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The class of elementary functions sine, cosine, tangent, cotangent, secant, cosecant. These are denoted, respectively, by:  $  \sin  x $, 
$  \cos  x $, 
$   \mathop{\rm tan}  x $(
or  $   \mathop{\rm tg}  x $), 
$   \mathop{\rm cot}  x $(
or  $   \mathop{\rm cotan}  x $), 
$   \mathop{\rm sec}  x $, 
$   \mathop{\rm csc}  x $(
or  $  \cosec  x $).

==Trigonometric functions of a real argument.==
Let  $  \alpha $
be a real number. Let  $  A = ( x _  \alpha  , y _  \alpha  ) $
be the end point of the arc on the unit circle  $  x  ^ {2} + y  ^ {2} = 1 $(
see Fig. a) having initial point  $  B = ( 1, 0) $
and length  $  | \alpha | $. 
The arc from  $  B $
to  $  A $
is taken in the counter-clockwise direction if  $  \alpha \geq  0 $, 
and in the clockwise direction if  $  \alpha < 0 $. 
If  $  \alpha = 0 $, 
then  $  A = B $; 
if, e.g.,  $  \alpha = (- 7 \pi )/ ( 2) $, 
then  $  A = ( 0, 1) $. 
$  B $, 
$  | BA | = \alpha > 0 $

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t094210a.gif" />

Figure: t094210a

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

$$ 
\sin  \alpha  = \ 
y _  \alpha  ,\ \ 
\cos  \alpha  = \ 
x _  \alpha  .
$$

The remaining trigonometric functions can be defined by the formulas

$$ 
 \mathop{\rm tan}  \alpha  = \ 

\frac{\sin  \alpha }{\cos  \alpha }
 ,\ \ 
 \mathop{\rm cot}  \alpha  = \ 

\frac{\cos  \alpha }{\sin  \alpha }
 ,
$$

$$ 
 \mathop{\rm sec}  \alpha  =  {
\frac{1}{\cos  \alpha }
 } ,\ \ 
 \mathop{\rm csc}  \alpha  =  {
\frac{1}{\sin  \alpha }
 } .
$$

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t094210b.gif" />

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.<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Function</td> <td colname="2" style="background-color:white;" colspan="1">Domain of definition</td> <td colname="3" style="background-color:white;" colspan="1">Range of values</td> <td colname="4" style="background-color:white;" colspan="1">Parity</td> <td colname="5" style="background-color:white;" colspan="1">Section of monotonicity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  \sin  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  - \infty < x <+ \infty $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  [- 1, + 1] $
</td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1"> $  \begin{array}{c}
\textrm{ increases  for   } x \in (( 4n - 1) \pi /2, ( 4n + 1) \pi /2) \\
 \textrm{ decreases  for   } x \in (( 4n + 1) \pi /2, ( 4n + 3) \pi /2) 
\end{array}
$
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  \cos  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  - \infty < x <+\infty $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  [- 1, + 1] $
</td> <td colname="4" style="background-color:white;" colspan="1">Even</td> <td colname="5" style="background-color:white;" colspan="1"> $  \begin{array}{c}
\textrm{ increases  for   } x \in (( 2n - 1) \pi , 2n \pi) \\
 \textrm{ decreases  for   } x \in ( 2n \pi , ( 2n + 1) \pi ) 
\end{array}
$
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $   \mathop{\rm tan}  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  x \neq \pi n + \pi / 2 $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  (- \infty , + \infty ) $
</td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1">increases for  $  x \in (( 2n - 1) \pi /2, ( 2n + 1) \pi /2) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $   \mathop{\rm cot}  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  x \neq \pi n $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  (- \infty , + \infty ) $
</td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1">decreases for  $  x \in ( n \pi , ( n + 1) \pi ) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $   \mathop{\rm sec}  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  x \neq \pi n + \pi / 2 $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  (- \infty , - 1 ] \cup [ + 1, + \infty ) $
</td> <td colname="4" style="background-color:white;" colspan="1">Even</td> <td colname="5" style="background-color:white;" colspan="1"> $  \begin{array}{c}
\textrm{ increases  for   } x \in ( 2n \pi , ( 2n + 1) \pi ) \\
 \textrm{ decreases  for   } x \in (( 2n - 1) \pi , 2n \pi ) 
\end{array}
$
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $   \mathop{\rm csc}  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  x \neq \pi n $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  (- \infty , - 1 ] \cup [ + 1, + \infty ) $
</td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1"> $  \begin{array}{c}
\textrm{ increases  for   } x \in (( 4n + 1) \pi /2, ( 4n + 3) \pi /2) \\
 \textrm{ decreases  for   } x \in (( 4n - 1) \pi /2, ( 4n + 1) \pi /2) 
\end{array}
$
</td> </tr> </tbody> </table>

