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The class of elementary functions sine, cosine, tangent, cotangent, secant, cosecant. These are denoted, respectively, by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942103.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942104.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942105.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942106.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942108.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t0942109.png" />).
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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.==
 
==Trigonometric functions of a real argument.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421010.png" /> be a real number. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421011.png" /> be the end point of the arc on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421012.png" /> (see Fig. a) having initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421013.png" /> and length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421014.png" />. The arc from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421016.png" /> is taken in the counter-clockwise direction if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421017.png" />, and in the clockwise direction if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421020.png" />; if, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421022.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421024.png" />
+
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t094210a.gif" />
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Figure: t094210a
 
Figure: t094210a
  
The basic trigonometric functions sine and cosine are defined at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421025.png" /> by the formulas
+
The basic trigonometric functions sine and cosine are defined at $  \alpha $
 +
by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421026.png" /></td> </tr></table>
+
$$
 +
\sin  \alpha  = \
 +
y _  \alpha  ,\ \
 +
\cos  \alpha  = \
 +
x _  \alpha  .
 +
$$
  
 
The remaining trigonometric functions can be defined by the formulas
 
The remaining trigonometric functions can be defined by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421027.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm tan}  \alpha  = \
 +
 
 +
\frac{\sin  \alpha }{\cos  \alpha }
 +
,\ \
 +
\mathop{\rm cot}  \alpha  = \
 +
 
 +
\frac{\cos  \alpha }{\sin  \alpha }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421028.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sec}  \alpha  = {
 +
\frac{1}{\cos  \alpha }
 +
} ,\ \
 +
\mathop{\rm csc}  \alpha  = {
 +
\frac{1}{\sin  \alpha }
 +
} .
 +
$$
  
 
All trigonometric functions are periodic. The graphs of the trigonometric functions are given in Fig. b.
 
All trigonometric functions are periodic. The graphs of the trigonometric functions are given in Fig. b.
Line 24: Line 80:
 
Figure: t094210b
 
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"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421029.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421030.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421031.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421032.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421033.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421034.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421035.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Even</td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421036.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421037.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421038.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421039.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1">increases for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421040.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421041.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421042.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421043.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1">decreases for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421044.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421045.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421046.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421047.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Even</td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421048.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421049.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421050.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421051.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Odd</td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421052.png" /></td> </tr> </tbody> </table>
+
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>
 
</td></tr> </table>
Line 30: Line 126:
 
Each trigonometric function is continuous and infinitely differentiable at each point of its domain of definition; the derivatives of the trigonometric functions are:
 
Each trigonometric function is continuous and infinitely differentiable at each point of its domain of definition; the derivatives of the trigonometric functions are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421053.png" /></td> </tr></table>
+
$$
 +
( \sin  x)  ^  \prime  = \cos  x,\ \
 +
( \cos  x)  ^  \prime  = - \sin  x,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421054.png" /></td> </tr></table>
+
$$
 +
(  \mathop{\rm tan}  x)  ^  \prime  = {
 +
\frac{1}{\cos  ^ {2}  x }
 +
} ,\  (
 +
\mathop{\rm cot}  x)  ^  \prime  = - {
 +
\frac{1}{\sin  ^ {2}  x }
 +
} .
 +
$$
  
 
The integrals of the trigonometric functions are:
 
The integrals of the trigonometric functions are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421055.png" /></td> </tr></table>
+
$$
 +
\int\limits \sin  x  dx  = - \cos  x + C,\ \
 +
\int\limits \cos  x  dx  = \sin  x + C,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421056.png" /></td> </tr></table>
+
$$
 +
\int\limits  \mathop{\rm tan}  x  dx  = - \mathop{\rm ln}  | \cos  x | + C,
 +
\int\limits  \mathop{\rm cot}  x  dx  =   \mathop{\rm ln}  | \sin  x | + C.
 +
$$
  
 
All trigonometric functions have a power series expansion:
 
All trigonometric functions have a power series expansion:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421057.png" /></td> </tr></table>
+
$$
 +
\sin  x  = x -
 +
 
 +
\frac{x  ^ {3} }{3! }
 +
+
 +
 
 +
\frac{x  ^ {5} }{5! }
 +
- \dots +
 +
(- 1)  ^ {n}
 +
 
 +
\frac{x ^ {2n + 1 } }{( 2n + 1)! }
 +
+ \dots
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421058.png" />;
+
for $  | x | < \infty $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421059.png" /></td> </tr></table>
+
$$
 +
\cos  x  = 1 -
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421060.png" />;
+
\frac{x  ^ {2} }{2! }
 +
+
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421061.png" /></td> </tr></table>
+
\frac{x  ^ {4} }{4! }
 +
-
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421062.png" /></td> </tr></table>
+
\frac{x  ^ {6} }{6! }
 +
+ \dots +
 +
(- 1)  ^ {n}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421063.png" />;
+
\frac{x  ^ {2n} }{( 2n)! }
 +
+ \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421064.png" /></td> </tr></table>
+
for  $  | x | < \infty $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421065.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm tan}  x  = \
 +
x + {
 +
\frac{1}{3}
 +
}
 +
x  ^ {3} + {
 +
\frac{2}{15}
 +
}
 +
x  ^ {5} + {
 +
\frac{17}{315}
 +
}
 +
x  ^ {7} + \dots
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421066.png" /> (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421067.png" /> are the [[Bernoulli numbers|Bernoulli numbers]]).
+
$$
 +
\dots
 +
+
 +
\frac{2  ^ {2n} ( 2  ^ {2n} - 1) | B _ {n} | }{( 2n)! }
 +
x ^ {2n - 1 } + \dots
 +
$$
  
