Details of Domain and Range of Inverse Trigonometric Functions, Understanding the Concepts and Properties

1. Physics Helpline
L K Satapathy
Inverse Trigonometry 1

2. Inverse Trigonometry 1
Physics Helpline
L K Satapathy
Function Domain Range
( Principal value branch )
1
siny x

1
cosy x

1
cosecy x

1
secy x

1
tany x

1
coty x

[ 1,1] ,
2 2
  
  
[ 1,1] [0 , ]
( 1, 1)R   , {0}
2 2
   
  
( 1, 1)R    [0 , ]
2
 
R
R
 ,
2 2
 
(0 , )

3. 1 1
1) sin sin sin sinx x and x x    
     
Physics Helpline
L K Satapathy
1 1 1
sin (sin ) sin sin(sin ) sinx and x x    
    
1 1
2) cos cos cos cosx x and x x    
     
1 1 1
cos (cos ) cos cos(cos ) cosx and x x    
    
1 1
3) tan tan tan tanx x and x x    
     
1 1 1
tan (tan ) tan tan(tan ) tanx and x x    
    
Concepts
Inverse Trigonometry 1

4. Physics Helpline
L K Satapathy
 1 111) sin cosec x
x
 

 1 112) cos sec x
x
 

 1 113) tan cot x
x
 

 1 1 11 1cosec cosec sin sin cosecx x x
x x
     
       
 1 1 11 1sec sec cos cos secx x x
x x
     
       
 1 1 11 1cot cot tan tan cotx x x
x x
     
       
Properties
Inverse Trigonometry 1

5. Physics Helpline
L K Satapathy
1 1
1) sin ( ) sinx x 
  
1
sin sinLet x x  
  
Properties
1 1 1 1
sin ( ) sin ( sin ) sin [sin( )] sinx x     
         
1 1
2) cos ( ) cosx x 
  
1
cos cosLet x x  
  
1 1 1 1
cos ( ) cos ( cos ) cos [cos( )] cosx x        
         
1 1
3) tan ( ) tanx x 
  
1
tan tanLet x x  
  
1 1 1 1
tan ( ) tan ( tan ) tan [tan( )] tanx x     
         
Inverse Trigonometry 1

6. Physics Helpline
L K Satapathy
1 1
1) sin cos
2
x x  
  1
sin sinLet x x 
  
Properties
 1 1 1 1
cos cos (sin ) cos cos sin
2 2 2
x x                 
1 1
sin cos
2
x x  
  
1 1
2) tan cot
2
x x  
  1
tan tanLet x x 
  
 1 1 1 1
cot cot (tan ) cot cot tan
2 2 2
x x                 
1 1
tan cot
2
x x  
  
1 1
3) sec cosec
2
x x  
  1
sec secLet x x 
  
 1 1 1 1
cosec cosec (sec ) cosec cosec sec
2 2 2
x x                 
1 1
sec cosec
2
x x  
  
Inverse Trigonometry 1

7. Physics Helpline
L K Satapathy
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