</td></tr> </table>

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

$$ 
( \sin  x)  ^  \prime   =  \cos  x,\ \ 
( \cos  x)  ^  \prime   =  - \sin  x,
$$

$$ 
(  \mathop{\rm tan}  x)  ^  \prime   =  {
\frac{1}{\cos  ^ {2}  x }
 } ,\  (
 \mathop{\rm cot}  x)  ^  \prime   =  - {
\frac{1}{\sin  ^ {2}  x }
 } .
$$

The integrals of the trigonometric functions are:

$$ 
\int\limits \sin  x  dx  =  - \cos  x + C,\ \ 
\int\limits \cos  x  dx  =  \sin  x + C,
$$

$$ 
\int\limits  \mathop{\rm tan}  x  dx  =  -  \mathop{\rm ln}  | \cos  x | + C,
\int\limits  \mathop{\rm cot}  x  dx  =   \mathop{\rm ln}  | \sin  x | + C.
$$

All trigonometric functions have a power series expansion:

$$ 
\sin  x  =  x -

\frac{x  ^ {3} }{3! }
 +

\frac{x  ^ {5} }{5! }
 - \dots +
(- 1)  ^ {n}

\frac{x ^ {2n + 1 } }{( 2n + 1)! }
 + \dots
$$

for  $  | x | < \infty $;

$$ 
\cos  x  =  1 -

\frac{x  ^ {2} }{2! }
 +

\frac{x  ^ {4} }{4! }
 -

\frac{x  ^ {6} }{6! }
 + \dots +
(- 1)  ^ {n}

\frac{x  ^ {2n} }{( 2n)! }
 + \dots
$$

for  $  | x | < \infty $;

$$ 
 \mathop{\rm tan}  x  = \ 
x + {
\frac{1}{3}
 }
x  ^ {3} + {
\frac{2}{15}
 }
x  ^ {5} + {
\frac{17}{315}
 }
x  ^ {7} + \dots
$$

$$ 
\dots 
+ 
\frac{2  ^ {2n} ( 2  ^ {2n} - 1) | B _ {n} | }{( 2n)! }
 x ^ {2n - 1 } + \dots
$$

for  $  | x | < {\pi / 2 } $;

$$ 
 \mathop{\rm cot}  x  =  {
\frac{1}{x}
 } -
$$

$$ 
- 
\left [ {
\frac{x}{3}
 } + 
\frac{x  ^ {3} }{45 }
 + 
\frac{2x
 ^ {5} }{945 }
 + 
\frac{x  ^ {7} }{4725 }
 + \dots + 
\frac{2  ^ {2n} | B _ {n} | }{( 2n)! }
 x ^ {2n - 1 } + \dots \right ]
$$

for  $  0 < | x | < \pi $(
the  $  B _ {n} $
are the [[Bernoulli numbers|Bernoulli numbers]]).

The function inverse to the function  $  x = \sin  y $
defines  $  y $
as a many-valued function of  $  x $, 
it is denoted by  $  y =  \mathop{\rm arc}  \sin  x $. 
The inverse functions of the other trigonometric functions are defined similarly; they are all called [[Inverse trigonometric functions|inverse trigonometric functions]].

==Trigonometric functions of a complex variable.==
The trigonometric functions for complex values of the variable  $  z = x + iy $
are defined as analytic continuations (cf. [[Analytic continuation|Analytic continuation]]) of the corresponding trigonometric functions of the real variable in the complex plane.

Thus,  $  \sin  z $
and  $  \cos  z $
can be defined by means of the power series for  $  \sin  x $
and  $  \cos  x $
given above. These series converge in the entire complex plane, therefore  $  \sin  z $
and  $  \cos  z $
are entire functions (cf. [[Entire function|Entire function]]).

