Function Domain Range

INVERSE TRIGONOMETRIC
Sin 1
x 1,1    
 2 , 2 
FUNCTION FORMULAE
Cos 1 x 1,1 0, 
Tan 1x R    
 2 ,2 
 
Cot 1 x R 0, 
1
Sec x (  , 1]  [1, )     
 0, 2    2 ,  
   
 Inverse of a function : Co sec1 x ( , 1]  [1, )     
 2 , 0    0, 2 
   
f : A  B is bijective  f 1 : B  A exists and
it is also bijective. All trigonometric functions
are not bijective functions. By restricting the
domains of the functions,we make them bijective
 Graphs of inverse circular functions:
  
 The function f :   ,    1,1 defined by 1.  
y = Sin 1 x, | x |  1, y   2 ,

 2 2  2


f  x   sin x is a bijection.

  
Then f 1 :  1,1    ,  is also a
 2 2
bijection. This function is called inverse sine
function and it is denoted by Arc sine x or
S in 1 x

 The function f :  0,     1,1 defined by

2. y = Cos 1 x , | x |  1, y  [0, ]
f  x   cos x is a bijection.

Then f 1 :  1,1   0,   is also a bijection.

This function is called inverse cosine function
and it is denoted by Arc cos x or C os1 x

  
 The function f :   ,   R defined by
 2 2
f  x   tan x is a bijection.

  
Then f 1 : R    ,  is also a bijection.
 2 2
This function is called inverse tangent function
  
and it is denoted by Arc tan x or Tan 1 x 3. y = Tan 1 x , x  R, y    2 ,
2


 Domains and Ranges of Inverse

trigonometric functions:
 Properties of inverse trigonometric
functions:
1 1
i) Sin x  C os ec , x  1,1 , x  0
1

x
1 1
1
ii) Cos x  Sec , x  1,1 , x  0
x
4. y = Cot 1
x , x  R, y  (0 , )
 1 1
 C ot x , x  0
iii) Tan x  
1

    C ot  1 1 ,  x  0
 x
 Some useful periodic graphs:

 3 
   x,  2 x
2

 x,  
 x
 2 2
1. y  Sin 1  sin x   
  x,  3
x
 2 2
 3 5
 2  x, x
 2 2

       
5. y = Sec 1 x,x 1, y  0 , 2    2 ,

 y is Periodic with period 2and y   ,
 2 2 

 x,   x  0
6. 1
y=Cosec x,x1,
    
y   , 0   0 ,   x, 0 x

 2   2
2. y  Cos 1  cos x   
2  x,   x  2
2  x, 2  x  3

y is periodic with period 2 and y   0,  

  3       
x   ,  2 x
2 x, x 0,   , 
   2 2 
 x,  
 x   3   3 
 6. y  Sec  sec x  2  x, x  ,   ,2 
1
3. y  Tan  1  tan x    2 2
   x ,  3   2 2 
x
 2 2  and so on
 3 5 
 x  2 , x 
 2 2

y is periodic with period 2 and y   0,  

   
y is periodic with period and y   , 
 2 2

 Some useful non-periodic graphs:

4. y = Cot1(Cot x) = x, x(0,) and so on. 1. y = Sin(Sin 1 x) = x, x[ 1 , 1], y[ 1 , 1]

y is periodic with period and y   0,  

1
-1
0 1

-1

2. y = Cos(Cos 1x) = x, x  [ 1 , 1], y[ 1 , 1]

     
x, x   ,0    0, 
  2   2

    3  1
5. y  Co sec  cos ecx     x, x   ,   , 
1

 2   2  -1
 and so on
 0 1

-1
   
y is periodic with period 2and y   , 
 2 2
3. y = Tan (Tan–1x) = x, x  R, y  R
 v) C o t -1  -x  = π -C o t -1 x ,  x  R
vi) Sec -1  -x  =π-Sec -1 x ,  x  (, 1]  1, )

1  1  1  1  1  1 
1: Cos    2Sin    3Cos  
 2  2  2
4Tan 1  1 
4. y = Cot (Cot–1x) = x, x  R, y  R 1  1  1  1  1  1 
Sol: Cos    2Sin    3Cos  
 2  2  2

