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:
with being the th power of the matrix , and 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, , yielding 
For example, taking to be a standard Pauli matrix,
one has
as well as, for the cardinal sine function,
Properties.
The analog of the Pythagorean trigonometric identity holds:
If is a diagonal matrix, and are also diagonal matrices with and , 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 :
Other functions.
The tangent, as well as inverse trigonometric functions, hyperbolic and inverse hyperbolic functions have also been defined for matrices:
and so on.