INVERSE
TRIGONOMETRIC
FUNCTIONS
DEFINITION: If f is a one-to-one function with
domain A and range B, then its inverse f 1 is
the function with domain B and range A defined
by
f 1 x   y  f  y   x
For a function to have an inverse, it must be
one-to-one. Since the trigonometric functions
are not one-to-one, they do not have inverses.
It is possible, however, to restrict the domains
of the trigonometric functions in such a way
that the resulting functions are one-to-one.
Inverse trigonometric functions are defined as follows:
1
 y  sin x if and only if sin y  x and 
y

o

2

2
 y

2
, where  1  x  1.

1




x
  
D :  , 
 2 2
R :  1, 1
2
1
y = sin x
y

2
D :  1, 1
  
R :  , 
 2 2
x
1
1


2
y = sin-1 x
 y  cos 1 x if and only if cos y  x and 0  y   , where  1  x  1.
y
1


o

2


2

1
y = cos x
2
D : 0 ,  
R : - 1, 1 
y


2
D :  1, 1
R :  0,  
1
y = cos-1 x
x
1
x
 y  tan1 x if and only if tan y  x and 

2
 y

2
, for all real nos. x.
y
1


o

2


2
1
y = tan x
x
  
D :  ,

 2 2
R :   ,   
y

2
x
D :   ,   
   
R : , 
 2 2
1
1


2
y = tan-1 x
 y  cot 1 x if and only if cot y  x and 0  y   , for all real nos. x.
y
1


o

2

x

D : 0 ,  
2
R :   ,   
1
y = cot x
y


2
D :   ,   
R : 0 ,  
1
y = cot-1 x
x
1
 y  sec 1 x if and only if sec y  x and 0  y   , y 

2
, x  -1, x  1.
y
1
o
1

3
2


2
y = sec x
x
2
   

D : 0 ,    ,  
 2 2

R :   , - 1  1,   
y


2
D :   , - 1  1,   
   

R : 0 ,    ,  
 2 2

1
y = sec-1 x
x
1
 y  csc 1 x if and only if csc y  x and 

2
 y

2
, y  0 , x  -1, x  1.
y

1


o

2


1
2
y = csc x
x
 
  
D :  , 0    0 , 
 2
  2
R :   , - 1  1,   
y

2
x
D :   , - 1  1,   

 
 
R :  , 0    0 , 
2
 2
 
1
1


2
y = csc-1 x
• In order to express θ as a function of x, we write:
θ = arc sin x or θ = sin-1 x
θ = arc cos x or θ = cos-1 x
θ = arc tan x or θ = tan-1 x
θ = arc csc x or θ = csc-1 x
θ = arc sec x or θ = sec-1 x
θ = arc cot x or θ = cot-1 x
• sin-1 x, cos-1 x, tan-1 x are read as “inverse sine of
x”, “inverse cosine of x”, “inverse tangent of x”.
EXAMPLES:
I. Find the exact values of the following:
3
1. Arc cos
2
 2 

2. Arc sin 

2


 3

3. Arc cos

2


Ans. 1.

6
Ans. 2. -

4
5
Ans. 3.
6

3
4. Arc sin
2
Ans. 4.
 1
5. Arc cos  
 2
2
Ans. 5.
3
3

2


6. Sin  

2


-1
 1
7. Sin   
 2
-1
-1
 
8. Cot - 3
9. Arcsec 2
Ans. 6. Ans. 7. -

4

6
5
Ans. 8.
6
Ans. 9.

3
10. Arctan 2.253
Ans. 10. 1.153 rad.
2 

11. Cos  cos

3 

2
Ans. 11.
3
-1

2 

12. Tan  tan

3 

Ans. 12. -
   
13. Sin  tan -  
  4 
Ans. 13. -

 3 

14. tan  Arcsin  

2



Ans. 14. - 3
15. tan Arc tan 0.6 
Ans. 15. 0.6
-1
-1
3

2