Tamil Nadu Board of Secondary EducationHSC Science Class 11th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 3 - Trigonometry [Latest edition]

## Chapter 3: Trigonometry

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9Exercise 3.10Exercise 3.11Exercise 3.12
Exercise 3.1 [Page 92]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.1 [Page 92]

Exercise 3.1 | Q 1. (i) | Page 92

Identify the quadrant in which an angle given measure lies
25°

Exercise 3.1 | Q 1. (ii) | Page 92

Identify the quadrant in which an angle given measure lies
825°

Exercise 3.1 | Q 1. (iii) | Page 92

Identify the quadrant in which an angle given measure lies
– 55°

Exercise 3.1 | Q 1. (iv) | Page 92

Identify the quadrant in which an angle given measure lies
328°

Exercise 3.1 | Q 1. (v) | Page 92

Identify the quadrant in which an angle given measure lies
– 230°

Exercise 3.1 | Q 2. (i) | Page 92

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
395°

Exercise 3.1 | Q 2. (ii) | Page 92

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
525°

Exercise 3.1 | Q 2. (iii) | Page 92

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
1150°

Exercise 3.1 | Q 2. (iv) | Page 92

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 270°

Exercise 3.1 | Q 2. (v) | Page 92

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 450°

Exercise 3.1 | Q 3 | Page 92

If a cos θ − b sin θ = c, show that a sin θ + b cos θ = +-  sqrt("a"^2 + "b"^2 - "c"^2)

Exercise 3.1 | Q 4 | Page 92

If sin θ + cos θ = m, show that cos6θ + sin6θ = (4 - 3("m"^2 - 1)^2)/4, where m2 ≤ 2

Exercise 3.1 | Q 5. (i) | Page 92

If (cos^4α)/(cos^2β) + (sin^4α)/(sin^2β) = 1, prove that sin4α + sin4β = 2 sin2α sin2β

Exercise 3.1 | Q 5.. (ii) | Page 92

If (cos^4alpha)/(cos^2beta) + (sin^4alpha)/(sin^2beta) = 1, prove that (cos^4beta)/(cos^2alpha) + (sin^4beta)/(sin^2alpha) = 1

Exercise 3.1 | Q 6 | Page 92

If y = (2sinalpha)/(1 + cosalpha + sinalpha), then prove that (1 - cosalpha + sinalpha)/(1 + sinalpha) = y

Exercise 3.1 | Q 7 | Page 92

If x = sum_("n" = 0)^oo cos^(2"n") theta, y = sum_("n" = 0)^oo sin^(2"n") theta and z = sum_("n" = 0)^oo cos^(2"n") theta, sin^(2"n") theta, 0 < theta < pi/2, then show that xyz = x + y + z. [Hint: Use the formula 1 + x + x2 + x3 + . . . = 1/(1 - x), where |x| < 1]

Exercise 3.1 | Q 8 | Page 92

If tan2 θ = 1 – k2, show that sec θ + tan3 θ cosec θ = (2 – k2)3/2. Also, find the values of k for which this result holds

Exercise 3.1 | Q 9 | Page 92

If sec θ + tan θ = p, obtain the values of sec θ, tan θ and sin θ in terms of p

Exercise 3.1 | Q 10 | Page 92

If cot θ(1 + sin θ) = 4m and cot θ(1 – sin θ) = 4n then prove that (m2 – n2)2 = m

Exercise 3.1 | Q 11 | Page 92

If cosec θ – sin θ = a3 and sec θ – cos θ = b3, then prove that a2b2 (a2 + b2) = 1

Exercise 3.1 | Q 12 | Page 92

Eliminate θ from the equations a sec θ – c tan θ = b and b sec θ + d tan θ = c

Exercise 3.2 [Pages 95 - 96]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.2 [Pages 95 - 96]

