#### Online Mock Tests

#### Chapters

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity And Differentiability

Chapter 6: Application Of Derivatives

Chapter 7: Integrals

Chapter 8: Application Of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

## Chapter 2: Inverse Trigonometric Functions

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

#### Short Answer

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

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

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

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

Evaluate tan (tan^{–1}(– 4)).

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

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

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

Find the value of `sec(tan^-1 y/2)`

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

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

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

#### Long Answer

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

Prove that cot^{–1}7 + cot^{–1}8 + cot^{–1}18 = cot^{–1}3

Which is greater, tan 1 or tan^{–1}1?

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

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

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

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

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

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)

The principal value branch of sec^{–1} is ______.

`[- pi/2, pi/2] - {0}`

`[0, pi] - {pi/2}`

(0, π)

`(- pi/2, pi/2)`

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]

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

`(3pi)/5`

`(-7pi)/5`

`pi/10`

`- pi/10`

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

`(2pi)/9`

`(-2pi)/9`

`(34pi)/9`

`pi/9`

The value of cot (sin^{–1}x) is ______.

`sqrt(1 + x^2)/x`

`x/sqrt(1 + x^2)`

`1/x`

`sqrt(1 - x^2)/x`

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

`pi/5`

`(2pi)/5`

`(3pi)/5`

`(4pi)/5`

The domain of sin^{–1 }2x is ______.

[0, 1]

[– 1, 1]

`[-1/2, 1/2]`

[–2, 2]

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

`- (2pi)/3`

`-pi/3`

`(4pi)/3`

`(5pi)/3`

The greatest and least values of (sin–^{1}x)^{2} + (cos^{–1}x)^{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

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

`pi/3`

`pi/2`

`(2pi)/3`

`(-2pi)/3`

The domain of the function y = sin^{–1} (– x^{2}) is ______.

[0, 1]

(0, 1)

[–1, 1]

φ

The domain of y = cos^{–1}(x^{2} – 4) is ______.

[3, 5]

[0, π]

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

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

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

[–1, 1]

[–1, π + 1]

`(– oo, oo)`

φ

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

.48

.96

1.2

sin 1.2

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

`pi/2`

π

0

`(2pi)/3`

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

`19/8`

`8/19`

`19/12`

`3/4`

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

0

1

`1/sqrt(3)`

`sqrt(2/3)`

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

No solution

Unique solution

Infinite number of solutions

Two solutions

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

α = `(-pi)/2`, β = `pi/2`

α = β = π

α = `(-pi)/2`, β = `(3pi)/2`

α = 0, β = 2π

The value of tan^{2} (sec^{–1}2) + cot^{2} (cosec^{–1}3) is ______.

5

11

13

15

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

#### Short Answer

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

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

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

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

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

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

Find the real solutions of the equation

`tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2`

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

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

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

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

#### Long Answer

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`

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

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

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

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

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

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?

If a_{1}, a_{2}, a_{3},...,a_{n} 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

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

`[(-pi)/2, pi/2]`

(0, π)

[0, π]

`(0, pi) - {pi/2}`

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

`((-pi)/2, pi/2)`

`[0, pi] - {pi/2}`

`[(-pi)/2, pi/2]`

`[(-pi)/2, pi/2] - {0}`

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

0

1

– 1

`1/2`

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

`(3pi)/5`

`(-7pi)/5`

`pi/10`

`(-pi)/10`

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

[0, 1]

[–1, 1]

( –1, 1)

[0, π]

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

[1, 2]

[–1, 1]

[0, 1]

None of these

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

`1/5`

`2/5`

0

1

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

0.75

1.5

0.96

sin 1.5

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

`pi/2`

`(3pi)/2`

`(5pi)/2`

`(7pi)/2`

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

`pi/6`

`(5pi)/6`

`(7pi)/6`

1

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

`pi/5`

`(2pi)/5`

`(3pi)/5`

π

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)`

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

`25/24`

`25/7`

`24/25`

`7/24`

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)`

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

`4 tan^-1x`

0

`pi/2`

π

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

0

1

6

12

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

If cos^{–1}x > sin^{–1}x, then ______.

`1/sqrt(2) < x ≤ 1`

`0 ≤ x < 1/2`

`-1 ≤ x < 1/2`

x > 0

#### Fill in the blanks 38 to 48

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

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

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

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

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

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

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

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

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

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

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

#### State True or False for the statement 49 to 55

All trigonometric functions have inverse over their respective domains.

True

False

The value of the expression (cos^{–1}x)^{2} is equal to sec^{2}x.

True

False

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

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

True

False

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

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

True

False

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

True

False

## Chapter 2: Inverse Trigonometric Functions

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

NCERT solutions for Mathematics Exemplar Class 12 chapter 2 (Inverse Trigonometric Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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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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