1  1   
 4Tan1  1    Cos    2  
2  6 
  1 
 3    Cos 1     4Tan 1
1

  2 
      
    3    4  
3 3  4 4
5. y  cos ec cos ec1 x  x , | x |  1, | y |  1,
  3  43
  3   
3  4  12

i)    0,   then Sin  cos    
1
1 
2
-1
   
0 1 ii)     ,  then Cos 1  sin     
 2 2 2
-1 
iii)    0,   then Tan  cot    
1

6. y = Sec(Sec–1x) = x, | x |  1, | y |  1    
iv)     ,  then Cot 1  tan     
 2 2 2

     
450 v)     , 0    0,  then
1  2   2

-1 Sec 1  cos ec   
0 1 2

-1     
vi)    0,    ,   then
 2 2 

 Important Results: Co sec 1  sec    
2
i) S in -1  -x  = -Sin -1 x ,  x   1,1

ii) C o s -1
 -x = π -C os -1
x ,  x   1,1  Cos 1 1  x 2 if 0  x  1

 i) S in 1 x    Cos 1 1  x 2 if  1  x  0
iii) T a n -1  -x  = -T an -1 x ,  x  R
 x
iv) Cosec -1  - x  =-Cosec -1 x ,  x  R   1,1  Tan 1 if x   1,1
 1  x2
  S in 1 1  x 2 if x   0,1

ii) C os 1 x  
   S in 1 1  x 2 if x   1, 0 

Sin1 x 1 y2  y 1 x2
 
0  x, y  1 and x2  y2  1 or
 x 
 S in
1
for x  0 1  x, y  1, xy  0 and x2  y2 1
 1  x2 
 
1
iii) T an x   1

Sin1x  Sin1 y   Sin1 x 1 y2  y 1 x2
C os 1 for x  0
 1  x2 
if 0  x  1, 1  y  1and x2  y2  1

2: The value of x , where x  0 and

 Sin1 x 1 y2  y 1 x2


 1
Tan  Sec 1   Sin Tan 1 2  is (EAM-2007) if 0<y  1,-1  x<0 and x2  y2  1
 x


Sol: Tan  Sec

1 1 

x
  Sin Tan 2 
1
 
C os1 xy  1 x2 1  y 2

if 1  x, y  1and x  y  0

C os1 x  C os1 y  

Tan  Tan 1

1  x2
x
 
  Sin  Sin 1

2
1 2 
2

 

2  C os1 xy  1  x2 1  y 2 
  
if 1 x, y  1and x  y  0
 x 
  T an x  S in
1 1


 
 1  x2 
C os1 xy  1 x2 1  y2

1 x 2
2 if 1  x, y  1, x  y
  5 1  x 2   4 x 2 1 1 
x C os x  C os y  
 
5
C os1 xy  1 x2 1 y2

5 5 if 1 y  0,0  x  1and x  y
 x2  x
9 3
1
 3: If  x  1 then
1 1 2
  i) S in x  C os x  ,  x  [-1,1]
2
x 1 
ii) T an 1 x  C ot 1 x   / 2 ,  x  R Cos 1   3  3x 2   Cos 1 x is equal to
2 2 

iii) S ec x  C os ec x  ,  x(, 1] 1, )
1 1
1 1  x 1 2 
2 Sol: Cos x  Cos  2  2 3  3 x 
 

 
Sin1 x 1  y2  y 1  x2
  1
 Cos  1 x  Cos  1  x . 
3 
1  x 2 
  2 2 
0  x, y , x  y  1 or
2 2

1  x, y  1, xy  0 and x2  y2  1 1 1 

 Cos 1 x  Cos 1    Cos 1 x  Cos 1   =
 2 2 3


Sin1x  Sin1 y    Sin1 x 1  y 2  y 1  x2


if 0  x, y  1 and x2  y2  1


  Sin1 x 1  y2  y 1  x2


if -1  x,y<0 and x2  y 2  1
  17   17  6
 1  x  y   cot cot 1    cot tan 1   
 6  6  17
Tan  1  xy  if x  0, y  0, xy  1
  
