Trigonometric functions of matrices

The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers:

${\begin{aligned}\sin X&=X-{\frac {X^{3}}{3!}}+{\frac {X^{5}}{5!}}-{\frac {X^{7}}{7!}}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}X^{2n+1}\\\cos X&=I-{\frac {X^{2}}{2!}}+{\frac {X^{4}}{4!}}-{\frac {X^{6}}{6!}}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}X^{2n}\end{aligned}}$

with $X^{n}$ being the $n$-th power of the matrix $X$ and $I$ being the identity matrix of appropriate dimensions. Equivalently, they can be defined using the matrix equivalent of Euler's formula along with the matrix exponential, $e^{iX}=\cos X+i\sin X$, yielding

${\begin{aligned}\sin X&={e^{iX}-e^{-iX} \over 2}\\\cos X&={e^{iX}+e^{-iX} \over 2i}.\end{aligned}}$

See also: Axis–angle representation §  Exponential map from so(3) to SO(3)

Properties

The analog of the Pythagorean trigonometric identity holds:

$\sin ^{2}X+\cos ^{2}X=I$

If $X$ is a diagonal matrix, $\sin X$ and $\cos X$ are also diagonal matrices with $(\sin X)_{nn}=\sin(X_{nn})$ and $(\cos X)_{nn}=\cos(X_{nn})$, that is, they can be calculated by simply taking the sines or cosines of the matrice's diagonal components.

The analogs of the trigonometric addition formulas are true if and only if $XY=YX$:

${\begin{aligned}\sin(X\pm Y)=\sin X\cos Y\pm \cos X\sin Y\\\cos(X\pm Y)=\cos X\cos Y\mp \sin X\sin Y\end{aligned}}$