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{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}$$ 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 exponential along with the matrix equivalent of Euler's formula, $e^{iX} = cos X + i sin X$, yielding
 * $$\begin{align}

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

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

\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\     1&0    \end{pmatrix} ~,$$ one has

\sin(\theta \sigma_1) =    \sin(\theta)~ \sigma_1, \qquad  \cos (\theta \sigma_1) =     \cos (\theta)~I~, $$ as well as, for the cardinal sine function,
 * $$\operatorname{sinc}( \theta \sigma_1) =\operatorname{sinc}( \theta) ~I.  $$

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 matrices's diagonal components.

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

Other functions
The tangent, as well as inverse trigonometric functions, hyperbolic and inverse hyperbolic functions have also been defined for matrices:
 * $$\arcsin X = -i \ln \left( iX + \sqrt{I-X^2} \right)$$ (see Inverse trigonometric functions, Matrix logarithm, Square root of a matrix)
 * $$\begin{align}

\sinh X & = {e^X - e^{-X} \over 2} \\ \cosh X & = {e^X + e^{-X} \over 2} \end{align}$$ and so on.