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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}
\begin{align}▼
\sin X & = X - \frac{X^3}{3!} + \frac{X^5}{5!} - \frac{X^7}{7!} + \cdots & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}X^{2n+1} \\
\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 a matrix|power]] of the matrix {{mvar|X}}, and {{mvar|I}} being the [[identity matrix]] of appropriate dimensions.
</math>▼
▲:<math>\begin{align}
\end{align}</math>▼
For example, taking {{mvar|X}} to be a standard [[Pauli matrices|Pauli matrix]],
\sigma_1 = \sigma_x =
0&1\\
1&0
\end{pmatrix} ~,</math>
one has
:<math>
\sin(\theta \sigma_1) = \sin(\theta)~ \sigma_1 , \qquad \cos (\theta \sigma_1) = \cos (\theta)~I~,
▲\begin{align}
▲\sin X & = {e^{iX} - e^{-iX} \over 2} \\
▲\cos X & = {e^{iX} + e^{-iX} \over 2i}.
▲\end{align}
</math>
as well as, for the [[Sinc function|cardinal sine function]],
:<math>\operatorname{sinc}( \theta \sigma_1) =\operatorname{sinc}( \theta) ~I. </math>
{{see also| Axis–angle representation # Exponential map from so(3) to SO(3)}}
==Properties==
The analog of the [[Pythagorean trigonometric identity]] holds:<ref name="Higham" />
:<math>\sin^2 X + \cos^2 X = I</math>▼
▲\sin^2 X + \cos^2 X = I
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 matrices's diagonal components.
The analogs of the [[trigonometric addition formulas]] hold as well:<ref name="Higham" />▼
▲The analogs of the [[trigonometric addition formulas]]
:<math>\begin{align}
\sin (X \pm Y) = \sin X \cos Y \pm \cos X \sin Y \\
\cos (X \pm Y) = \cos X \cos Y \mp \sin X \sin Y
\end{align}</math>
==Other functions==
The tangent, as well as [[inverse trigonometric functions]], [[hyperbolic function|hyperbolic]] and [[inverse hyperbolic function]]s have also been defined for matrices:<ref>[https://help.scilab.org/docs/5.5.2/en_US/section_99038107015b1d789de50bf92f154a85.html Scilab trigonometry].</ref>
:<math>\arcsin X = -i \ln \left( iX + \sqrt{I-X^2} \right)</math> (see [[Inverse trigonometric functions#Logarithmic forms]], [[Matrix logarithm]], [[Square root of a matrix]])
:<math>\begin{align}
\sinh X & = {e^X - e^{-X} \over 2} \\
\cosh X & = {e^X + e^{-X} \over 2}
\end{align}</math>
and so on.
==References==
{{reflist}}
[[Category:Matrices]]▼
[[Category:Trigonometry]]
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