Download Inverse Trigonometric Functions 1. Find the principal values of the

Inverse Trigonometric Functions
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1.
Find the principal values of the following
(i)
1
Sin -1  



 6 
(ii)
Tan -1 ( -3 )



 3 
2



 4 
Tan -1 ( - )
1
Sec -1 ( - )
 2 


 3 
(i)
3
If Tan -1   = find Sin
3
 
5
(ii)
1
If Cot -1   = find Cos
 1 


5 2 
(iii)
4
 
5
(ii)
 17 
 
 6 
(iv)
4
7
 1 


 3
2
Evaluate
(i)
3
5
Sin ( Cos -1 )
Cos ( Tan -1
(iii)
4
 
5
3
)
4
4
2
Tan  Cos 1  Tan 1 

3
5
3
 
5
1
3
2 Tan -1 = Tan-1x, x =
(v)
4.
Cosec -
(v)
1
If Sin -1   = Tan -1x, find x
3.
(iv)



 6 
( - )
2.
(iii)
 1 


 10 
1
4
Sin  Cos 1 
2
5
Prove that
1
a)
Sin -1
b)
2 Tan -1
5
+ Sin -1
1
= Sin
x
2
5
-1
=

2
 2x 
 2

 x 1 
5.
6.
c)
 x
Cosx 
Tan -1 
=
d)
Sin -1
e)
ab 
 bc 
 ca 
Tan -1 
 + Tan -1 
 + Tan -1 
 =0
 1  Sinx 
4
2

2 
-1
-1
 x 1  x  x 1  x  = Sin x - Sin
 1  ab 
 1  ab 
x
 1  ca 
Solve for x
2x
Tan -1
b)
Tan -1 (x + 1) + Tan -1 (x-1) = Tan -1
1 x2
Prove that if Sin -1
then
x=
+ Cot -1
1 x2

= , (x > 0)
3
2x
a)
2a
1 a
ab
.
1  ab
2
+ Sin -1
2b
1 b2
8
13
, (x > 0)
2  3 
1
 
4
= 2 Tan -1 x
7.
If two angles of a triangle are Tan -1 2 and Tan -1 3, prove that the third angle is
8.
Prove that Sec2 (Tan -1 2) + Cosec 2 (Cot -1 3) = 15.
9.
Solve :
Sin -1 x + Sin -1 ( 1-x ) = Cos -1 x
( 0, ½ ).
10.
Simplify :
a) Sec -1 


 2 x 1 
1
(2 Cos -1 x)
2

x
 1 x
b) Sin -1 




( Tan -1 x )
c) Cot -1  1  x 2  x 



1

Tan 2 Tan 1    
5
4
11.
Evaluate:
12.
If Sin -1x - Cos -1x =
13.
1
1
2 Tan -1   + Tan -1   is _________
14.
Evaluate : Sin 2 Cos 1 
 

3



6

, find x
 1
1 
  Tan x 
4 2

7


 17 
 3


 2 


7

 
4
 3 

 5 
  24 


 25 

.
4
15.
Prove that
 2 
1
2
Tan -1   + Tan -1   = Cos -1  
16.
Prove that
Tan -1 
17.
Prove that
Tan -1 
18.
Simplify
Sin -1 
4
9
 5
 1  Cosx  1  Cosx   x
= + , 0 < x < 

2
 1  Cosx  1  Cosx  4 2
 1 x2  1 x2 
 =  + 1 Cos-1 x2, -1 < x < 1.
2
2 
4 2
 1 x  1 x 
 Sinx  Cosx 

2


,


<x<
4
4


x  
4

 Sinx  Cosx  

 =  x   .
4

2


Cos -1 
19.
Prove that
20.
Solve the equation : 2 Tan -1 (Cos x) = Tan -1 (2 Cosec x)

 
4
21.
Write in simplest form
a) Tan -1  x  1  x 2  , x  R


 1
1 
  Cot x 
2 2

b) Tan -1  1  x 2  x  , x  R
1
1 
 Cot x 
2

 1  x 2 1 
, x  0
c) Tan -1 

1
1 
 Tan x 
2




x

 1 x 1
 , x 0
d) Tan -1 

x
2


 ax
 ax

 , -a < x < a


e) Tan -1 


,

2
2 
a a x 
f) Tan -1 
x
 x  1 x2
g) Sin -1 

2
-a < x < a

,




< x < Hint: (let x = sin α)
4
4
 1 x  1 x 
 , 0 < x < 1 Hint: (let x = cos 2α)
h) Sin -1 


2

1  x 
i) Sin  2 Tan 1
,


1 x 

 1
1 
  Tan x 
2 2

1
1 x 
 Cos

a
2
1
1 x 
 Sin

a
2

1 
  Sin x 
4

 1
1 
  Cos x 
4 2

 1 x2 




j) Cot -1 

a

,

2
2 
x

a


k) Tan -1 

x

x

,
 2
2 
 a x 

1 x 
 Sec

a

 x > a

1 x 
 Sin

a

-a < x < a



2
2 
 x a 
l) Sin -1 
22.

1 x 
 Tan

a

Solve for x
a) Cos -1 x + Sin -1
x

=
6
2
(1)
 1 
1 x  1

b) Tan -1 
  Tan -1 x = 0 , x > 0 Hint: (let x = cos α) 
1 x  2
23.
Prove that
Tan -1
24.
 3
x
xy


- Tan -1
xy
4
y
The value of Cos -1  Cos

13  

6 
is ----------

 
6

 1 
Evaluate: sin   sin 1    (Ans:√3/2 )
(CBSE 2008)
 2 
3

26. Solve for x: tan 1 (2 x)  tan 1 3 x  
[ Ans : 1/6 , -1] (CBSE 2008, 2009)
4
1
1
1
1 
27.
Prove tan 1  tan 1  tan 1  tan 1 
( CBSE 2008)
3
5
7
8 4

 x 1 
1  x  1 
28.
Solve for x: tan 1 
  tan 
  [ Ans : +/-(1/√2)]
 x 2
 x  2 4
2008)
3
2
) [ Ans :
29.
Using principal value, evaluate sin 1 (sin
] (CBSE 2009)
5
5
25.
30.
What is the principal value of
 1
 1
sin 1    cos 1   (CBSE 2010)
 2 
 2 
[ Ans :

]
2
(CBSE
31.
32.
33.
34.
35.


3
 ? [ Ans : - ]
What is the principal value of sin 1  
(CBSE 2010)

3
 2 
1
1 x 
Prove tan 1 x  cos 1 
,.....x  0,1 (CBSE 2010)
2
1 x 
Prove that tan 1 1  tan 1 2  tan 1 (3)  
(CBSE 2010)

 x 1 
1  x  1 
If tan 1 
  tan 
  , find the value of x. (CBSE 2010)
 x 2
 x  2 4
(Same as Q28)
 12 
3
 56 
Prove cos 1    sin 1    sin 1   (CBSE 2010)
 13 
5
 65 

36. Write the value of sin   sin1
3
 1 
  2   (CBSE


2011)
37.Prove the following:
 1  sin x 
cot-1 
 1  sin x 
1  sin x 
x
 
  , x   0,  (CBSE 2011)
2
1  sin x 
 4
x
xy

)(CBSE 2011)
 (Sol:
4
xy
38.Find the value of tan-1   - tan-1 
y