 Transformation of Inverse functions by
 1  x  y 
   Tan   if x  0, y  0, xy  1 elementry substitution and their graphs:
1 1   1  xy 
Tan x  Tan y  
  Tan1  x  y  if x  0, y  0, xy  1   2 Tan 1 x x  1
   2x  1
 1  xy  i) Sin–1 = 2Tan x  1 x  1
 1 x2   2Tan 1 x
 , if xy  1  x 1
 2

 1  x  y 
Tan   for xy 1
  1 xy 
  x y 
Tan1x Tan1y   T an1   if x  0, y  0and xy 1
 1 xy 
  x y 
 T an1   if x  0, y  0and xy 1
 1 xy   2 Tan 1 x
1 x2 x 0
 (i) If x, y , z have same sign and ii) Cos –1
=  
  2 Tan x x  0
1
1 x2
xy  yz  zx  1 then

 x  y  z  xyz 
Tan 1 x  Tan 1 y  Tan 1 z  Tan 1  
 1  xy  yz  zx 

(ii) T an 1 x1  T an 1 x 2  ......T an 1 x n 

 S  S  S  .... 
T an 1  1 3 5 
 1  S2  S4  S6 .... 

where S1 = sum of values , S 2 = sum of product   2 Tan 1 x x  1


of taken two elements at a time and so on.., 2x 1
2 Tan x  1 x  1
iii) Tan–1 = 
1 x2
   2 Tan x
1
S n = product of values. x 1

 1 1 
5 2
4: The value of Cot  Co sec 3  Tan 3  
 

 1 5 1 2   1 3 1 2 
Sol: Cot  Co sec 3  Tan 3   Cot  Sin  Tan 
   5 3

 3 2  x 
 Cot  Tan1  Tan1   Sin x  Tan
1 1

 4 3  1  x2 

  3 2    17      3Sin1x if 1  x  1/ 2
       
 Cot  Tan1  4 3   Cot  Tan1  12  i) Sin1  3x  4x3   3Sin 1x if 1/ 2  x  1/2
  1   3  2    1
      3Sin1x
  
   4  3     2   if 1/ 2  x  1
 C os 1  2 x 2  1 if 0  x  1
ii) 2C os1 x  
2  C os
1
 2x 2
 1 if  1  x  0

-3/2 -½  1  2x 
Tan  1  x 2  if  1  x  1
½ 3/2  

1  1  2x 
iii) 2Tan x     Tan  1  x 2  if x  1

 1  2x 
3cos1 x  2 if 1 x  1/2     Tan  2 
if x   1
 1 x 

ii) Cos1  4x3 3x  2 3Cos 1 x if 1/2  x 1/ 2 ;
3C os1 x if 1/2  x 1  1  2x 
    Sin  1  x 2  if x   1
  
1  1  2 x 
iv) 2Tan x   Sin  2 
if  1  x  1
  1  x 
 1  2x 
  Sin  2 
if x  1
 1 x 

 1  1  x 
2

  C os   if    x  0
  1  x2 
½ v) 2 T an x  
1
-3/2 -½ 3/2
 C os  1  1  x  if 0  x  
2

  2 
 1 x 
 1 1 1
3Tan x if -  x  1 4 1
 3 3 5: Sin  2Tan 1 
 3x  x3
  1 5 3
iii) Tan1  2 
   3Tan1 x if x 
 1  3x   3 1 4 1
Sol: Sin  2Tan 1
 1 5 3
  3Tan x if x  
1

 3
 1 
 2    2Tan1x  
1 4 1   3   
 Sin  Tan  1  2 x 
5   1 2  Tan  2 
/2  1       1  x 
  3 
-3 -1/3 1/3 3 4 3
 Sin 1  Tan 1
5 4
-/2 
4 4 1  1 
 Sin 1  Cos 1  Tan x  Cos 
1

5 5   1 x
2
 
 1
 
  Sin 2 x 1  x if  1  x  
2 1
2 
  
 Sin x  Cos x  2 
1 1

 2
1  1


i) 2Sin x  Sin 2 x 1  x
2
if 
1
2
 x
1
2  Some important facts:


