Inverse Trigonometric Function

The inverse trigonometric functions play an important role in calculus for they serve to define m  any integrals. The concepts of inverse trigonometric functions is also used in science and engineering. 

Basic Trigonometric functions: 

We have already studied about six trigonometric functions, details of all six trigonometric functions is given as:

         Domain range and period of trigonometric functions:

Domain and Range

Inverse of Trigonometric functions is not possible in their natural Domain and Range because they are not ONE-ONE and ONTO functions in their domain.
But if we fix Domain and range of Trigonometric functions then we can find inverse easily 
 

Inverse Trigonometric

          Note:

1. All trigonometric functions are periodic functions.
2. All trigonometric functions are Continuous in its domain.
3. Sin–1(x) should not be confused with (sin x)^–1. In fact (sin x)^–1 =  and similarly for other trigonometric functions.
4. The value of an inverse trigonometric functions which lies in the range of principal branch is called the principal value of that inverse trigonometric functions.
5. If y = f (x) is an invertible function, then x = f^-1(y). Thus, the graph of sin^–1 function can be obtained from the graph of original function by interchanging x and y axes, i.e., if (a,b) is a point on the graph of sine function, then (b, a) becomes the corresponding point on the graph of inverse of sine function.  
6. Thus, the graph of the function y = sin^–1(x) can be obtained from the graph of y = sin x by interchanging x and y axes.

Graph of Inverse trigonometric functions: 

Graph of inverse trigonometric functions are obtained by making then ONE-ONE and ONTO functions by fixing their domains, because inverse of a function does not exist if functions is not ONE-ONE and ONTO functions.

   If you know what is ONE-ONE and ONTO functions then you can proceed to next otherwise it's better to go and read functions available on our blog.

NCERT Graph1

Graph of inverse trigonometric functions

  Graph of y = Cos^–1(x):-

NCERT GRAPH2

 Graph of inverse trigonometric functions

NCERT GRAPH3

 Graph of inverse trigonometric functions

NCERT GRAPH

 Graph of inverse trigonometric functions

NCERT Graph

 Graph of inverse trigonometric functions

NCERT Graph

inverse trigonometric functions examples :

1.         Find the principal value of sin^-1{1 }, cos^-1(0 ), cosec^-1(2).
2.         Find the value of tan^-1(1) + cos^-1(1 ) + sin^-1 (1/2 ).

inverse trigonometric functions formulas

      Properties of inverse functions:-

      1.      Relation between inverse functions:

i.          Sin^-1 x = cosec^-1 (1/x);  x ɛ (-∞  -1] U [1 ∞ )

ii.       Cos^-1 x = sec^-1 (1/x) ; x ɛ (-∞  -1] U [1 ∞ )

iii.      tan^-1 x = cot^-1 (1/x) ; x > 0

      2.      Negative values :

i.          Sin^-1( - x ) = – Sin^-1( x ), x ɛ [-1   1]

ii.       tan^-1( - x ) = – tan^-1( x ), x ɛ R

iii.     Cosec^-1( - x ) = – cosec^-1( x ), x ɛ [1   ∞)

iv.      Cos^-1( - x )   = π – Cos^-1( x ) ; x ɛ [-1   1]

v.        sec^-1( - x ) = π – sec^-1( x ) ; x ɛ [1   ∞)

vi.      cot^-1( - x ) = π  – cot^-1( x ) ; x ɛ R.

      3.      Inverse formulas : -

i.          Sin^-1( x ) + Cos^-1( x ) = π/2  ,  x ɛ [-1   1]

ii.       tan^-1( x ) + cot^-1( x ) = π/2,  x ɛ R

iii.     sec^-1( x ) + cosec^-1( x ) = π/2,  x ɛ [1  ∞]

iv.       tan^-1( x ) + tan^-1 (y ) = tan^-1[ (x+y)/(1 - xy) ] ;  xy < 1, if xy > 1 the π + tan^-1[(x+y)/(1 - xy)]

inverse trigonometric functions formulas are easy to remember, all formulas are derived from trigonometric formulas, just plug some trigonometric functions in places of x and y then you will realize how easy to remember. 

        inverse trigonometric functions examples :

       1.       Find the value of tan^-12 + tan^-13.

       2.       Find the value of tan^-1(1/2) + tan^-1 (1/5) + tan^-1(1/8).

       3.      Find the value of Sin^-1(2/3) + Sin^-1(8/17).

       4.      Find the value of x in Sin^-1(1 – x) – 2Sin^-1x = π/2 .