The function inverse to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421068.png" /> defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421069.png" /> as a many-valued function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421070.png" />, it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421071.png" />. The inverse functions of the other trigonometric functions are defined similarly; they are all called [[Inverse trigonometric functions|inverse trigonometric functions]].
+
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.==
 
==Trigonometric functions of a complex variable.==
The trigonometric functions for complex values of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421072.png" /> are defined as analytic continuations (cf. [[Analytic continuation|Analytic continuation]]) of the corresponding trigonometric functions of the real variable in the complex plane.
+
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421074.png" /> can be defined by means of the power series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421076.png" /> given above. These series converge in the entire complex plane, therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421078.png" /> are entire functions (cf. [[Entire function|Entire function]]).
+
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
 
The trigonometric functions tangent and cotangent are defined by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421079.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm tan}  z  = \
  
The trigonometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421081.png" /> are meromorphic functions (cf. [[Meromorphic function|Meromorphic function]]). The poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421082.png" /> are simple (of order one) and are situated at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421084.png" />.
+
\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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421086.png" /> take all complex values: The equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421088.png" /> each have infinitely many solutions for any complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421089.png" />:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421090.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm arc}  \sin  a  = \
 +
- i  \mathop{\rm ln} ( ia \pm  \sqrt {1 - a  ^ {2} } ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421091.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm arc}  \cos  a  = - i  \mathop{\rm ln} ( a \pm  \sqrt {a  ^ {2} - 1 } ).
 +
$$
  
The trigonometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421093.png" /> take all complex values except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421094.png" />: The equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421096.png" /> each have infinitely many solutions for any complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421097.png" />:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421098.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm arc}  \mathop{\rm tan}  a  = \
 +
{
 +
\frac{i}{2}
 +
}  \mathop{\rm ln} 
 +
\frac{1 - ia }{1 + ia }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t09421099.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm arc}  \mathop{\rm cot}  a  = {
 +
\frac{i}{2}
 +
}  \mathop{\rm ln} 
 +
\frac{ia + 1 }{ia - 1 }
 +
.
 +
$$
  
 
The trigonometric functions can be expressed in terms of the [[Exponential function|exponential function]]:
 
The trigonometric functions can be expressed in terms of the [[Exponential function|exponential function]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210100.png" /></td> </tr></table>
+
$$
 +
\sin  z  = {
 +
\frac{1}{2i}
 +
} ( e  ^ {iz} - e  ^ {-} iz ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210101.png" /></td> </tr></table>
+
$$
 +
\cos  z  = {
 +
\frac{1}{2}
 +
} ( e  ^ {iz} + e  ^ {-} iz ),\ \
 +
\mathop{\rm tan}  z  = {
 +
\frac{1}{i}
 +
}
 +
\frac{e  ^ {iz} - e  ^ {-} iz }{e  ^ {iz} + e  ^ {-} iz }
 +
,
 +
$$
  
 
and the [[Hyperbolic functions|hyperbolic functions]]:
 
and the [[Hyperbolic functions|hyperbolic functions]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210102.png" /></td> </tr></table>
+
$$
 
+
\sin  z  = - i  \sinh  iz,\ \
 
+
\cos  z  = \cosh  iz,\ \
 +
\mathop{\rm tan}  z  = - i  \mathop{\rm tanh}  iz.
 +
$$
  
 
====Comments====
 
====Comments====
 
The trigonometric functions are also called circular functions.
 
The trigonometric functions are also called circular functions.
  
A formal definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210104.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210106.png" /> follows:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210107.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm arc}  \sin  x  = \int\limits _ { 0 } ^ { x } 
 +
\frac{dx}{\sqrt {1- t ^ {2} } }
 +
.
 +
$$
  
This can be taken as a formal definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210108.png" /> and the inverse function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210109.png" /> can be taken as a formal definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210110.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210111.png" /> is a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210112.png" />, with real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210114.png" />, one can define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210115.png" />, and then for complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210116.png" /> define:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094210/t094210117.png" /></td> </tr></table>
+
$$
 +
\sin  z  =
 +
\frac{e  ^ {iz} - e  ^ {-} iz }{2i}
 +
,\ \
 +
\cos  z  =
 +
\frac{e  ^ {iz} + e  ^ {-} iz }{2}
 +
.
 +
$$
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:26, 6 June 2020


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 $

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.

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
$ \sin x $ $ - \infty < x <+ \infty $ $ [- 1, + 1] $ Odd $ \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} $
$ \cos x $ $ - \infty < x <+\infty $ $ [- 1, + 1] $ Even $ \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} $
$ \mathop{\rm tan} x $ $ x \neq \pi n + \pi / 2 $ $ (- \infty , + \infty ) $ Odd increases for $ x \in (( 2n - 1) \pi /2, ( 2n + 1) \pi /2) $
$ \mathop{\rm cot} x $ $ x \neq \pi n $ $ (- \infty , + \infty ) $ Odd decreases for $ x \in ( n \pi , ( n + 1) \pi ) $
$ \mathop{\rm sec} x $ $ x \neq \pi n + \pi / 2 $ $ (- \infty , - 1 ] \cup [ + 1, + \infty ) $ Even $ \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} $
$ \mathop{\rm csc} x $ $ x \neq \pi n $ $ (- \infty , - 1 ] \cup [ + 1, + \infty ) $ Odd $ \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} $

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).

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.

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) 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).

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). 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:

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

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

and the hyperbolic functions:

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

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

[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: http://encyclopediaofmath.org/index.php?title=Trigonometric_functions&oldid=14919
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article