CHAPTER 6: INVERSE CIRCULAR FUNCTIONS
6.1 Inverse Function
If is oneone on to function then corresponding
to every element of B, we can get a unique element of A. This defines a function
from B to A. Such a function is called an inverse function and is defined by
Inverse Circular Functions
Consider the equation This gives a unique value
of x, for a given value of For example if But when x is given , the equation may have no solution or many
(infinite) solutions . If x = 2, there is no solution
We get infinite values of q but for x >
1, there is no solution.
For example if sin=1/2 ,there are many values of satisfying
the equation, namely = 30^{0},^{ }150^{0} , 390^{0}, 510^{0},.....
To express as a function of x, we write = arc sinx or = sin^{1}x (read
as sine inverse x).
It should be noted that sin^{1}x is entirely different from (sin x)^{ 1}
. The former is the measure of an angle in radians whose sine is x while
the latter is
We can similarly define cos1 x as an angle in radians whose
cosine is x . The other functions of this kind are tan1x , cot1x, sec1x ,
cosec1x (i.e. csc1x). These functions are called inverse circular functions
or inverse trigonometric functions.
Principal values
It is at times necessary to consider inverse trigonometric functions as single
valued.
To do this, we select only one value out of many values of angles corresponding
to the given value of x .
This selected value is called the principal value.
1) For sin = x or = sin^{1}x, among all values of satisfying
this relation, there is only one value between This value of q
is called the principal value of sin^{ }1 x. For example for
the principal value is If x is + ve, it lies between 0
and and if x is ve it lies between  and 0. Therefore the principal
value of sin^{1}(1/2) is
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