# RD Sharma Solution CLass 12 Mathematics Chapter 4 Inverse Trigonometric Functions

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.1

Question 1

Find the value of each of the following :

Solution 1

Question 2

Find the value of each of the following :

Solution 2

Question 3

Find the value of each of the following :

Solution 3

Question 4

Find the value of each of the following :

Solution 4

Question 5

Find the value of each of the following :

Solution 5

Question 6

Find the value of each of the following :

Solution 6

Question 7

Find the value of each of the following :

Solution 7

Question 8

Find the value of each of the following :

Solution 8

Question 9

Find the domain of each of the following functions:

F(x) = sin-1x2

Solution 9

Question 10

Find the domain of each of the following functions:

F(x) = sin-1x + sin x

Solution 10

Question 11

Find the domain of each of the following functions:

Solution 11

Question 12

Find the domain of each of the following functions:

f(x) = sin-1x + sin-12x

Solution 12

Question 13

If sin-1x + sin-1y + sin-1z
+ sin-1t = 2
π, then find the
value x2 + y2 + z2 + t2.

Solution 13

Question 14

Solution 14

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.10

Question 1

Evaluate:

Solution 1

Question 2

Evaluate:

Solution 2

Question 3

Evaluate:

Solution 3

Question 4

Evaluate:

Solution 4

Question 5

Evaluate:

Solution 5

Question 6

Solution 6

[ ∏/2 – sin-1
x ] + [ ∏/2 – sin-1y ] = ∏/4

sin-1x
+ sin-1y = ∏ – ∏/4

sin-1x
+ sin-1y = 3∏/4

Question 7

Solution 7

the equation

Π/2 + sin-1y –
cos-1y = Π/2

[ Π/2-
cos-1y ] – cos-1y = 0

cos-1y
=
Π/4

y = 1/√2

on putting y=1/√2
in 2nd equation

cos-1x
Π/4
= Π/6

cos-1x
=
Π/4
+ Π/6

x = cos(Π/4 + Π/6)

x = cos(Π/4)cos(Π/6)-sin(Π/4)sin(Π/6)

x = (√3-1)/2√2

Question 8

Solution 8

cot(z)
= 0 means z = Π/2, 3Π/2,
5Π/2
………..

cos-1(3/5)
+ sin-1x = +
Π/2

sin-1x
= +
Π/2
– cos-1(3/5)

sin-1x
= + sin-1(3/5)

x = sin(
+ sin-1(3/5)) = (-1)n sin (sin-1(3/5))

x = (-1)n
3/5

Question 9

Solution 9

[ Π/2
– cos-1x ]2 + (cos-1x)2 =17Π2/36

Π2/4 – Πcos-1x
+ 2(cos-1x)2 =17Π2/36

Let, cos-1x=u

2u2
Πu + Π2/4
– 17Π2/36
= 0

2u2
Πu – 2Π2/9
= 0

18u2
– 9
Πu
-2Π2
= 0

On factorizing

18u2
– 12
Πu
+ 3Πu
-2Π2
= 0

6u( 3u -2Π
) + Π(
3u -2Π
) = 0

( 3u -2Π
)(6u + Π)
= 0

u = –Π/6,
2Π/3

i.e. cos-1x
= –
Π/6,
2Π/3

but range of cos-1x
is [0, π]

x = cos(Π/2 + Π/6)

x = -1/2

Question 10

Solve:

Solution 10

sin-1(1/5)
+ [
Π/2
– sin-1x ] = sin-11

sin-1(1/5)
+
Π/2
– sin-1x = Π/2

sin-1(1/5)
– sin-1x = 0

x = 1/5

Question 11

Solve:

Solution 11

Π/2 – cos-1x = Π/6
+ cos-1x

Π/3 = 2cos-1x

cos-1x
=
Π/6

x = √3/2

Question 12

Solve:

4 sin-1x = Π
– cos-1x

Solution 12

4sin-1x+cos-1x=Π

3sin-1x+sin-1x+cos-1x=Π

3sin-1x=Π/2 [sin-1x+cos-1x=Π/2]

sin-1x=Π/6

x = sinΠ/6=0.5

Question 13

Solve:

Solution 13

tan-1x+cot-1x=Π/2
so the above equation reduces to

cot-1x
=2
Π/3-Π/2
=Π/6

x= cotΠ/6
=√3

Question 14

Solve:

5 tan-1x + 3 cot-1x = 2Π

Solution 14

2tan-1x+3(Π/2)=2Π

2tan-1x=2Π-3Π/2=Π/3

tan-1x=Π/6

x=tanΠ/6=1/√3

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.11

Question 1

Solution 1

Question 2

Solution 2

Question 3

Prove the following result:

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solve the following equations for x :

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solve the following equations for x:

Solution 12

Question 13

Solve the following equations for x:

Solution 13

Question 14

Sum the following series:

Solution 14

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.12

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solve the following:

Solution 5

Question 6

Solve the following :

Solution 6

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.13

Question 1

Solution 1

Question 2

Solve the equation:

Solution 2

Question 3

Solve :

Solution 3

Question 4

Prove that:

Solution 4

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.14

Question 1

Solution 1

Question 2

Evaluate the following

Solution 2

Question 3

Solution 3

Question 4

Evaluate the following:

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Prove the following result :

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Thus, the solution is

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.2

Question 1

Find the domain of definition of f(x) = cos-1 (x2
– 4).

Solution 1

Question 2

Find the domain of f(x) = 2cos-12x + sin-1 x.

Solution 2

Question 3

Find the domain of f(x) = cos-1x + cos
x.

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Find the principal value of each of the following :

Solution 6

Question 7

Find the principal value of each of the following :

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

?

Question 11

Solution 11

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.3

Question 1

Solution 1

Question 2

Find the principal value of each of the following:

Solution 2

Question 3

Find the principal value of each of the following:

Solution 3

Question 4

Find the principal value of each of the following:

Solution 4

Question 5

Solution 5

Question 6

For the principal values, evaluate each of the following :

Solution 6

Question 7

Solution 7

Question 8

Evaluate each of the following:

Solution 8

Question 9

Evaluate each of the following:

Solution 9

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Find the principal value of each of the following:

Solution 3

Question 4

Find the principal value of each of the following:

Solution 4

Question 5

Solution 5

Question 6

Find the principal value of each of the following:

Solution 6

Question 7

Find the domain of

sec-1 (3x – 1)

Solution 7

Question 8

Find the domain of

sec-1 x – tan-1x

Solution 8

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.5

Question 1

Solution 1

Question 2

Find the principal value of each of the following:

cosec-1 (-2)

Solution 2

Question 3

Solution 3

Question 4

Find the principal value of each of the following:

Solution 4

Question 5

Find the set of values of

Solution 5

Question 6

For the principal value evaluate the following:

Solution 6

Question 7

For the principal value evaluate the following:

Solution 7

Question 8

For the principal value evaluate the following:

Solution 8

Question 9

For the principal value evaluate the following:

Solution 9

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.6

Question 1

Solution 1

Question 2

Find the principal value of each of the following:

Solution 2

Question 3

Find the principal value of each of the following:

Solution 3

Question 4

Find the principal value of each of the following:

Solution 4

Question 5

Find the domain of f(x) = cot x + cot-1 x.

