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Chapters
▶ 2: Inverse Trigonometric Functions
3: Matrices
4: Determinants
5: Continuity And Differentiability
6: Application Of Derivatives
7: Integrals
8: Application Of Integrals
9: Differential Equations
10: Vector Algebra
11: Three Dimensional Geometry
12: Linear Programming
13: Probability
![NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 2 - Inverse Trigonometric Functions - Shaalaa.com](/images/mathematics-english-class-12_6:5f2b1b2038084cf381bfa42c826a928c.jpg "NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 2 - Inverse Trigonometric Functions")
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Solutions for Chapter 2: Inverse Trigonometric Functions
Below listed, you can find solutions for Chapter 2 of CBSE NCERT Exemplar for Mathematics [English] Class 12.
NCERT Exemplar solutions for Mathematics [English] Class 12 2 Inverse Trigonometric Functions Solved Examples [Pages 20 - 35]
Short Answer
Find the principal value of cos–1x, 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–1x) = cot(tan–1x). 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–1x) 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–17 + cot–18 + cot–118 = cot–13
Which is greater, tan 1 or tan–11?
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–1x + sin–1(1 – x) = cos–1x.
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–1x) 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–1x 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–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
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 (– x2) is ______.
[0, 1]
(0, 1)
[–1, 1]
φ
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)]`
The domain of the function defined by f(x) = sin–1x + 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–1x + sin–1y = `pi/2`, then value of cos–1x + cos–1y 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–11))] is ______.
0
1
`1/sqrt(3)`
`sqrt(2/3)`
The equation tan–1x – cot–1x = `(1/sqrt(3))` has ______.
No solution
Unique solution
Infinite number of solutions
Two solutions
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
The value of tan2 (sec–12) + cot2 (cosec–13) is ______.
5
11
13
15
NCERT Exemplar solutions for Mathematics [English] Class 12 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 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
Which of the following is the principal value branch of cos–1x?
`[(-pi)/2, pi/2]`
(0, π)
[0, π]
`(0, pi) - {pi/2}`
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}`
If 3 tan–1x + cot–1x = 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–1x + tan–1y = `(4pi)/5`, then cot–1x + cot–1y 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–1α + cos–1β + cos–1γ = 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–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
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–1x + cos–1x), |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–1x 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–1x)2 is equal to sec2x.
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
Solutions for 2: Inverse Trigonometric Functions
NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 2 - Inverse Trigonometric Functions
Shaalaa.com has the CBSE Mathematics Mathematics [English] Class 12 CBSE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT Exemplar solutions for Mathematics Mathematics [English] Class 12 CBSE 2 (Inverse Trigonometric Functions) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT Exemplar textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.
Concepts covered in Mathematics [English] Class 12 chapter 2 Inverse Trigonometric Functions are Inverse Trigonometric Functions, Inverse Trigonometric Functions (Simplification and Examples), Properties of Inverse Trigonometric Functions, Graphs of Inverse Trigonometric Functions, Inverse Trigonometric Functions - Principal Value Branch.
Using NCERT Exemplar Mathematics [English] Class 12 solutions Inverse Trigonometric Functions exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Exemplar Solutions are essential questions that can be asked in the final exam. Maximum CBSE Mathematics [English] Class 12 students prefer NCERT Exemplar Textbook Solutions to score more in exams.
Get the free view of Chapter 2, Inverse Trigonometric Functions Mathematics [English] Class 12 additional questions for Mathematics Mathematics [English] Class 12 CBSE, and you can use Shaalaa.com to keep it handy for your exam preparation.
