Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th
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Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 4 - Trigonometry [Latest edition]

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Class 11th Business Mathematics and Statistics Answers Guide - Shaalaa.com
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Chapter 4: Trigonometry

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Problems
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Exercise 4.1 [Pages 80 - 81]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 4 TrigonometryExercise 4.1 [Pages 80 - 81]

Exercise 4.1 | Q 1. (i) | Page 80

Convert the following degree measure into radian measure.

60°

Exercise 4.1 | Q 1. (ii) | Page 80

Convert the following degree measure into radian measure.

150°

Exercise 4.1 | Q 1. (iii) | Page 80

Convert the following degree measure into radian measure.

240°

Exercise 4.1 | Q 1. (iv) | Page 80

Convert the following degree measure into radian measure.

- 320°

Exercise 4.1 | Q 2. (i) | Page 80

Find the degree measure corresponding to the following radian measure.

`pi/8`

Exercise 4.1 | Q 2. (ii) | Page 80

Find the degree measure corresponding to the following radian measure.

`(9pi)/5`

Exercise 4.1 | Q 2. (iii) | Page 80

Find the degree measure corresponding to the following radian measure.

-3

Exercise 4.1 | Q 2. (iv) | Page 80

Find the degree measure corresponding to the following radian measure.

`(11pi)/18`

Exercise 4.1 | Q 3. (i) | Page 81

Determine the quadrant in which the following degree lie.

380°

Exercise 4.1 | Q 3. (ii) | Page 81

Determine the quadrant in which the following degree lie.

-140°

Exercise 4.1 | Q 3. (iii) | Page 81

Determine the quadrant in which the following degree lie.

1195°

Exercise 4.1 | Q 4. (i) | Page 81

Find the values of the following trigonometric ratio.

sin 300°

Exercise 4.1 | Q 4. (ii) | Page 81

Find the values of the following trigonometric ratio.

cos (-210°)

Exercise 4.1 | Q 4. (iii) | Page 81

Find the values of the following trigonometric ratio.

sec 390°

Exercise 4.1 | Q 4. (iv) | Page 81

Find the values of the following trigonometric ratio.

tan(-855°)

Exercise 4.1 | Q 4. (v) | Page 81

Find the values of the following trigonometric ratio.

cosec 1125°

Exercise 4.1 | Q 5. (i) | Page 81

Prove that:

tan(-225°) cot(-405°) – tan(-765°) cot(675°) = 0.

Exercise 4.1 | Q 5. (ii) | Page 81

Prove that:

2 sin2 `pi/6` + cosec`(7pi)/6`  cos2 `pi/3 = 3/2`

Exercise 4.1 | Q 5. (iii) | Page 81

Prove that:

`sec((3pi)/2 - theta) sec(theta - (5pi)/2) + tan((5pi)/2 + theta) tan(theta - (5pi)/2)` = - 1

Exercise 4.1 | Q 6 | Page 81

If A, B, C, D are angles of a cyclic quadrilateral, prove that: cos A + cos B + cos C + cos D = 0.

Exercise 4.1 | Q 7. (i) | Page 81

Prove that:

`(sin(180^circ - theta)cos(90^circ + theta)tan(270^circ - theta)cot(360^circ - theta))/(sin(360^circ - theta)cos(360^circ + theta)sin(270^circ - theta)cosec (-theta))` =  -1

Exercise 4.1 | Q 7. (ii) | Page 81

Prove that:

`sin theta * cos theta {sin(pi/2 - theta) * "cosec" theta + cos (pi/2 - theta) * sec theta}` = 1

Exercise 4.1 | Q 8 | Page 81

Prove that: cos 510° cos 330° + sin 390° cos 120° = -1.

Exercise 4.1 | Q 9. (i) | Page 81

Prove that:

tan(π + x) cot(x – π) – cos(2π – x) cos(2π + x) = sinx.

Exercise 4.1 | Q 9. (ii) | Page 81

Prove that:

`(sin(180^circ + "A")cos(90^circ - "A")tan(270^circ - "A"))/(sec(540^circ - "A") cos(360^circ + "A") "cosec"(270^circ + "A"))` = - sin A cos2 A.

