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6.1 Inverse Function

If is one-one 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 = 300, 1500 , 3900, 5100,.....

To express as a function of x, we write = arc sinx or = sin-1x (read as sine inverse x).

It should be noted that sin-1x 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 cos-1 x as an angle in radians whose cosine is x . The other functions of this kind are tan-1x , cot-1x, sec-1x , cosec-1x (i.e. csc-1x). 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-1x, 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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6.1 Inverse Functions
6.2 Trignometric Equations
Supplementary Problems

Chapter 7

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