Solution 5

Question 6

Evaluate each of the following:

Solution 6

Question 7

Evaluate each of the following:

Solution 7

Question 8

Evaluate each of the following:

Solution 8

Question 9

Evaluate each of the following:

Solution 9

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.7

Question 1

Solution 1

Question 2

Evaluate the following :

Solution 2

Question 3

Evaluate the following :

Solution 3

Question 4

Evaluate the following :

Solution 4

Question 5

Evaluate the following :

Solution 5

Question 6

Evaluate the following :

Solution 6

Question 7

Evaluate the following :

sin-1(sin 3)

Solution 7

Question 8

Evaluate the following :

sin-1 (sin 4)

Solution 8

Question 9

Evaluate the following :

sin-1 (sin 12)

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Evaluate the following :

cos-1 (cos 3)

Solution 13

Question 14

Evaluate the following :

cos-1 (cos 4)

Solution 14

Question 15

Evaluate the following :

cos-1 (cos 5)

Solution 15

Question 16

Evaluate the following :

cos-1 (cos 12)

Solution 16

Question 17

Evaluate the following :

Solution 17

Question 18

Evaluate the following :

Solution 18

Question 19

Solution 19

Question 20

Evaluate the following :

Solution 20

Question 21

Evaluate the following :

tan-1 (tan 1)

Solution 21

Question 22

Evaluate the following :

tan-1 (tan 2)

Solution 22

Question 23

Evaluate the following :

tan-1 (tan 4)

Solution 23

Question 24

Evaluate the following :

tan-1 (tan 12)

Solution 24

Question 25

Evaluate the following :

Solution 25

Question 26

Evaluate the following :

Solution 26

Question 27

Evaluate the following :

Solution 27

Question 28

Evaluate the following :

Solution 28

Question 29

Evaluate the following :

Solution 29

Question 30

Evaluate the following :

Solution 30

Question 31

Evaluate the following :

Solution 31

Question 32

Evaluate the following :

Solution 32

Question 33

Evaluate the following :

Solution 33

Question 34

Evaluate the following :

Solution 34

Question 35

Evaluate the following :

Solution 35

Question 36

Evaluate the following :

Solution 36

Question 37

Evaluate the following :

Solution 37

Question 38

Evaluate the following :

Solution 38

Question 39

Evaluate the following :

Solution 39

Question 40

Evaluate the following :

Solution 40

Question 41

Evaluate the following :

Solution 41

Question 42

Evaluate the following :

Solution 42

Question 43

Evaluate the following :

Solution 43

Question 44

Evaluate the following :

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Write each of the following in the simplest form:

Solution 52

Question 53

Solution 53

Question 54

Solution 54

## Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.8

Question 1

Evaluate the following:

Solution 1

Question 2

Evaluate the following:

Solution 2

Question 3

Evaluate the following:

Solution 3

Question 4

Evaluate the following:

Solution 4

Question 5

Evaluate the following:

Solution 5

Question 6

Evaluate the following:

Solution 6

Question 7

Solution 7

Question 8

Evaluate the following:

Solution 8

Question 9

Evaluate the following:

Solution 9

Question 10

Prove the following result:

Solution 10

Question 11

Prove the following result:

Solution 11

Question 12

Evaluate the following:

Solution 12

Question 13

Evaluate the following:

Solution 13

Question 14

Solve:

Solution 14

Question 15

Solve:

Solution 15

Question 1

Evaluate:

Solution 1

Question 2

Evaluate:

Solution 2

Question 3

Evaluate:

Solution 3

Question 4

Evaluate:

Solution 4

Question 5

Evaluate:

Solution 5

Question 6

Evaluate:

Solution 6

Question 7

Evaluate:

Solution 7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

## Chapter 4 – Inverse Trigonometric Functions Exercise MCQ

Question 1

a. sin 2α

b. sin α

c. cos

d. cos α

Solution 1

Correct option: (a)

Question 2

a.

b.

c.

d.