Exercise 3.2 | Q 1. (i) | Page 95

Express the following angles in radian measure:
30°

Exercise 3.2 | Q 1. (ii) | Page 95

Express the following angles in radian measure:
135°

Exercise 3.2 | Q 1. (iii) | Page 95

Express the following angles in radian measure:
– 205°

Exercise 3.2 | Q 1. (iv) | Page 95

Express the following angles in radian measure:
150°

Exercise 3.2 | Q 1. (v) | Page 95

Express the following angles in radian measure:
330°

Exercise 3.2 | Q 2. (i) | Page 95

Find the degree measure corresponding to the following radian measures
pi/3

Exercise 3.2 | Q 2. (ii) | Page 95

Find the degree measure corresponding to the following radian measures
pi/9

Exercise 3.2 | Q 2. (iii) | Page 95

Find the degree measure corresponding to the following radian measures
(2pi)/5

Exercise 3.2 | Q 2. (iv) | Page 95

Find the degree measure corresponding to the following radian measures
(7pi)/3

Exercise 3.2 | Q 2. (v) | Page 95

Find the degree measure corresponding to the following radian measures
(10pi)/9

Exercise 3.2 | Q 3 | Page 95

What must be the radius of a circular running path, around which an athlete must run 5 times in order to describe 1 km?

Exercise 3.2 | Q 4 | Page 95

In a circle of diameter 40 cm, a chord is of length 20 cm. Find the length of the minor arc of the chord

Exercise 3.2 | Q 5 | Page 95

Find the degree measure of the angle subtended at the centre of circle of radius 100 cm by an arc of length 22 cm

Exercise 3.2 | Q 6 | Page 95

What is the length of the arc intercepted by a central angle of measure 41° in a circle of radius 10 ft?

Exercise 3.2 | Q 7 | Page 95

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii

Exercise 3.2 | Q 8 | Page 96

The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius. Express the angle of the sector in degrees, minutes and seconds

Exercise 3.2 | Q 9 | Page 96

An airplane propeller rotates 1000 times per minute. Find the number of degrees that a point on the edge of the propeller will rotate in 1 second

Exercise 3.2 | Q 10 | Page 96

A train is moving on a circular track of 1500 m radius at the rate of 66 km/hr. What angle will it turn in 20 seconds?

Exercise 3.2 | Q 11 | Page 96

A circular metallic plate of radius 8 cm and thickness 6 mm is melted and molded into a pie (a sector of the circle with thickness) of radius 16 cm and thickness 4 mm. Find the angle of the sector.

Exercise 3.3 [Page 104]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.3 [Page 104]

Exercise 3.3 | Q 1. (i) | Page 104

Find the values of sin(480°)

Exercise 3.3 | Q 1. (ii) | Page 104

Find the values of sin (– 1110°)

Exercise 3.3 | Q 1. (iii) | Page 104

Find the values of cos(300°)

Exercise 3.3 | Q 1. (iv) | Page 104

Find the values of tan(1050°)

Exercise 3.3 | Q 1. (v) | Page 104

Find the values of cot(660°)

Exercise 3.3 | Q 1. (vi) | Page 104

Find the values of tan ((19pi)/3)

Exercise 3.3 | Q 1. (vii) | Page 104

Find the values of sin (-(11pi)/3)

Exercise 3.3 | Q 2 | Page 104

(5/7, (2sqrt(6))/7) is a point on the terminal side of an angle θ in standard position. Determine the six trigonometric function values of angle θ

Exercise 3.3 | Q 3. (i) | Page 104

Find the value of the trigonometric functions for the following:
cos θ = - 1/2, θ lies in the III quadrant

Exercise 3.3 | Q 3. (ii) | Page 104

Find the value of the trigonometric functions for the following:
cos θ = 2/3, θ lies in the I quadrant

Exercise 3.3 | Q 3. (iii) | Page 104

Find the value of the trigonometric functions for the following:
cos θ = - 2/3, θ lies in the IV quadrant

Exercise 3.3 | Q 3. (iv) | Page 104

Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant

Exercise 3.3 | Q 3. (v) | Page 104

Find the value of the trigonometric functions for the following:
sec θ = 13/5, θ lies in the IV quadrant

Exercise 3.3 | Q 4 | Page 104

Prove that (cot(180^circ + theta) sin(90^circ - theta) cos(- theta))/(sin(270^circ + theta) tan(- theta) "cosec"(360^circ + theta)) = cos2θ cotθ

Exercise 3.3 | Q 5 | Page 104

Find all the angles between 0° and 360° which satisfy the equation sin2θ = 3/4

Exercise 3.3 | Q 6 | Page 104

Show that sin^2  pi/18 + sin^2  pi/9 + sin^2  (7pi)/18 + sin^2  (4pi)/9 = 2

Exercise 3.4 [Pages 109 - 110]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.4 [Pages 109 - 110]

