Chapter 4 : Inverse Trigonometric Functions - Rd Sharma Solutions for Class 12-science Maths CBSE

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

On TopperLearning, you can find study resources such as sample papers, mock tests, Class 12 Maths NCERT solutions and more. These learning materials can help you understand concepts such as differentiation of functions, direction cosines, integrals, and more. Also, you can practise the Maths problems by going through the solutions given by our experts.

Maths is considered as one of the most difficult subjects in CBSE Class 12 Science. Our Maths experts simplify complex Maths problems by assisting you with the right methods to solve problems and score full marks. You may still have doubts while referring to the Maths revision notes or Maths NCERT solutions. Solve those doubts by asking an expert through the “Undoubt” feature on the student dashboard.

Read  more

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 = 2π, then find the value x2 + y2 + z2 + t2.

Solution 13

Question 14

Solution 14

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:

begin mathsize 14px style sin minus 1 open curly brackets fraction numerator straight x plus square root of 1 minus straight x 2 end root over denominator square root of 2 end fraction close curly brackets comma negative 1 half space less than space straight x space less than fraction numerator 1 over denominator square root of 2 end fraction end style

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

Chapter 4 - Inverse Trigonometric Functions Exercise Ex. 4.9

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

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

 

 

On adding both 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)/22

 

Question 8

Solution 8

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

 

cos-1(3/5) + sin-1x = nΠ + Π/2

 

sin-1x = nΠ + Π/2 - cos-1(3/5)

 

sin-1x = nΠ + sin-1(3/5)

 

x = sin(nΠ + 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 - 9Πu -2Π2 = 0

 

On factorizing

 

18u2 - 12Πu + 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 =2Π/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

F i n d space t h e space v a l u e space o f space tan to the power of minus 1 end exponent open parentheses x over y close parentheses minus tan to the power of minus 1 end exponent open parentheses fraction numerator x minus y over denominator x plus y end fraction close parentheses

Solution 4

W e space k n o w space t h a t comma space tan to the power of minus 1 end exponent A minus tan to the power of minus 1 end exponent B equals tan to the power of minus 1 end exponent open parentheses fraction numerator A minus B over denominator 1 plus A B end fraction close parentheses comma space i f space A B greater than minus 1
C o n s i d e r space t h e space g i v e n space e x p r e s s i o n space tan to the power of minus 1 end exponent open parentheses x over y close parentheses minus tan to the power of minus 1 end exponent open parentheses fraction numerator x minus y over denominator x plus y end fraction close parentheses :
tan to the power of minus 1 end exponent open parentheses x over y close parentheses minus tan to the power of minus 1 end exponent open parentheses fraction numerator x minus y over denominator x plus y end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator x over y minus fraction numerator x minus y over denominator x plus y end fraction over denominator 1 plus open parentheses x over y close parentheses open parentheses fraction numerator x minus y over denominator x plus y end fraction close parentheses end fraction close parentheses
equals tan to the power of minus 1 end exponent open parentheses fraction numerator x open parentheses x plus y close parentheses minus y open parentheses x minus y close parentheses over denominator y open parentheses x plus y close parentheses plus x open parentheses x minus y close parentheses end fraction close parentheses
equals tan to the power of minus 1 end exponent open parentheses fraction numerator x squared plus y squared over denominator x squared plus y squared end fraction close parentheses
equals tan to the power of minus 1 end exponent open parentheses 1 close parentheses
equals straight pi over 4


Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solve the following equations for x :

begin mathsize 14px style tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses minus 1 half tan to the power of negative 1 end exponent straight x equals 0 comma space where space straight x space greater than space 0 end style

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 x equals n straight pi plus straight pi over 4

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Chapter 4 - Inverse Trigonometric Functions Exercise MCQ

Question 1

 

a. sin 2α

b. sin α

c. cos 2α

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)

Chapter 4 - Inverse Trigonometric Functions Exercise Ex. 4VSAQ

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

Loading...

Why CBSE Class 12 Science Maths solutions are important?

Maths is a subject which requires practising a variety of problems to understand concepts clearly. By solving as many problems as you can, you’ll be able to train your brain in thinking the logical way to solve maths problems. For practising problems, study materials such as sample papers, previous year papers, and NCERT solutions are needed.

Some of the best Maths experts work with us to give you the best solutions for Maths textbook questions and sample paper questions. Chapter-wise NCERT solutions for Class 12 Science Maths can be easily accessible on TopperLearning. Use these solutions to practise problems based on concepts such as direction ratios, probability, area between lines, inverse trigonometric functions, and more.

To prepare for your Maths exam, you need to attempt solving different kinds of Maths questions. One of the best ways to assess your problem-solving abilities is to attempt solving previous year papers with a set timer. Our Maths solutions will come in handy to help you with checking your answers and thus, improving your learning experience. So, to score more marks in your Class 12 board exams, use our Maths solutions that will enable you with the appropriate preparation.