Exercise 4.1 | Q 10 | Page 81

If sin θ = `3/5`, tan φ = `1/2 and pi/2` < θ < π < φ < `(3pi)/2,`, then find the value of 8 tan θ – `sqrt5` sec φ.

Exercise 4.2 [Pages 84 - 85]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 4 TrigonometryExercise 4.2 [Pages 84 - 85]

Exercise 4.2 | Q 1. (i) | Page 84

Find the value of the following:

cosec 15º

Exercise 4.2 | Q 1. (ii) | Page 84

Find the value of the following:

sin (-105°)

Exercise 4.2 | Q 1. (iii) | Page 84

Find the value of the following:

cot 75°

Exercise 4.2 | Q 2. (i) | Page 84

Find the value of the following:

sin 76° cos 16° – cos 76° sin 16°

Exercise 4.2 | Q 2. (ii) | Page 84

Find the value of the following:

`sin  pi/4 cos  pi/12 + cos  pi/4 sin  pi/12`

Exercise 4.2 | Q 2. (iii) | Page 84

Find the value of the following:

cos 70° cos 10° – sin 70° sin 10°

Exercise 4.2 | Q 2. (iv) | Page 84

Find the value of the following:

cos2 15° – sin2 15°

Exercise 4.2 | Q 3 | Page 84

If sin A = `3/5`, 0 < A < `pi/2` and cos B = `(-12)/13`, π < B < `(3pi)/2`, find the values of the following:

  1. cos(A + B)
  2. sin(A – B)
  3. tan(A – B)
Exercise 4.2 | Q 4 | Page 84

If cos A = `13/14` and cos B = `1/7` where A, B are acute angles prove that A – B = `pi/3` 

Exercise 4.2 | Q 5 | Page 84

Prove that 2 tan 80° = tan 85° – tan 5°.

Exercise 4.2 | Q 6 | Page 84

If cot α = `1/2`, sec β = `(-5)/3`, where π < α < `(3pi)/2 and pi/2` < β < π, find the value of tan(α + β). State the quadrant in which α + β terminates.

Exercise 4.2 | Q 7 | Page 84

If A + B = 45°, prove that (1 + tan A) (1 + tan B) = 2 and hence deduce the value of tan 22`1/2`.

Exercise 4.2 | Q 8. (i) | Page 84

Prove that:

sin(A + 60°) + sin(A – 60°) = sin A.

Exercise 4.2 | Q 8. (ii) | Page 84

Prove that:

tan 4A tan 3A tan A + tan 3A + tan A – tan 4A = 0

Exercise 4.2 | Q 9. (i) | Page 84

If tan θ = 3 find tan 3θ

Exercise 4.2 | Q 9. (ii) | Page 84

If sin A = `12/13`, find sin 3A.

Exercise 4.2 | Q 10 | Page 84

If sin A = `3/5`, find the values of cos 3A and tan 3A.

Exercise 4.2 | Q 11 | Page 84

Prove that `(sin ("B - C"))/(cos "B" cos "C") + (sin ("C - A"))/(cos "C" cos "A") + (sin ("A - B"))/(cos "A" cos "B")` = 0

Exercise 4.2 | Q 12 | Page 85

If tan A – tan B = x and cot B – cot A = y prove that cot(A – B) = `1/x + 1/y`.

Exercise 4.2 | Q 13 | Page 85

If sin α + sin β = a and cos α + cos β = b, then prove that cos(α – β) = `(a^2 + b^2 - 2)/2`

Exercise 4.2 | Q 14 | Page 85

Find the value of tan `pi/8`.

Exercise 4.2 | Q 15 | Page 85

If tan α = `1/7`, sin β = `1/sqrt10`. Prove that α + 2β = `pi/4`  where 0 < α < `pi/2` and 0 < β < `pi/2`.

