
Trigonometry 



Inverse Trigonometric Functions or
Arcfunctions and their Graphs 
Inverse
functions

The
arcsine function and the arccosine function 
The
arcsine function 
The
arccosine function 
The
graph of the
arcsine
function and the arccosine
function 
The
arctangent function and the
arccotangent function 
The
arctangent function 
The
arccotangent function 
The
graph of the
arctangent function and the
arccotangent function 
The
arccosecant function and the arcsecant function 
The
graph of the
arccosecant and the
arcsecant function 





Inverse Trigonometric Functions or
Arcfunctions and their Graphs 
Inverse
functions

The inverse function, usually written
f ^{1}, is the function whose
domain and the range are respectively the range and domain of a given function
f, that is 
f
^{1}(x)
= y if and only
if f
(y)
= x . 
Thus, the
composition of the inverse function and the given function returns
x, which is called the
identity function,
i.e., 
f
^{1}(ƒ(x))
= x
and f
(f
^{1}(x))
= x. 
The inverse of a function undoes the procedure
(or function) of the given function. 
A pair of inverse functions is in
inverse relation. 
Example: If
given
f (x)
= log_{2 }x
then f ^{1}(x)
= 2^{x}
since, 


The
arcsine function and the arccosine function 
 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
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
graph of the
arcsine
function and the arccosine
function 


The
arctangent function and the
arccotangent function 
 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
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. 

The
graph of the
arctangent function and the
arccotangent function 


The
arccosecant function and the arcsecant function 
 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
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
graph of the
arccosecant and the
arcsecant function 









Precalculus contents
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