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Trigonometric functions are the class of elementary transcendental functions: sine, cosine, tangent, cotangent, secant, cosecant.

For a real value, arguments of a trigonometric function are defined in geometrical terms. Set B is the point of a unit circle with center at the origin, φ is the polar angle of the point B.

If xφ and yφ are the rectangular Cartesian coordinates, then the trigonometric functions sine and cosine are defined by

The other trigonometric functions: tangent, cotangent, secant, cosecant, can be determined by

(1a)

All trigonometric functions are periodic; sine and cosine have a period of 2π, tangent and cotangent, a period π.

Sine, tangent and cotangent are odd functions, cosine is an even function.

All trigonometric functions in their fields of definition are continuous and infinitely differentiable. The derivatives of the trigonometric functions are given by:

(1)

The integrals of trigonometric functions are

(2)

All trigonometric functions can be expanded in power series

(3)

Ban are Bernoulli numbers.

A trigonometric system

(4)

is one of the important orthogonal function systems, and it is widely employed in solving problems of heat conduction. A trigonometric system is complete and closed in the space of the continuous 2π periodic functions.

Trigonometric functions of a complex argument z = x + iy are defined as analytic continuation of corresponding trigonometric functions of a real variable into a complex piane. All formulas valid for trigonometric functions of a real argument remain also valid for a complex argument. Trigonometric functions of a complex argument can be expressed in terms of an exponential function by the Euler formulas

(5)
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