
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 





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 ƒ
(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 ƒ
(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
ƒ (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 









Trigonometry
contents B 



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