Trigonometric functions
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
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The remaining trigonometric functions can be defined by the formulas
 |
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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.'
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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 
;
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 |
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
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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 
:
 |
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The trigonometric functions can be expressed in terms of the exponential function:
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and the hyperbolic functions:
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Comments
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:
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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:
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References
| [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) |
Trigonometric functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_functions&oldid=49035