NCERT solutions for Mathematics Exemplar Class 12 chapter 2 - Inverse Trigonometric Functions [Latest edition]

Chapters Chapter 2: Inverse Trigonometric Functions

Solved ExamplesExercise
Solved Examples [Pages 20 - 35]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 2 Inverse Trigonometric Functions Solved Examples [Pages 20 - 35]

Solved Examples | Q 1 | Page 20

Find the principal value of cos–1x, for x = sqrt(3)/2.

Solved Examples | Q 2 | Page 21

Evaluate tan^-1(sin((-pi)/2)).

Solved Examples | Q 3 | Page 21

Find the value of cos^-1(cos  (13pi)/6).

Solved Examples | Q 4 | Page 21

Find the value of tan^-1 (tan  (9pi)/8).

Solved Examples | Q 5 | Page 21

Evaluate tan (tan–1(– 4)).

Solved Examples | Q 6 | Page 21

Evaluate: tan^-1 sqrt(3) - sec^-1(-2).

Solved Examples | Q 7 | Page 22

Evaluate: sin^-1 [cos(sin^-1 sqrt(3)/2)]

Solved Examples | Q 8 | Page 22

Prove that tan(cot–1x) = cot(tan–1x). State with reason whether the equality is valid for all values of x.

Solved Examples | Q 9 | Page 22

Find the value of sec(tan^-1  y/2)

Solved Examples | Q 10 | Page 22

Find value of tan (cos–1x) and hence evaluate tan(cos^-1  8/17)

Solved Examples | Q 11 | Page 23

Find the value of sin[2cot^-1 ((-5)/12)]

Solved Examples | Q 12 | Page 23

Evaluate cos[sin^-1  1/4 + sec^-1  4/3]

Solved Examples | Q 13 | Page 24

Prove that 2sin^-1  3/5 - tan^-1  17/31 = pi/4

Solved Examples | Q 14 | Page 24

Prove that cot–17 + cot–18 + cot–118 = cot–13

Solved Examples | Q 15 | Page 25

Which is greater, tan 1 or tan–11?

Solved Examples | Q 16 | Page 25

Find the value of sin(2tan^-1  2/3) + cos(tan^-1 sqrt(3))

Solved Examples | Q 17 | Page 26

Solve for x tan^-1((1 - x)/(1 + x)) = 1/2 tan^-1x, x > 0

Solved Examples | Q 18 | Page 26

Find the values of x which satisfy the equation sin–1x + sin–1(1 – x) = cos–1x.

Solved Examples | Q 19 | Page 26

Solve the equation sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2

Solved Examples | Q 20 | Page 27

Show that 2tan^-1 {tan  alpha/2 * tan(pi/4 - beta/2)} = tan^-1  (sin alpha cos beta)/(cosalpha + sinbeta)

Objective type questions Examples 21 to 41

Solved Examples | Q 21 | Page 28

Which of the following corresponds to the principal value branch of tan–1?

• (- pi/2, pi/2)

• [- pi/2, pi/2]

• (- pi/2, pi/2) - {0}

• (0, π)

Solved Examples | Q 22 | Page 28

The principal value branch of sec–1 is ______.

• [- pi/2, pi/2] - {0}

• [0, pi] - {pi/2}

• (0, π)

• (- pi/2, pi/2)

Solved Examples | Q 23 | Page 29

One branch of cos–1 other than the principal value branch corresponds to ______.

• [pi/2, (3pi)/2]

• [pi, 2pi]- {(3pi)/2}

• (0, π)

• [2π, 3π]

Solved Examples | Q 24 | Page 29

The value of sin^-1 (cos((43pi)/5)) is ______.

• (3pi)/5

• (-7pi)/5

• pi/10

• - pi/10

Solved Examples | Q 25 | Page 29

The principal value of the expression cos–1[cos (– 680°)] is ______.

• (2pi)/9

• (-2pi)/9

• (34pi)/9

• pi/9

Solved Examples | Q 26 | Page 29

The value of cot (sin–1x) is ______.

• sqrt(1 + x^2)/x

• x/sqrt(1 + x^2)

• 1/x

• sqrt(1 - x^2)/x

Solved Examples | Q 27 | Page 30

If tan^-1x = pi/10 for some x ∈ R, then the value of cot–1x is ______.

