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.[1] They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers:[2]

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

with Xⁿ being the nth power of the matrix X, and 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, e^iX = cos X + i sin X, yielding

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

For example, taking X to be a standard Pauli matrix,

${\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}~,}$

one has

${\displaystyle \sin(\theta \sigma _{1})=\sin(\theta )~\sigma _{1},\qquad \cos(\theta \sigma _{1})=\cos(\theta )~I~,}$

as well as, for the cardinal sine function,

${\displaystyle \operatorname {sinc} (\theta \sigma _{1})=\operatorname {sinc} (\theta )~I.}$