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

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

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 hold as well:

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