Trigonometric functions

From Encyclopedia of Mathematics
Revision as of 17:10, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

The class of elementary functions sine, cosine, tangent, cotangent, secant, cosecant. These are denoted, respectively, by: , , (or ), (or ), , (or ).

Trigonometric functions of a real argument.

Let be a real number. Let be the end point of the arc on the unit circle (see Fig. a) having initial point and length . The arc from to is taken in the counter-clockwise direction if , and in the clockwise direction if . If , then ; if, e.g., , then . ,

Figure: t094210a

The basic trigonometric functions sine and cosine are defined at by the formulas

The remaining trigonometric functions can be defined by the formulas

All trigonometric functions are periodic. The graphs of the trigonometric functions are given in Fig. b.

Figure: t094210b

The main properties of the trigonometric functions — the domain of definition, the range, the parity, and sections of monotonicity — are given in the table below.'

<tbody> </tbody>
Function Domain of definition Range of values Parity Section of monotonicity
Odd increases for
Odd decreases for

Each trigonometric function is continuous and infinitely differentiable at each point of its domain of definition; the derivatives of the trigonometric functions are:

The integrals of the trigonometric functions are:

All trigonometric functions have a power series expansion:

for ;

for ;

for ;

for (the are the Bernoulli numbers).

The function inverse to the function defines as a many-valued function of , it is denoted by . The inverse functions of the other trigonometric functions are defined similarly; they are all called inverse trigonometric functions.

Trigonometric functions of a complex variable.

The trigonometric functions for complex values of the variable are defined as analytic continuations (cf. Analytic continuation) of the corresponding trigonometric functions of the real variable in the complex plane.

Thus, and can be defined by means of the power series for and given above. These series converge in the entire complex plane, therefore and are entire functions (cf. Entire function).

The trigonometric functions tangent and cotangent are defined by the formulas

The trigonometric functions and are meromorphic functions (cf. Meromorphic function). The poles of are simple (of order one) and are situated at the points , .

All formulas for the trigonometric functions of a real argument remain true for a complex argument as well.

In contrast to the trigonometric functions of a real variable, the functions and take all complex values: The equations and each have infinitely many solutions for any complex :

The trigonometric functions and take all complex values except : The equations , each have infinitely many solutions for any complex number :

The trigonometric functions can be expressed in terms of the exponential function:

and the hyperbolic functions:


The trigonometric functions are also called circular functions.

A formal definition of and (independent of a picture) can be given by power series and as follows. First of all it can easily be proved that from the previous, visual definition of and follows:

This can be taken as a formal definition of and the inverse function of can be taken as a formal definition of .

If is a complex number , with real and , one can define , and then for complex define:


[a1] T.M. Apostol, "Calculus" , I , Blaisdell (1967)
[a2] A.R.F. Verhey, "Complex variables and applications" , McGraw-Hill (1974)
[a3] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972)
How to Cite This Entry:
Trigonometric functions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article