Exercise 4.3 [Pages 88 - 89]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 4 TrigonometryExercise 4.3 [Pages 88 - 89]

Exercise 4.3 | Q 1. (i) | Page 88

Express the following as the sum or difference of sine or cosine:

`sin  "A"/8  sin  (3"A")/8`

Exercise 4.3 | Q 1. (ii) | Page 88

Express the following as the sum or difference of sine or cosine:

cos(60° + A) sin(120° + A)

Exercise 4.3 | Q 1. (iii) | Page 88

Express the following as the sum or difference of sine or cosine:

`cos  (7"A")/3 sin  (5"A")/3`

Exercise 4.3 | Q 1. (iv) | Page 88

Express the following as the sum or difference of sine or cosine:

cos 7θ sin 3θ

Exercise 4.3 | Q 2. (i) | Page 88

Express the following as the product of sine and cosine.

sin A + sin 2A

Exercise 4.3 | Q 2. (ii) | Page 88

Express the following as the product of sine and cosine.

cos 2A + cos 4A

Exercise 4.3 | Q 2. (iii) | Page 88

Express the following as the product of sine and cosine.

sin 6θ – sin 2θ

Exercise 4.3 | Q 2. (iv) | Page 88

Express the following as the product of sine and cosine.

cos 2θ – cos θ

Exercise 4.3 | Q 3. (i) | Page 88

Prove that:

cos 20° cos 40° cos 80° = `1/8`

Exercise 4.3 | Q 3. (ii) | Page 88

Prove that:

tan 20° tan 40° tan 80° = `sqrt3`.

Exercise 4.3 | Q 4. (i) | Page 88

Prove that:

(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`

Exercise 4.3 | Q 4. (ii) | Page 88

Prove that:

sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A

Exercise 4.3 | Q 5. (i) | Page 88

Prove that:

sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0

Exercise 4.3 | Q 5. (ii) | Page 88

Prove that:

2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0

Exercise 4.3 | Q 6. (i) | Page 89

Prove that:

`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan  "A"/2`

Exercise 4.3 | Q 6. (ii) | Page 89

Prove that:

`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A

Exercise 4.3 | Q 7 | Page 89

Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.

Exercise 4.3 | Q 8. (i) | Page 89

Evaluate-

cos 20° + cos 100° + cos 140°

Exercise 4.3 | Q 8. (ii) | Page 89

Evaluate:

sin 50° – sin 70° + sin 10°

Exercise 4.3 | Q 9 | Page 89

If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`

Exercise 4.3 | Q 10 | Page 89

If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.

Exercise 4.3 | Q 11 | Page 89

If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.

Exercise 4.4 [Page 92]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 4 TrigonometryExercise 4.4 [Page 92]

Exercise 4.4 | Q 1. (i) | Page 92

Find the principal value of the following:

`sin^-1 (- 1/2)`

Exercise 4.4 | Q 1. (ii) | Page 92

Find the principal value of the following:

tan-1 (-1)

Exercise 4.4 | Q 1. (iii) | Page 92

Find the principal value of the following:

cosec-1 (2)

Exercise 4.4 | Q 1. (iv) | Page 92

Find the principal value of the following:

`sec^-1 (-sqrt2)`

Exercise 4.4 | Q 2. (i) | Page 92

Prove that:

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

Exercise 4.4 | Q 2. (ii) | Page 92

Prove that:

`tan^-1 (4/3) + tan^-1 (1/7) = pi/4`

Exercise 4.4 | Q 3 | Page 92

Show that `tan^-1 (1/2) + tan^-1 (2/11) = tan^-1 (3/4)`

Exercise 4.4 | Q 4 | Page 92

Solve `tan^-1 2x + tan^-1 3x = pi/4`

Exercise 4.4 | Q 5 | Page 92

Solve: tan-1 (x + 1) + tan-1 (x – 1) = `tan^-1 (4/7)`

Exercise 4.4 | Q 6. (i) | Page 92

Evaluate:

`cos[tan^-1 (3/4)]`

Exercise 4.4 | Q 6. (ii) | Page 92

Evaluate: sin`[1/2 cos^-1 (4/5)]`

Exercise 4.4 | Q 7 | Page 92

Evaluate: `cos (sin^-1 (4/5) + sin^-1 (12/13))`

Exercise 4.4 | Q 8 | Page 92

Prove that `tan^-1 (m/n) - tan^-1 ((m - n)/(m + n)) = pi/4`

Exercise 4.4 | Q 9 | Page 92

Show that `sin^-1 (- 3/5) - sin^-1 (- 8/17) = cos^-1 (84/85)`

Exercise 4.4 | Q 10 | Page 92

Express `tan^-1 [(cos x)/(1 - sin x)], - pi/2 < x < (3pi)/2` in the simplest form.

