Tamil Nadu Board of Secondary EducationHSC Science Class 12th

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 4 - Inverse Trigonometric Functions [Latest edition]

Chapter 4: Inverse Trigonometric Functions

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6
Exercise 4.1 [Page 137]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 4 Inverse Trigonometric Functions Exercise 4.1 [Page 137]

Exercise 4.1 | Q 1. (i) | Page 137

Find all the values of x such that – 10π ≤ x ≤ 10π and sin x = 0

Exercise 4.1 | Q 1. (ii) | Page 137

Find all the values of x such that Find all the values of x such that −3π ≤ x ≤ 3π and sin x = −1

Exercise 4.1 | Q 2. (i) | Page 137

Find the period and amplitude of y = sin 7x

Exercise 4.1 | Q 2. (ii) | Page 137

Find the period and amplitude of y = - sin(1/3 x)

Exercise 4.1 | Q 2. (iii) | Page 137

Find the period and amplitude of y = 4 sin(– 2x)

Exercise 4.1 | Q 3 | Page 137

Sketch the graph of y = sin(1/3 x) for 0 ≤ x ≤ 6π

Exercise 4.1 | Q 4. (i) | Page 137

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

Exercise 4.1 | Q 4. (ii) | Page 137

Find the value of sin^-1 (sin((5pi)/4))

Exercise 4.1 | Q 5 | Page 137

For what value of x does sin x = sin–1x?

Exercise 4.1 | Q 6. (i) | Page 137

Find the domain of the following

f(x) = sin^-1 ((x^2 + 1)/(2x))

Exercise 4.1 | Q 6. (ii) | Page 137

Find the domain of the following

g(x) = 2sin^-1(2x - 1) - pi/4

Exercise 4.1 | Q 7 | Page 137

Find the value of sin^-1(sin  (5pi)/9 cos  pi/9 + cos  (5pi)/9 sin  pi/9)

Exercise 4.2 [Pages 142 - 143]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 4 Inverse Trigonometric Functions Exercise 4.2 [Pages 142 - 143]

Exercise 4.2 | Q 1. (i) | Page 142

Find all values of x such that – 6π ≤ x ≤ 6π and cos x = 0

Exercise 4.2 | Q 1. (ii) | Page 142

Find all values of x such that – 5π ≤ x ≤ 5π and cos x = 1

Exercise 4.2 | Q 2 | Page 142

State the reason for  cos^-1 [cos(- pi/6)] ≠ - pi/6

Exercise 4.2 | Q 3 | Page 142

Exercise 4.2 | Q 4 | Page 142

Find the principal value of cos^-1 (1/2)

Exercise 4.2 | Q 5. (i) | Page 142

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

Exercise 4.2 | Q 5. (ii) | Page 142

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

Exercise 4.2 | Q 5. (iii) | Page 142

Find the value of cos-1 [cos  pi/7 cos  pi/17 - sin  pi/7 sin  pi/17]

Exercise 4.2 | Q 6. (i) | Page 143

Find the domain of f(x) = sin^-1 ((|x| - 2)/3) + cos^-1 ((1 - |x|)/4)

Exercise 4.2 | Q 6. (ii) | Page 143

Find the domain of g(x) = sin^-1x + cos^-1x

Exercise 4.2 | Q 7 | Page 143

For what value of x, the inequality pi/2 < cos^-1 (3x - 1) < pi holds?

Exercise 4.2 | Q 8. (i) | Page 143

Find the value of cos[cos^-1 (4/5) + sin^-1(4/5)]

Exercise 4.2 | Q 8. (ii) | Page 143

Find the value of cos^-1(cos((4pi)/3)) + cos^-1 (cos((5pi)/4))

Exercise 4.3 [Pages 147 - 148]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 4 Inverse Trigonometric Functions Exercise 4.3 [Pages 147 - 148]

Exercise 4.3 | Q 1. (i) | Page 147

Find the domain of the following functions:

tan^-1 (sqrt(9 - x^2))

Exercise 4.3 | Q 1. (ii) | Page 147

Find the domain of the following functions:

1/2 tan^-1 (1 - x^2) - pi/4

Exercise 4.3 | Q 2. (i) | Page 147

Find the value of tan^-1(tan  (5pi)/4)

Exercise 4.3 | Q 2. (ii) | Page 147

Find the value of tan^-1 (tan(- pi/6))

Exercise 4.3 | Q 3. (i) | Page 148

Find the value of tan(tan^-1((7pi)/4))

Exercise 4.3 | Q 3. (ii) | Page 148

Find the value of tan(tan^-1(1947))

Exercise 4.3 | Q 3. (iii) | Page 148

Find the value of tan(tan^-1(- 0.2021))

Exercise 4.3 | Q 4. (i) | Page 148

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

Exercise 4.3 | Q 4. (ii) | Page 148

Find the value of sin(tan^-1 (1/2) - cos^-1 (4/5))

Exercise 4.3 | Q 4. (iii) | Page 148

Find the value of cos(sin^-1 (4/5) - tan^-1 (3/4))

