The trigonometric functions (especially

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

and cosine) for real or complex

square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...

occur in solutions of second-order systems of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s. They are defined by the same

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

that hold for the trigonometric functions of real and

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

: :

\begin
\sin X & = X - \frac + \frac - \frac + \cdots & = \sum_^\infty \fracX^ \\
\cos X & = I - \frac + \frac - \frac + \cdots & = \sum_^\infty \fracX^
\end

with being the th

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

of the matrix , and being the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

of appropriate dimensions. Equivalently, they can be defined using the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...

along with the matrix equivalent of

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...

, , yielding :

\begin
\sin X & =  \\
\cos X & = .
\end

For example, taking to be a standard

Pauli matrix In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in c ...

, :

\sigma_1 = \sigma_x =
    \begin
      0&1\\
      1&0
    \end ~,

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( \theta \sigma_1) =\operatorname( \theta) ~I.

Properties

The analog of the

Pythagorean trigonometric identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations b ...

holds: :

\sin^2 X + \cos^2 X = I

If is a

diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...

, 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 In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...

are true

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

: :

\begin
\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

Other functions

The tangent, as well as

inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

and

inverse hyperbolic function In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. Th ...

s have also been defined for matrices:Scilab trigonometry
:

\arcsin X = -i \ln \left( iX + \sqrt \right)

(see Inverse trigonometric functions#Logarithmic forms,

Matrix logarithm In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exp ...

Square root of a matrix In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix is said to be a square root of if the matrix product is equal to . Some authors use the name ''square root'' or the notation onl ...

) :

\begin
\sinh X & =  \\
\cosh X & = 
\end

and so on.

References

{{reflist Trigonometry Matrix theory