X''}} are also diagonal matrices with {{math|(sin ''X'')<sub>''nn''</sub> {{=}} sin(''X<sub>nn</sub>'')}} and {{math|(cos ''X'')<sub>''nn''</sub> {{=}} cos(''X<sub>nn</sub>'')}}, that is, they can be calculated by simply taking the sines or cosines of the
The analogs of the [[trigonometric addition formulas]] are true [[if and only if]] {{mvar|XY {{=}} YX}}:<ref name="Higham" />
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