1

  Sin 2 x 1  x
2
if
1

2
 x 1
1 1 1 
1) Tan x  Tan y  Tan z  , if xy  yz  zx  1
2
2) Tan x  Tan y  Tan z   ,
1 1 1

if x  y  z  xyz
 1 a b    
3) Tan  Tan1  ,then x  ab i) a  x , put x  a sin  ,    , 
2 2
x x 2  2 2
a b 
4) Sin
1
 Sin 1  , then x  a 2  b2   
ii) a  x , put x  a tan  ,    , 
2 2
x x 2
 2 2
1  p  1  q  p  
5) Tan    Tan   iii) x 2  a 2 , put x  a sec ,   0,  
q q p 4
 Range of some special inverse
1  1  x  
6) T an 
1
   tan x if x  1 Trigonometric Functions :
 1  x  4
3 7 3
  Sin x    Cos x  
1 3 1 3

1  1  x   i)
7) T an 
1
   tan x if x  1 32 8
1 x  4
2 5 2
  Sin 1 x    Cos1 x  
2 2
 ii)
1 1
8) If Tan x  Tan y  then xy=1. 8 4
2
2 3 2
  Cos x   Sin x  
2 2
1 1  iii) 
1 1

9) Cot x  Cot y  then xy=1 4 4

2
6 : The set of values of x such that Sin1 x Cos1 x  0
10) If Cos1 x  Cos1 y  Cos1 z  3
are
then xy  yx  zx  3

3
Sol: Sin 1 x  Cos1 x  0  Sin 1 x 
1 1 1
11) If Sin x  Sin y  Sin z  4
2
 1 
then xy  yx  zx  3  x ,1 as x  1
 2 
12) If Sin1 xSin1 y  then Cos1 xCos1 y 
7: The sum to the n terms of the series
13) If Cos1 x  Cos1 y  then Sin1 x Sin1 y  
Co sec 1 10  Co sec1 50  Co sec 1 70  .....
1 1
14) If aSin x  bC os x  c then
 ab  c  a  b
...  Co sec1 n 2
 1 n2  2n  2  is
a Sin1 x  b Cos1 x 
a b Sol: Let y  Co sec  1 n 2
 1 n 2  2 n  2 
1 x 1 y
15) If Cos  Cos
a b
  then Co sec 2 y   n 2  1  n 2  1  2n  1

  n 2  1  2n  n 2  1  n 2  1
2
x 2 2 xy y2
2
 cos   2  Sin2
a ab b
  n2  n 1 1
2

x y
cot 2 y   n 2  n  1
2
16) If Sin
1
 Sin 1   then  cos ec 2  cot 2   1
a b

x 2 2 xy y2 Tany 
1

 n  1  n
 cos    Sin 2 n  n  1 1   n  1 n
2
a 2 ab b2
1  1  1  1    n  1  n 
17) Tan  1 x( x  1)   Tan  1 ( x  1)( x  2)   ....  y  Tan 1  
1   n  1 n 
   

 1 
Tan1   Tan  xn Tan x,nN
1 1
y  Tan 1  n  1  Tan 1n
1 xn1 xn  Thus sum of n terms of the given series
 If an expression contains
y  T a n  1 2  T an  11    T a n  1 3  T a n  1 2  

T a n 1
4  T an  1 3   ......  T an  1  n  1   T a n  1 n 


 Tan1  n  1  Tan11  Tan  n  1 
1

4
8: Sin 1  sin 5  
Sol: Here   5 rad , Clearly it does not lie between
 
and . But 2  5 and 5  2 both lies
2 2
 
between and .
2 2

Sin  5  2   Sin   2  5   Sin  2  5  Sin5

 Sin 1  Sin5   Sin1  Sin  5  2    5  2

9 : Cos 1  cos10  is equal to....

Sol: We know that Cos 1  cos     if 0    

Here   10 rad , clearly, it does not lie between
0 and  . But 4  10 lies between 0 and 
 Cos 1  cos10   Cos 1  cos  4  10    4  10

10: Tan1 Tan  6   is equal to.....

 
Sol: We know that Tan 1 Tan    if  
2 2
Here   6 rad , does not lie between
 
and
2 2
 
But 2  6 lies between and
2 2
Now Tan  2  6   Tan6  Tan  6 

Tan 1 Tan  6    Tan 1 Tan  2  6    2  6