Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
INVERSE TRIGONOMETRIC FUNCTIONS
Inverse of a function:
• Inverse of a function f, is denoted by f
-1
, exists only if f is one-one and onto.
• Domain of f = Range of f
-1
and Range of f = domain of f
-1
, provided f
-1
remains one to one and onto.
• If (a, b) lies on f, then (b, a) lies on f
-1
• The graph of the inverse of a function is the reflection on the line y = x
Domain and range of Trigonometric Functions:
Function
Domain
Range
sin x
R
[-1, 1]
cos x
R
[-1, 1]
cosec x
R - n
(since, the denominator sin x is 0 for x = n)
R - (-1, 1)
sec x
R - (2n+1) /2
[since, the denominator cos x is 0 for x = (2n+1) /2]
R - (-1,1)
tan x
R - (2n+1) /2
[since, the denominator cos x is 0 for x = (2n+1) /2]
R
cot x
R – n
(since, the denominator sin x is 0 for x = n)
R
Domain and range of Inverse Trigonometric functions:
• Since, trigonometric functions are not one to one and onto, hence, by definition of a function, their
inverses shouldn’t exist, BUT, by restricting the domain, the existence of inverse can be ensured.
• The below table is obtained by swapping the domain and ranges of the corresponding original
trigonometric functions and restricting the domain of original function (or range of the inverse
function) such that the corresponding inverse is a function by definition (i.e. one to one and onto).
• The range below only includes the principal value branch.
Function
Domain
Range (Principal value branch)
sin
-1
x
[-1, 1]
[-/2, /2]
cos
-1
x
[-1, 1]
[0, ]
cosec
-1
x
R - (-1, 1)
[-/2, /2] – {0}
sec
-1
x
R - (-1,1)
[0, ] – {/2}
tan
-1
x
R
(-/2, /2)
cot
-1
x
R
(0, )
R – The Real number set
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
Properties of Inverse Trigonometric Functions (for suitable values of domains):
1. sin (sin
-1
x) = x and sin
-1
(sin x) = x
2. sin
-1
= cosec
-1
x
3. cos
-1
= sec
-1
x
4. tan
-1
= cot
-1
x
5. sin
-1
(-x) = - sin
-1
x
6. cosec
-1
(-x) = - cosec
-1
x
7. tan
-1
(-x) = - tan
-1
x
8. cos
-1
(-x) = - cos
-1
x
9. sec
-1
(-x) = - sec
-1
x
10. cot
-1
(-x) = - cot
-1
x
11. sin
-1
x + cos
-1
x = /2
12. tan
-1
x + cot
-1
x = /2
13. sec
-1
x + cosec
-1
x = /2
14. tan
-1
x + tan
-1
y = tan
-1
, xy < 1
15. tan
-1
x - tan
-1
y = tan
-1
, xy > -1
16. tan
-1
x + tan
-1
y = + tan
-1
, xy > -1; x, y > 0
17. 2 tan
-1
x = sin
-1
18. 2 tan
-1
x = cos
-1
19. 2 tan
-1
x = tan
-1