• pi/5

• (2pi)/5

• (3pi)/5

• (4pi)/5

Solved Examples | Q 28 | Page 30

The domain of sin–1 2x is ______.

• [0, 1]

• [– 1, 1]

• [-1/2, 1/2]

• [–2, 2]

Solved Examples | Q 29 | Page 30

The principal value of sin^-1 ((-sqrt(3))/2) is ______.

• - (2pi)/3

• -pi/3

• (4pi)/3

• (5pi)/3

Solved Examples | Q 30 | Page 31

The greatest and least values of (sin–1x)2 + (cos–1x)2 are respectively ______.

• (5pi^2)/4 and pi^2/8

• pi/2 and (-pi)/2

• pi^2/4 ad (-pi^2)/4

• pi^2/4 and 0

Solved Examples | Q 31 | Page 31

Let θ = sin–1 (sin (– 600°), then value of θ is ______.

• pi/3

• pi/2

• (2pi)/3

• (-2pi)/3

Solved Examples | Q 32 | Page 32

The domain of the function y = sin–1 (– x2) is ______.

• [0, 1]

• (0, 1)

• [–1, 1]

• φ

Solved Examples | Q 33 | Page 32

The domain of y = cos–1(x2 – 4) is ______.

• [3, 5]

• [0, π]

• [-sqrt(5), -sqrt(3)] ∩ [-sqrt(5), sqrt(3)]

• [-sqrt(5), -sqrt(3)] ∪ [-sqrt(3), sqrt(5)]

Solved Examples | Q 34 | Page 32

The domain of the function defined by f(x) = sin–1x + cosx is ______.

• [–1, 1]

• [–1, π + 1]

• (– oo, oo)

• φ

Solved Examples | Q 35 | Page 33

The value of sin (2 sin–1 (.6)) is ______.

• .48

• .96

• 1.2

• sin 1.2

Solved Examples | Q 36 | Page 33

If sin–1x + sin–1y = pi/2, then value of cos–1x + cos–1y is ______.

• pi/2

• π

• 0

• (2pi)/3

Solved Examples | Q 37 | Page 33

The value of tan(cos^-1  3/5 + tan^-1  1/4) is ______.

• 19/8

• 8/19

• 19/12

• 3/4

Solved Examples | Q 38 | Page 34

The value of the expression sin [cot–1 (cos (tan–11))] is ______.

• 0

• 1

• 1/sqrt(3)

• sqrt(2/3)

Solved Examples | Q 39 | Page 34

The equation tan–1x – cot–1x = (1/sqrt(3)) has ______.

• No solution

• Unique solution

• Infinite number of solutions

• Two solutions

Solved Examples | Q 40 | Page 34

If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.

• α = (-pi)/2, β = pi/2

• α = β = π

• α = (-pi)/2, β = (3pi)/2

• α = 0, β = 2π

Solved Examples | Q 41 | Page 35

The value of tan2 (sec–12) + cot2 (cosec–13) is ______.

• 5

• 11

• 13

• 15

Exercise [Pages 35 - 41]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 2 Inverse Trigonometric Functions Exercise [Pages 35 - 41]

Exercise | Q 1 | Page 35

Find the value of tan^-1 (tan  (5pi)/6) +cos^-1(cos  (13pi)/6)

Exercise | Q 2 | Page 35

Evaluate cos[cos^-1 ((-sqrt(3))/2) + pi/6]

Exercise | Q 3 | Page 35

Prove that cot(pi/4 - 2cot^-1 3) = 7

Exercise | Q 4 | Page 35

Find the value of tan^-1 (- 1/sqrt(3)) + cot^-1(1/sqrt(3)) + tan^-1(sin((-pi)/2))

Exercise | Q 5 | Page 35

Find the value of tan^-1 (tan  (2pi)/3)

Exercise | Q 6 | Page 35

Show that 2tan^-1 (-3) = (-pi)/2 + tan^-1 ((-4)/3)

Exercise | Q 7 | Page 36

Find the real solutions of the equation
tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2

Exercise | Q 8 | Page 36

Find the value of the expression sin(2tan^-1  1/3) + cos(tan^-1 2sqrt(2))

Exercise | Q 9 | Page 36

If 2 tan–1(cos θ) = tan–1(2 cosec θ), then show that θ = π 4, where n is any integer.