Solution 2

Correct option: (d)

Question 3

2 tan-1 {cosec(tan-1x) – tan (cot-1x)} is equal to

a. cot-1x

b. cot-1

c. tan-1x

d. None of these

Solution 3

Correct option: (c)

Question 4

a. sin2α

b. cos2α

c. tan2α

d. cot2α

Solution 4

Correct option: (a)

Question 5

The positive integral solution of the equation

a. x = 1, y = 2

b. x = 2, y = 1

c. x = 3, y = 2

d. x = -2, y = -1

Solution 5

Correct option: (a)

Question 6

a.

b.

c.

d. None of these

Solution 6

Correct option: (b)

Question 7

a. x

b.

c.

d. None of these

Solution 7

Correct option: (a)

Question 8

The number of solutions of the equation

a. 2

b. 3

c. 1

d. None of these

Solution 8

Correct option: (a)

Question 9

a. 4α = 3β

b. 3α = 4 β

c.

d. None of these

Solution 9

Correct option: (a)

Question 10

The number of real solutions of the equation

a. 0

b. 1

c. 2

d. Infinite

Solution 10

Correct option: (c)

Question 11

If x < 0, y < 0 such that xy = 1, then tan-1x + tan-1y equals

a.

b.

c. – π

d. None of these

Solution 11

Correct option: (b)

Question 12

a.

b.

c. Tan θ

d. Cot θ

Solution 12

Correct option: (a)

Question 13

a. 36

b. 36 – 36 cos θ

c. 18 – 18 cos θ

d. 18 + 18 cos θ

Solution 13

Correct option: (c)

Question 14

a.

b.

c.

d.

Solution 14

Correct option: (a)

Question 15

Let f(x) = ecos-1{sin(x+π/3)}. Then, f(8π/9) =

a. e5π/18

b. e13π/18

c. e-2π/18

d. None of these

Solution 15

Correct option: (b)

Question 16

a. 0

b. 1/2

c. -1

d. none of these

Solution 16

Correct option: (d)

Question 17

a. 36

b. -36 sin2θ

c. 36 sin2θ

d. 36cos2θ

Solution 17

Correct option: (c)

Question 18

If tan-13 + tan-1x= tan-18, then x =

a. 5

b. 1/5

c. 5/14

d. 14/5

Solution 18

Correct option: (b)

Question 19

a.

b.

c.

d.

Solution 19

Correct option: (b)

Question 20

a.

b.

c.

d. 0

Solution 20

Correct option: (d)

Question 21

a.

b.

c.

d.

Solution 21

Correct option: (d)

Question 22

If θ = sin-1 {sin (-600°)}, then one of the possible values of θ is

a.

b.

c.

d.

Solution 22

Correct option: (a)

Question 23

a.

b.

c.

d.

Solution 23

Correct option: (a)

Question 24

If 4 cos-1x + sin-1 x = π, then the value of x is

a.

b.

c.

d.

Solution 24

Correct option: (c)

Question 25

a. 0

b. -2

c. 1

d. 2

Solution 25

Correct option: (d)

Question 26

If cos -1 x > sin-1x, then

a.

b.

c.

d. X > 0

Solution 26

Correct option: (a)

Question 27

a.

b.

c.

d.

Solution 27

Correct option: (b)

Question 28

a.

b.

c.

d.

Solution 28

Correct option: (c)

Question 29

a. 7

b. 6

c. 5

d. None of these

Solution 29

Correct option:(a)

Question 30

If tan-1(cotθ) = 2θ, then θ =

a.

b.

c.

d. none of these

Solution 30

Correct option: (c)

Question 31

a. 0

b.

c. a

d.

Solution 31

Correct option: (d)

Question 32

The value of sin (2(tan-10.75)) is equal to

a. 0.75

b. 1.5

c. 0.96

d. Sin-11.5

Solution 32

Correct option: (c)

Question 33

a. 4tan-1x

b. 0

c.

d. π

Solution 33

Correct option: (a)

Question 34

The domain of cos-1(x2 – 4) is

a. [3, 5]

b. [-1, 1]

c.

d.

Solution 34

Correct option: (c)

Question 35

a.

b.

c.

d.

Solution 35

Correct option: (a)

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