Exercise 4.4 [Pages 154 - 155]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 4 Inverse Trigonometric Functions Exercise 4.4 [Pages 154 - 155]

Exercise 4.4 | Q 1. (i) | Page 154

Find the principal value of sec^-1 (2/sqrt(3))

Exercise 4.4 | Q 1. (ii) | Page 154

Find the principal value of cot^-1 (sqrt(3))

Exercise 4.4 | Q 1. (iii) | Page 154

Find the principal value of "cosec"^-1 (- sqrt(2))

Exercise 4.4 | Q 2. (i) | Page 155

Find the value of tan^-1 (sqrt(3)) - sec^-1 (- 2)

Exercise 4.4 | Q 2. (ii) | Page 155

Find the value of  sin^-1 (- 1) + cos^-1 (1/2) + cot^-1 (2)

Exercise 4.4 | Q 2. (iii) | Page 155

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

Exercise 4.5 [Page 166]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 4 Inverse Trigonometric Functions Exercise 4.5 [Page 166]

Exercise 4.5 | Q 1. (i) | Page 166

Find the value, if it exists. If not, give the reason for non-existence

sin^-1 (cos pi)

Exercise 4.5 | Q 1. (ii) | Page 166

Find the value, if it exists. If not, give the reason for non-existence

tan^-1(sin(- (5pi)/2))

Exercise 4.5 | Q 1. (iii) | Page 166

Find the value, if it exists. If not, give the reason for non-existence

sin^-1 [sin 5]

Exercise 4.5 | Q 2. (i) | Page 166

Find the value of the expression in terms of x, with the help of a reference triangle

sin (cos–1(1 – x))

Exercise 4.5 | Q 2. (ii) | Page 166

Find the value of the expression in terms of x, with the help of a reference triangle

cos (tan–1 (3x – 1))

Exercise 4.5 | Q 2. (iii) | Page 166

Find the value of the expression in terms of x, with the help of a reference triangle

tan(sin^-1(x + 1/2))

Exercise 4.5 | Q 3. (i) | Page 166

Find the value of sin^-1[cos(sin^-1 (sqrt(3)/2))]

Exercise 4.5 | Q 3. (ii) | Page 166

Find the value of cot[sin^-1  3/5 + sin^-1  4/5]

Exercise 4.5 | Q 3. (iii) | Page 166

Find the value of  tan(sin^-1  3/5 + cot^-1  3/2)

Exercise 4.5 | Q 4. (i) | Page 166

Prove that tan^-1  2/11 + tan^-1  7/24 = tan^-1  1/2

Exercise 4.5 | Q 4. (ii) | Page 166

Prove that sin^-1  3/5 - cos^-1  12/13 = sin^-1  16/65

Exercise 4.5 | Q 5 | Page 166

Prove that tan^-1x + tan^-1y + tan^-1z = tan^-1[(x + y + z - xyz)/(1 - xy - yz - zx)]

Exercise 4.5 | Q 6 | Page 166

If tan–1x + tan1y + tan1z = π, show that x + y + z = xyz

Exercise 4.5 | Q 7 | Page 166

Prove that tan^-1x + tan^-1  (2x)/(1 - x^2) = tan^-1  (3x - x^3)/(1 - 3x^2), |x| < 1/sqrt(3)

Exercise 4.5 | Q 8 | Page 166

Simplify: tan^-1  x/y - tan^-1  (x - y)/(x + y)

Exercise 4.5 | Q 9. (i) | Page 166

Solve: sin^-1  5/x + sin^-1  12/x = pi/2

Exercise 4.5 | Q 9. (ii) | Page 166

Solve: tan^-1x = cos^-1  (1 - "a"^2)/(1 + "a"^2) - cos^-1  (1 - "b"^2)/(1 + "b"^2), "a" > 0, "b" > 0

Exercise 4.5 | Q 9. (iii) | Page 166

Solve: 2tan^-1 (cos x) = tan^-1 (2"cosec"  x)

Exercise 4.5 | Q 9. (iv) | Page 166

Solve: cot^-1 x - cot^-1 (x + 2) = pi/12, x > 0

Exercise 4.5 | Q 10 | Page 166

Find the number of solutions of the equation tan^-1 (x - 1) + tan^-1x + tan^-1(x + 1) = tan^-1(3x)

Exercise 4.6 [Pages 166 - 168]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 4 Inverse Trigonometric Functions Exercise 4.6 [Pages 166 - 168]

MCQ

Exercise 4.6 | Q 1 | Page 166

Choose the correct alternative:

The value of sin–1(cos x), 0 ≤ x ≤ π is

• pi - x

• x - pi/2

• pi/2 - x

• x - pi

Exercise 4.6 | Q 2 | Page 166

Choose the correct alternative:

If sin^-1x + sin^-1y = (2pi)/3 ; then cos^-1x + cos^-1y is equal to

• (2pi)/3

• pi/3

• pi/6

• pi

Exercise 4.6 | Q 3 | Page 167

Choose the correct alternative:

sin^-1  3/5 - cos^-1  13/13 + sec^-1  5/3 - "cosec"^-1  13/12 is equal to

• 2pi

• pi

• 0

• tan^-1  12/65

Exercise 4.6 | Q 4 | Page 167

Choose the correct alternative:

If sin–1x = 2sin1 α has a solution, then

• |alpha| ≤ 1/sqrt(2)

• |alpha| ≥ 1/sqrt(2)

• |alpha| < 1/sqrt(2)

• |alpha| > 1/sqrt(2)

Exercise 4.6 | Q 5 | Page 167

Choose the correct alternative:

sin^-1(cos x) = pi/2 - x is valid for

• pi ≤ x ≤ 0

• 0 ≤ x ≤ π

• - pi/2 ≤ x ≤ pi/2

• - pi/4 ≤ x ≤ (3pi)/4

Exercise 4.6 | Q 6 | Page 167

Choose the correct alternative:

If sin-1 x + sin-1 y + sin-1 z = (3pi)/2, the value of show that x^2017 + y^2018 + z^2019 - 9/(x^101 + y^101 + z^101) is

• 0

• 1

• 2

• 3

Exercise 4.6 | Q 7 | Page 167

Choose the correct alternative:

If cot^-1x = (2pi)/5 for some x ∈ R, the value of tan-1 x is

• - pi/10

• pi/5

• pi/10

• - pi/5

Exercise 4.6 | Q 8 | Page 167

Choose the correct alternative:

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

• [1, 2]

• [– 1, 1]

• [0, 1]

• [– 1, 0]

Exercise 4.6 | Q 9 | Page 167

Choose the correct alternative:

If x = 1/5, the value of cos(cos^-1x + 2sin^-1x) is

• - sqrt(24/25)

• sqrt(24/25)

• 1/5

• - 1/5

Exercise 4.6 | Q 10 | Page 167

Choose the correct alternative:

tan^-1 (1/4) + tan^-1 (2/9) is equal to

• 1/2 cos^-1 (3/5)

• 1/2 sin^-1 (3/5)

• 1/2 tan^-1 (3/5)

• tan^-1 (1/2)

Exercise 4.6 | Q 11 | Page 167

Choose the correct alternative:

If the function f(x) = sin^-1 (x^2 - 3), then x belongs to

• [– 1, 1]

• [sqrt(2), 2]

• [-2, sqrt(2)] ∪ [sqrt(2), 2]

• [- 2 -sqrt(2)]

Exercise 4.6 | Q 12 | Page 167

Choose the correct alternative:

If cot–12 and cot13 are two angles of a triangle, then the third angle is

• pi/4

• (3pi)/4

• pi/6

• pi/3

Exercise 4.6 | Q 13 | Page 167

Choose the correct alternative:

sin^-1 (tan  pi/4) - sin^-1 (sqrt(3/x)) = pi/6. Then x is a root of the equation

• x2 – x – 6 = 0

• x2 – x – 12 = 0

• x2 + x – 12 = 0

• x2 + x – 6 = 0

Exercise 4.6 | Q 14 | Page 168

Choose the correct alternative:

sin–1(2 cos2x – 1) + cos1(1 – 2 sin2x) =

• pi/2

• pi/3

• pi/4

• pi/6

Exercise 4.6 | Q 15 | Page 168

Choose the correct alternative:

If cot^-1(sqrt(sin alpha)) + tan^-1(sqrt(sin alpha)) = u, then cos 2u is equal to

• tan2α

• 0

• – 1

• tan 2α

Exercise 4.6 | Q 16 | Page 168

Choose the correct alternative:

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

• tan–1x

• sin1x

• 0

• π

Exercise 4.6 | Q 17 | Page 168

Choose the correct alternative:

The equation tan–1x – cot1x = tan^-1 (1/sqrt(3)) has

• no solution

• unique solution

• two solutions

• infinite number of solutions

Exercise 4.6 | Q 18 | Page 168

Choose the correct alternative:

If sin^-1x + cot^-1 (1/2) = pi/2, then x is equal to

• 1/2

• 1/sqrt(5)

• 2/sqrt(5)

• sqrt(3)/5

Exercise 4.6 | Q 19 | Page 168

Choose the correct alternative:

If sin^-1x + "cosec"^-1  5/4 = pi/2, then the value of x is

• 4

• 5

• 2

• 3

Exercise 4.6 | Q 20 | Page 168

Choose the correct alternative:

sin(tan–1x), |x| < 1 is equal to

• x/sqrt(1 - x^2)

• 1/sqrt(1 - x^2)

• 1/sqrt(1 + x^2)

• x/sqrt(1 + x^2)

Chapter 4: Inverse Trigonometric Functions

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 4 - Inverse Trigonometric Functions

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 4 Inverse Trigonometric Functions are Some Fundamental Concepts, Sine Function and Inverse Sine Function, The Cosine Function and Inverse Cosine Function, The Tangent Function and the Inverse Tangent Function, The Cosecant Function and the Inverse Cosecant Function, The Secant Function and Inverse Secant Function, The Cotangent Function and the Inverse Cotangent Function, Principal Value of Inverse Trigonometric Functions, Properties of Inverse Trigonometric Functions, Inverse Trigonometric Functions.

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