Exercise | Q 10 | Page 36

Show that cos(2tan^-1  1/7) = sin(4tan^-1  1/3)

Exercise | Q 11 | Page 36

Solve the following equation cos(tan^-1x) = sin(cot^-1  3/4)

Exercise | Q 12 | Page 36

Prove that tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/((1 + x^2) - sqrt(1 - x^2))) = pi/2 + 1/2 cos^-1x^2

Exercise | Q 13 | Page 36

Find the simplified form of cos^-1 (3/5 cosx + 4/5 sin x), where x ∈ [(-3pi)/4, pi/4]

Exercise | Q 14 | Page 36

Prove that sin^-1  8/17 + sin^-1  3/5 = sin^-1  7/85

Exercise | Q 15 | Page 36

Show that sin^-1  5/13 + cos^-1  3/5 = tan^-1  63/16

Exercise | Q 16 | Page 36

Prove that tan^-1  1/4 + tan^-1  2/9 = sin^-1  1/sqrt(5)

Exercise | Q 17 | Page 36

Find the value of 4tan^-1  1/5 - tan^-1  1/239

Exercise | Q 18 | Page 37

Show that tan(1/2 sin^-1  3/4) = (4 - sqrt(7))/3 and justify why the other value (4 + sqrt(7))/3 is ignored?

Exercise | Q 19 | Page 37

If a1, a2, a3,...,an is an arithmetic progression with common difference d, then evaluate the following expression.

tan[tan^-1("d"/(1 + "a"_1 "a"_2)) + tan^-1("d"/(21 + "a"_2 "a"_3)) + tan^-1("d"/(1 + "a"_3 "a"_4)) + ... + tan^-1("d"/(1 + "a"_("n" - 1) "a""n"))]

Objective Type Questions from 20 to 37

Exercise | Q 20 | Page 37

Which of the following is the principal value branch of cos–1x?

• [(-pi)/2, pi/2]

• (0, π)

• [0, π]

• (0, pi) - {pi/2}

Exercise | Q 21 | Page 37

Which of the following is the principal value branch of cosec–1x?

• ((-pi)/2, pi/2)

• [0, pi] - {pi/2}

• [(-pi)/2, pi/2]

• [(-pi)/2, pi/2] - {0}

Exercise | Q 22 | Page 37

If 3 tan–1x + cot–1x = π, then x equals ______.

• 0

• 1

• – 1

• 1/2

Exercise | Q 23 | Page 37

The value of sin^-1 [cos((33pi)/5)] is ______.

• (3pi)/5

• (-7pi)/5

• pi/10

• (-pi)/10

Exercise | Q 24 | Page 38

The domain of the function cos–1(2x – 1) is ______.

• [0, 1]

• [–1, 1]

• ( –1, 1)

• [0, π]

Exercise | Q 25 | Page 38

The domain of the function defined by f(x) = sin^-1 sqrt(x- 1) is ______.

• [1, 2]

• [–1, 1]

• [0, 1]

• None of these

Exercise | Q 26 | Page 38

If cos(sin^-1  2/5 + cos^-1x) = 0, then x is equal to ______.

• 1/5

• 2/5

• 0

• 1

Exercise | Q 27 | Page 38

The value of sin (2 tan–1(0.75)) is equal to ______.

• 0.75

• 1.5

• 0.96

• sin 1.5

Exercise | Q 28 | Page 38

The value of cos^-1 (cos  (3pi)/2) is equal to ______.

• pi/2

• (3pi)/2

• (5pi)/2

• (7pi)/2

Exercise | Q 29 | Page 38

The value of the expression 2 sec^-1 2 + sin^-1 (1/2) is ______.

• pi/6

• (5pi)/6

• (7pi)/6

• 1

Exercise | Q 30 | Page 38

If tan–1x + tan–1y = (4pi)/5, then cot–1x + cot–1y equals ______.