Exercise 4.5 [Pages 92 - 94]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 4 TrigonometryExercise 4.5 [Pages 92 - 94]

Choose the correct answer

Exercise 4.5 | Q 1 | Page 92

The degree measure of `pi/8` is

  • 20°60′

  • 22°30′

  • 22°60′

  • 20°30′

Exercise 4.5 | Q 2 | Page 92

The radian measure of 37°30′ is:

  • `(5pi)/24`

  • `(3pi)/24`

  • `(7pi)/24`

  • `(9pi)/24`

Exercise 4.5 | Q 3 | Page 92

If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:

  • `1/sqrt6`

  • `(-1)/sqrt6`

  • `sqrt5/sqrt6`

  • `(-sqrt5)/sqrt6`

Exercise 4.5 | Q 4 | Page 92

The value of sin 15° is:

  • `(sqrt3 + 1)/(2sqrt2)`

  • `(sqrt3 - 1)/(2sqrt2)`

  • `sqrt3/sqrt2`

  • `(sqrt3)/(2sqrt2)`

Exercise 4.5 | Q 5 | Page 92

The value of sin (-420°)

  • `sqrt3/2`

  • `- sqrt3/2`

  • `1/2`

  • `- 1/2`

Exercise 4.5 | Q 6 | Page 92

The value of cos(-480°) is:

  • `sqrt3`

  • `- sqrt3/2`

  • `1/2`

  • `(-1)/2`

Exercise 4.5 | Q 7 | Page 93

The value of sin 28° cos 17° + cos 28° sin 17°

  • `1/sqrt2`

  • 1

  • `(-1)/sqrt2`

  • 0

Exercise 4.5 | Q 8 | Page 93

The value of sin 15° cos 15° is:

  • 1

  • `1/2`

  • `sqrt3/2`

  • `1/4`

Exercise 4.5 | Q 9 | Page 93

The value of sec A sin(270° + A) is:

  • -1

  • cos2 A

  • sec2 A

  • 1

Exercise 4.5 | Q 10 | Page 93

If sin A + cos A = 1 then sin 2A is equal to:

  • 1

  • 2

  • 0

  • \[\frac{1}{2}\]

Exercise 4.5 | Q 11 | Page 93

The value of cos2 45° – sin2 45° is:

  • `sqrt3/2`

  • \[\frac{1}{2}\]

  • 0

  • `1/sqrt1`

Exercise 4.5 | Q 12 | Page 93

The value of 1 – 2 sin2 45° is:

  • 1

  • `1/2`

  • `1/4`

  • 0

Exercise 4.5 | Q 13 | Page 93

The value of 4 cos3 40° – 3 cos 40° is

  • `sqrt3/2`

  • `- 1/2`

  • `1/2`

  • `1/sqrt2`

Exercise 4.5 | Q 14 | Page 93

The value of `(2 tan 30^circ)/(1 + tan^2 30^circ)` is:

  • `1/2`

  • `1/sqrt3`

  • `sqrt3/2`

  • `sqrt3`

Exercise 4.5 | Q 15 | Page 93

If sin A = `1/2` then 4 cos3 A – 3 cos A is:

  • 1

  • 0

  • `sqrt3/2`

  • `1/sqrt2`

Exercise 4.5 | Q 16 | Page 93

The value of `(3 tan 10^circ - tan^3 10^circ)/(1 - 3 tan^2 10^circ)` is:

  • `1/sqrt3`

  • `1/2`

  • `sqrt3/2`

  • `1/sqrt2`

Exercise 4.5 | Q 17 | Page 93

The value of cosec-1 `(2/sqrt3)` is:

  • `pi/4`

  • `pi/2`

  • `pi/3`

  • `pi/6`

Exercise 4.5 | Q 18 | Page 93

sec-1 `(2/3)` + cosec-1 `(2/3)`=

  • `(- pi)/2`

  • `pi/2`

  • π

  • - π

Exercise 4.5 | Q 19 | Page 93

If α and β be between 0 and `pi/2`  and if cos(α + β) = `12/13` and sin (α – β) = `3/5` then sin 2α is:

  • `16/15`

  • 0

  • `56/65`

  • `64/65`

Exercise 4.5 | Q 20 | Page 93

If tan A = `1/2` and tan B = `1/3` then tan(2A + B) is equal to:

  • 1

  • 2

  • 3

  • 4

Exercise 4.5 | Q 21 | Page 93

tan`(pi/4 - x)` is:

  • `((1 + tan x)/(1 - tan x))`

  • `((1 - tan x)/(1 + tan x))`

  • 1 - tan x

  • 1 + tan x

Exercise 4.5 | Q 22 | Page 93

`sin (cos^-1  3/5)` is

  • `3/5`

  • `5/3`

  • `4/5`

  • `5/4`

Exercise 4.5 | Q 23 | Page 93

The value of `1/("cosec" (-45^circ))` is:

  • `(-1)/sqrt2`

  • `1/sqrt2`

  • `sqrt2`

  • `-sqrt2`

Exercise 4.5 | Q 24 | Page 94

If p sec 50° = tan 50° then p is:

  • cos 50°

  • sin 50°

  • tan 50°

  • sec 50°

Exercise 4.5 | Q 25 | Page 94

`((cos x)/(cosec x)) - sqrt(1 - sin^2x) sqrt(1 - cos^2 x)` is:

  • cos2 x – sin2 x

  • sin2 x – cos2 x

  • 1

  • 0

Miscellaneous Problems [Page 94]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 4 TrigonometryMiscellaneous Problems [Page 94]

Miscellaneous Problems | Q 1 | Page 94

Prove that:

`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x)` = cot 3x

Miscellaneous Problems | Q 2 | Page 94

Prove that `sqrt3  "cosec"  20^circ - sin 20^circ` = 4

Miscellaneous Problems | Q 3 | Page 94

Prove that cot 4x (sin 5x + sin 3x) = cot x(sin 5x - sin 3x).

Miscellaneous Problems | Q 4 | Page 94

If tan x = `3/4` and `pi < x < (3pi)/2`, then find the value of sin `x/2` and cos `x/2`.

Miscellaneous Problems | Q 5 | Page 94

Prove that `2 sin^2  (3pi)/4 + 2 cos^2  pi/4 + 2 sec^2  pi/3` = 10

Miscellaneous Problems | Q 6. (i) | Page 94

Find the value of sin 75°.

Miscellaneous Problems | Q 6. (ii) | Page 94

Find the value of tan 15°.

Miscellaneous Problems | Q 7 | Page 94

If sin A = `1/3`, sin B = `1/4` then find the value of sin (A + B) where A and B are acute angles.

Miscellaneous Problems | Q 8 | Page 94

Show that `cos^-1 (12/13) + sin^-1 (3/5) = sin^-1 (56/65)`

Miscellaneous Problems | Q 9 | Page 94

If cos (α + β) = `4/5` and sin (α - β) = `5/13` where (α + β) and (α - β) are acute, then find tan 2α.

Miscellaneous Problems | Q 10 | Page 94

Express `tan^-1 ((cos x - sin x)/(cos x + sin x))`, 0 < x < π in the simplest form.

Chapter 4: Trigonometry

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Problems
Class 11th Business Mathematics and Statistics Answers Guide - Shaalaa.com

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 4 - Trigonometry

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 4 (Trigonometry) 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 Tamil Nadu Board of Secondary Education Class 11th Business Mathematics and Statistics Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11th Business Mathematics and Statistics Answers Guide chapter 4 Trigonometry are Trigonometric Ratios, Trigonometric Ratios of Compound Angles, Transformation Formulae, Inverse Trigonometric Functions.

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Get the free view of chapter 4 Trigonometry Class 11th extra questions for Class 11th Business Mathematics and Statistics Answers Guide and can use Shaalaa.com to keep it handy for your exam preparation

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