Exercise 3.4 | Q 1. (i) | Page 109

If sin x = 15/17 and cos y = 12/13, 0 < x < pi/2, 0 < y < pi/2 find the value of sin(x + y)

Exercise 3.4 | Q 1. (ii) | Page 109

If sin x = 15/17 and cos y = 12/13, 0 < x < pi/2, 0 < y < pi/2, find the value of cos(x − y)

Exercise 3.4 | Q 1. (iii) | Page 109

If sin x = 15/17 and cos y = 12/13, 0 < x < pi/2, 0 < y < pi/2, find the value of tan(x + y)

Exercise 3.4 | Q 2. (i) | Page 109

If sin A = 3/5 and cos B = 9/41 0 < "A" < pi/2, 0 < "B" < pi/2, find the value of sin(A + B)

Exercise 3.4 | Q 2. (ii) | Page 109

If sin A = 3/5 and cos B = 9/41, 0 < "A" < pi/2, 0 < "B" < pi/2, find the value of cos(A – B)

Exercise 3.4 | Q 3 | Page 109

Find cos(x − y), given that cos x = - 4/5 with pi < x < (3pi)/2  and sin y = - 24/25 with pi < y < (3pi)/2

Exercise 3.4 | Q 4 | Page 109

Find sin(x – y), given that sin x = 8/17 with 0 < x < pi/2, and cos y = - 24/25, x < y < (3pi)/2

Exercise 3.4 | Q 5. (i) | Page 109

Find the value of cos 105°

Exercise 3.4 | Q 5. (ii) | Page 109

Find the value of sin 105°

Exercise 3.4 | Q 5. (iii) | Page 109

Find the value of tan (7pi)/12

Exercise 3.4 | Q 6. (i) | Page 109

Prove that cos(30° + x) = (sqrt(3) cos x - sin x)/2

Exercise 3.4 | Q 6. (ii) | Page 109

Prove that cos(π + θ) = − cos θ

Exercise 3.4 | Q 6. (iii) | Page 109

Prove that sin(π + θ) = − sin θ.

Exercise 3.4 | Q 7 | Page 109

Find a quadratic equation whose roots are sin 15° and cos 15°

Exercise 3.4 | Q 8 | Page 109

Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = pi/2

Exercise 3.4 | Q 9. (i) | Page 109

Prove that sin(45° + θ) – sin(45° – θ) = sqrt(2) sin θ

Exercise 3.4 | Q 9. (ii) | Page 109

Prove that sin(30° + θ) + cos(60° + θ) = cos θ

Exercise 3.4 | Q 10 | Page 109

If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y

Exercise 3.4 | Q 11 | Page 109

Prove that sin 105° + cos 105° = cos 45°

Exercise 3.4 | Q 12 | Page 109

Prove that sin 75° – sin 15° = cos 105° + cos 15°

Exercise 3.4 | Q 13 | Page 109

Show that tan 75° + cot 75° = 4

Exercise 3.4 | Q 14 | Page 109

Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)

Exercise 3.4 | Q 15 | Page 109

Prove that sin(n + 1) θ sin(n – 1) θ + cos(n + 1) θ cos(n – 1)θ = cos 2θ, n ∈ Z

Exercise 3.4 | Q 16 | Page 109

If x cos θ = y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3). find the value of xy + yz + zx

Exercise 3.4 | Q 17. (i) | Page 109

Prove that sin(A + B) sin(A – B) = sin2A – sin2B

Exercise 3.4 | Q 17. (ii) | Page 109

Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A

Exercise 3.4 | Q 17. (iii) | Page 109

Prove that sin2(A + B) – sin2(A – B) = sin2A sin2B

Exercise 3.4 | Q 17. (iv) | Page 109

Prove that cos 8θ cos 2θ = cos25θ – sin2

Exercise 3.4 | Q 18 | Page 109

Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)

Exercise 3.4 | Q 19 | Page 109

If cos(α – β) + cos(β – γ) + cos(γ – α) = - 3/2, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ = 0

Exercise 3.4 | Q 20. (i) | Page 110

Show that tan(45° + A) =  (1 + tan"A")/(1 - tan"A")

Exercise 3.4 | Q 20. (ii) | Page 110

Show that tan(45° − A) = (1 - tan "A")/(1 + tan "A")

Exercise 3.4 | Q 21 | Page 110

Prove that cot(A + B) = (cot "A" cot "B" - 1)/(cot "A" + cot "B")

