The trigonometric functions (especially sine and cosine) for 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 complex numbers:

sin ⁡ X = X − X 3 3 ! + X 5 5 ! − X 7 7 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! X 2 n + 1 cos ⁡ X = I − X 2 2 ! + X 4 4 ! − X 6 6 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! X 2 n {\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, eiX = cos X + i sin X, yielding

sin ⁡ X = e i X − e − i X 2 i cos ⁡ X = e i X + e − i X 2 . {\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,

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

one has

sin ⁡ ( θ σ 1 ) = sin ⁡ ( θ )   σ 1 , cos ⁡ ( θ σ 1 ) = cos ⁡ ( θ )   I   , {\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,

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