Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

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

## Chapter 4: Trigonometry

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Problems
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 221/2.

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]

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

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

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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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