• pi/5

• (2pi)/5

• (3pi)/5

• π

Exercise | Q 31 | Page 38

If sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2)). where a, x ∈ ] 0, 1, then the value of x is ______.

• 0

• "a"/2

• a

• (2"a")/(1 - "a"^2)

Exercise | Q 32 | Page 39

The value of cot[cos^-1 (7/25)] is ______.

• 25/24

• 25/7

• 24/25

• 7/24

Exercise | Q 33 | Page 39

The value of the expression tan (1/2 cos^-1  2/sqrt(5)) is ______.

• 2 + sqrt(5)

• sqrt(5) - 2

• (sqrt(5) + 2)/2

• 5 + sqrt(2)

Exercise | Q 34 | Page 39

If |x| ≤ 1, then 2 tan^-1x + sin^-1 ((2x)/(1 + x^2)) is equal to ______.

• 4 tan^-1x

• 0

• pi/2

• π

Exercise | Q 35 | Page 39

If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.

• 0

• 1

• 6

• 12

Exercise | Q 36 | Page 39

The number of real solutions of the equation sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x) in [pi/2, pi] is ______.

• 0

• 1

• 2

• Infinite

Exercise | Q 37 | Page 39

If cos–1x > sin–1x, then ______.

• 1/sqrt(2) < x ≤ 1

• 0 ≤ x < 1/2

• -1 ≤ x  < 1/2

• x > 0

Fill in the blanks 38 to 48

Exercise | Q 38 | Page 40

The principal value of cos^-1 (- 1/2) is ______.

Exercise | Q 39 | Page 40

The value of sin^-1 (sin  (3pi)/5) is ______.

Exercise | Q 40 | Page 40

If cos(tan^-1x + cot^-1 sqrt(3)) = 0, then value of x is ______.

Exercise | Q 41 | Page 40

The set of values of sec^-1 (1/2) is ______.

Exercise | Q 42 | Page 40

The principal value of tan^-1 sqrt(3) is ______.

Exercise | Q 43 | Page 40

The value of cos^-1 (cos  (14pi)/3) is ______.

Exercise | Q 44 | Page 40

The value of cos (sin–1x + cos–1x), |x| ≤ 1 is ______.

Exercise | Q 45 | Page 40

The value of expression tan((sin^-1x + cos^-1x)/2), when x = sqrt(3)/2 is ______.

Exercise | Q 46 | Page 40

If y = 2 tan^-1x + sin^-1 ((2x)/(1 + x^2)) for all x, then ______ < y < ______.

Exercise | Q 47 | Page 40

The result tan^1x - tan^-1y = tan^-1 ((x - y)/(1 + xy)) is true when value of xy is ______.

Exercise | Q 48 | Page 40

The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.

State True or False for the statement 49 to 55

Exercise | Q 49 | Page 40

All trigonometric functions have inverse over their respective domains.

• True

• False

Exercise | Q 50 | Page 40

The value of the expression (cos–1x)2 is equal to sec2x.

• True

• False

Exercise | Q 51 | Page 40

The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions.

• True

• False

Exercise | Q 52 | Page 40

The least numerical value, either positive or negative of angle θ is called principal value of the inverse trigonometric function.

• True

• False

Exercise | Q 53 | Page 40

The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging x and y axes.

• True

• False

Exercise | Q 54 | Page 41

The minimum value of n for which tan^-1  "n"/pi > pi/4, n ∈ N, is valid is 5.

• True

• False

Exercise | Q 55 | Page 41

The principal value of sin^-1 [cos(sin^-1  1/2)] is pi/3.

• True

• False

Chapter 2: Inverse Trigonometric Functions

Solved ExamplesExercise NCERT solutions for Mathematics Exemplar Class 12 chapter 2 - Inverse Trigonometric Functions

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Concepts covered in Mathematics Exemplar Class 12 chapter 2 Inverse Trigonometric Functions are Inverse Trigonometric Functions (Simplification and Examples), Properties of Inverse Trigonometric Functions, Graphs of Inverse Trigonometric Functions, Inverse Trigonometric Functions - Principal Value Branch, Basic Concepts of Trigonometric Functions.

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