The trigonometric functions tangent and cotangent are defined by the formulas

$$ 
 \mathop{\rm tan}  z  = \ 

\frac{\sin  z }{\cos  z }
 ,\ \ 
 \mathop{\rm cot}  z  = \ 

\frac{\cos  z }{\sin  z }
 .
$$

The trigonometric functions  $   \mathop{\rm tan}  z $
and  $   \mathop{\rm cot}  z $
are meromorphic functions (cf. [[Meromorphic function|Meromorphic function]]). The poles of  $   \mathop{\rm tan}  z $
are simple (of order one) and are situated at the points  $  z = \pi / 2 + \pi n $, 
$  n = 0, \pm  1 ,\dots $.

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  $  \sin  z $
and  $  \cos  z $
take all complex values: The equations  $  \sin  z = a $
and  $  \cos  z = a $
each have infinitely many solutions for any complex  $  a $:

$$ 
z  =   \mathop{\rm arc}  \sin  a  = \ 
- i   \mathop{\rm ln} ( ia \pm  \sqrt {1 - a  ^ {2} } ),
$$

$$ 
z  =   \mathop{\rm arc}  \cos  a  =  - i   \mathop{\rm ln} ( a \pm  \sqrt {a  ^ {2} - 1 } ).
$$

The trigonometric functions  $   \mathop{\rm tan}  z $
and  $   \mathop{\rm cot}  z $
take all complex values except  $  \pm  i $: 
The equations  $   \mathop{\rm tan}  z = a $, 
$   \mathop{\rm cot}  z = a $
each have infinitely many solutions for any complex number  $  a \neq \pm  i $:

$$ 
z  =   \mathop{\rm arc}   \mathop{\rm tan}  a  = \ 
{
\frac{i}{2}
 }   \mathop{\rm ln}  
\frac{1 - ia }{1 + ia }
 ,
$$

$$ 
z  =   \mathop{\rm arc}   \mathop{\rm cot}  a  =  {
\frac{i}{2}
 }   \mathop{\rm ln}  
\frac{ia + 1 }{ia - 1 }
 .
$$

The trigonometric functions can be expressed in terms of the [[Exponential function|exponential function]]:

$$ 
\sin  z  =  {
\frac{1}{2i}
 } ( e  ^ {iz} - e  ^ {-} iz ),
$$

$$ 
\cos  z  =  {
\frac{1}{2}
 } ( e  ^ {iz} + e  ^ {-} iz ),\ \ 
 \mathop{\rm tan}  z  =  {
\frac{1}{i}
 } 
\frac{e  ^ {iz} - e  ^ {-} iz }{e  ^ {iz} + e  ^ {-} iz }
 ,
$$

and the [[Hyperbolic functions|hyperbolic functions]]:

$$ 
\sin  z  =  - i  \sinh  iz,\ \ 
\cos  z  =  \cosh  iz,\ \ 
 \mathop{\rm tan}  z  =  - i   \mathop{\rm tanh}  iz.
$$

====Comments====
The trigonometric functions are also called circular functions.

A formal definition of  $  \sin  z $
and  $  \cos  z $(
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  $  \sin  x $
and  $  \cos  x $
follows:

$$ 
 \mathop{\rm arc}  \sin  x  =  \int\limits _ { 0 } ^ { x }  
\frac{dx}{\sqrt {1- t  ^ {2} } }
 .
$$

This can be taken as a formal definition of  $   \mathop{\rm arc}  \sin  x $
and the inverse function of  $   \mathop{\rm arc}  \sin  x $
can be taken as a formal definition of  $  \sin  x $.

If  $  z $
is a complex number  $  x + iy $, 
with real  $  x $
and  $  y $, 
one can define  $  e  ^ {z} = e  ^ {x} ( \cos  y + i  \sin  y ) $, 
and then for complex  $  z $
define:

$$ 
\sin  z  =  
\frac{e  ^ {iz} - e  ^ {-} iz }{2i}
 ,\ \ 
\cos  z  =  
\frac{e  ^ {iz} + e  ^ {-} iz }{2}
 .
$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Calculus" , '''I''' , Blaisdell  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.R.F. Verhey,   "Complex variables and applications" , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1972)</TD></TR></table>