

Transcendental functions 
Exponential
and logarithmic functions, inverse functions 
Trigonometric
(cyclometric) functions and inverse trigonometric functions (arcfunctions) 






Transcendental functions: 
· Exponential
and logarithmic functions are mutually inverse functions 
 Exponential
function 
y
= e^{x}
<=>
x = ln y,
e = 2.718281828...the
base of the natural logarithm, 

exponential
function is inverse
of the natural logarithm
function, so that e^{ln
}^{x} = x. 
 Logarithmic
function 
y
= ln x
= log _{e } x
<=>
x = e ^{y},^{ }
where x
> 0


the
natural logarithm
function is inverse
of the exponential
function, so that
ln(e^{x}) =
x. 



 Exponential
function 
y =
a^{x}
<=>
x = log_{a}
y,
where a > 0 and
a is not
1 

exponential
function with base a is
inverse of the logarithmic
function, so that 


 Logarithmic
function 
y
=
log _{a
}
x
<=>
x = a ^{y},
where a
> 0,
a is
not
1 and x
> 0 

the
logarithmic
function with base a
is inverse of the exponential
function, so that
log_{a }(a^{x}) =
x. 


·
Trigonometric
(cyclometric) functions and
inverse trigonometric functions
(arc
functions) 
Trigonometric functions are defined as the ratios of the sides of a right
triangle containing the angle equal to the argument of the
function in radians.

Or
more generally for real arguments, trigonometric
functions are defined in terms of the coordinates of the
terminal point Q of
the arc
(or angle) of the unit circle with the initial point at P(1,
0). 



sin^{2}x
+ cos^{2}x
= 1 



 The
sine function
y
= sin x
is the ycoordinate
of the terminal point of the arc x
of the unit circle. The
graph of the sine function is the sine curve or sinusoid. 
In
a rightangled triangle the
sine function is equal to the ratio of the length of the side
opposite the given angle to the length of the hypotenuse. 
 The
arcsine function
y
= sin^{}^{1}x
or y
= arcsin x
is the inverse of the sine function, so that its value for any
argument is an arc (angle) whose sine equals the given argument. 
That
is, y
= sin^{}^{1}x
if and only if x
= sin y.
For
example, 


Thus, the arcsine
function is defined for arguments between 1
and 1, and its principal
values are by
convention taken to be those between p/2
and p/2. 

  
   
 The
cosine function
y
= cos x
is the xcoordinate
of the terminal point of the arc x
of the unit circle. The
graph of the cosine function is the cosine curve or cosinusoid. 
In
a rightangled triangle the cosine function is equal to the
ratio of the length of the side adjacent the given angle to the
length of the hypotenuse. 
 The
arccosine function
y
= cos^{}^{1}x
or y
= arccos x
is the inverse of the cosine function, so that its value for any
argument is an arc (angle) whose cosine equals the given
argument. 
That
is, y
= cos^{}^{1}x
if and only if x
= cos y.
For
example, 


Thus, the arccosine
function is defined for arguments between 1
and 1, and its principal
values are by
convention taken to be those between 0 and p. 

  
   
 The
tangent function
y
= tan x
is the ratio of the ycoordinate to
the xcoordinate
of the terminal point of the arc x
of the unit circle, or it is the ratio of the sine function to the cosine function. 
In
a rightangled triangle the
tangent function is equal to the ratio of the length of the side
opposite the given angle to that of the adjacent side. 
 The
arctangent function
y
= tan^{}^{1}x
or y
= arctan x
is the inverse of the tangent function, so that its value for any
argument is an arc (angle) whose tangent equals the given
argument. 
That
is, y
= tan^{}^{1}x
if and only if x
= tan
y.
For
example, 


Thus, the arctangent
function is defined for all real arguments, and its principal
values are by
convention taken to be those strictly between p/2
and p/2. 

  
   
 The
cosecant function
y
= csc x
is the reciprocal of the sine function. 
In
a rightangled triangle the
cosecant function is equal to the ratio of the length of the
hypotenuse to that of the side opposite to the given angle. 
 The
arccosecant function
y
= csc^{}^{1}x
or y
= arccsc x
is the inverse of the cosecant function, so that its value for any
argument is an arc (angle) whose cosecant equals the given
argument. 
That
is, y
= csc^{}^{1}x
if and only if x
= csc
y.
For
example, 


Thus, the arccosecant
function is defined for arguments less than 1
or greater than 1, and its principal
values are by
convention taken to be those between p/2
and p/2. 

  
   
 The
secant function
y
= sec x
is the reciprocal of the cosine function. 
In
a rightangled triangle the secant function is equal to the
ratio of the length of the hypotenuse to that of the side
adjacent to the given angle. 
 The
arcsecant function
y
= sec^{}^{1}x
or y
= arcsec x
is the inverse of the secant function, so that its value for any
argument is an arc (angle) whose secant equals the given
argument. 
That
is, y
= sec^{}^{1}x
if and only if x
= sec
y.
For
example, 


Thus, the arcsecant
function is defined for arguments less than 1
or greater than 1, and its principal
values are by
convention taken to be those between 0
and p. 

  
   
 The
cotangent function
y
= cot x
is the reciprocal of the tangent function, or it is the ratio of the cosine function to the
sine function. 
In
a rightangled triangle the cotangent function is equal to the ratio of the length of the side adjacent
to the given angle to that of the side opposite it. 
 The
arccotangent function
y
= cot^{}^{1}x
or y
= arccot x
is the inverse of the cotangent function, so that its value for any
argument is an arc (angle) whose cotangent equals the given
argument. 
That
is, y
= cot^{}^{1}x
if and only if x
= cot
y.
For
example, 


Thus, the arccotangent
function is defined for all real arguments, and its principal
values are by
convention taken to be those strictly between 0
and p. 








Calculus
contents A 



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