The classical(geometrical) definition of is as follows:

For an angle (that is for a real number that lies between 0 and ) is the ratio

where is the opposite side and is the hypotenuse of the right triangle with the angle . \hr

Clearly, this definition cannot be expanded to incorporate matrices or complex numbers instead of angles.

** Universal definition**

Also, the function can be defined by the Taylor series:

This definition returns the same value of as the classical definition everywhere where the last one is applicable, but the definition through the Taylor series is applicable to a broader set of arguments.

** sin of a complex argument**

Now one can compute of any real or complex number.
Particularly, one
can easily prove the selebrated Euler's formula

and other trigonometric formulas for complex numbers, expanding the definition of trigonometric functions.

Notice that does not need to be smaller than
one any more; on the contrary, the equation

where is any real or complex number has a solution in the complex plane.

** sin of matrices**

Recall, that the Taylor series uses only a sequence of additions and multiplications
to define a function. These operations are well defined for any square
matrix , where is an arbitrary integer.
Therefore, we have

Now we may prove all trigonomeric formulas for matrix arguments or investigate the transform of eigensystem of a matrix.

Similarly, we can define any other analytic function of a square matrix :

and so on.