Exercise 3.4 | Q 22 | Page 110

If tan x = "n"/("n" + 1) and tan y = 1/(2"n" + 1), find tan(x + y)

Exercise 3.4 | Q 23 | Page 110

Prove that tan(pi/4 + theta) tan((3pi)/4 + theta) = – 1

Exercise 3.4 | Q 24 | Page 110

Find the value of tan(α + β), given that cot α = 1/2, α ∈ (pi, (3pi)/2) and sec β = - 5/3 β ∈ (pi/2, pi)

Exercise 3.4 | Q 25 | Page 110

If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = ("k" - 1)/("k" + 1) sin α

Exercise 3.5 [Pages 117 - 118]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.5 [Pages 117 - 118]

Exercise 3.5 | Q 1. (i) | Page 117

Find the value of cos 2A, A lies in the first quadrant, when cos A = 15/17

Exercise 3.5 | Q 1. (ii) | Page 117

Find the value of cos 2A, A lies in the first quadrant, when sin A = 4/5

Exercise 3.5 | Q 1. (iii) | Page 117

Find the value of cos 2A, A lies in the first quadrant, when tan A  16/63

Exercise 3.5 | Q 2. (i) | Page 118

If θ is an acute angle, then find sin (pi/4 - theta/2), when sin θ = 1/25

Exercise 3.5 | Q 2. (ii) | Page 118

If θ is an acute angle, then find cos (pi/4 + theta/2), when sin θ = 8/9

Exercise 3.5 | Q 3 | Page 118

If cos θ = 1/2 ("a" + 1/"a"), show that cos 3θ = 1/2 ("a"^3 + 1/"a"^3)

Exercise 3.5 | Q 4 | Page 118

Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ

Exercise 3.5 | Q 5 | Page 118

Prove that sin 4α = 4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2

Exercise 3.5 | Q 6 | Page 118

If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2

Exercise 3.5 | Q 7 | Page 118

Prove that (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ..... (1 + tan 44°) is a multiple of 4

Exercise 3.5 | Q 8 | Page 118

Prove that tan (pi/4 + theta) - tan(pi/4 - theta) = 2 tan 2θ

Exercise 3.5 | Q 9 | Page 118

Show that cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)

Exercise 3.5 | Q 10 | Page 118

Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ) = tan 2nθ

Exercise 3.5 | Q 11 | Page 118

Prove that 32(sqrt(3)) sin  pi/48  cos  pi/48  cos  pi/24  cos  pi/12  cos  pi/6 = 3

Exercise 3.6 [Pages 121 - 122]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.6 [Pages 121 - 122]

Exercise 3.6 | Q 1. (i) | Page 121

Express the following as a sum or difference
sin 35° cos 28°

Exercise 3.6 | Q 1. (ii) | Page 121

Express the following as a sum or difference
sin 4x cos 2x

Exercise 3.6 | Q 1. (iii) | Page 121

Express the following as a sum or difference
2 sin 10θ  cos 2θ

Exercise 3.6 | Q 1. (iv) | Page 121

Express the following as a sum or difference
cos 5θ  cos 2θ

Exercise 3.6 | Q 1. (v) | Page 121

Express the following as a sum or difference
sin 5θ  sin 4θ

Exercise 3.6 | Q 2. (i) | Page 121

Express the following as a product
sin 75° sin 35°

Exercise 3.6 | Q 2. (ii) | Page 121

Express the following as a product
cos 65° + cos 15°

Exercise 3.6 | Q 2. (iii) | Page 121

Express the following as a product
sin 50° + sin 40°

Exercise 3.6 | Q 2. (iv) | Page 121

Express the following as a product
cos 35° – cos 75°

Exercise 3.6 | Q 3 | Page 121

Show that sin 12° sin 48° sin 54° = 1/8

Exercise 3.6 | Q 4 | Page 121

Show that cos  pi/15  cos  (2pi)/15  cos  (3pi)/15  cos  (4pi)/15  cos  (5pi)/15  cos  (6pi)/15  cos  (7pi)/15 = 1/128

Exercise 3.6 | Q 5 | Page 121

Show that (sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x) = tan 2x

Exercise 3.6 | Q 6 | Page 121

Show that ((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta)) = 1

Exercise 3.6 | Q 7 | Page 121

Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)

Exercise 3.6 | Q 8 | Page 121

Prove that (sin 4x + sin 2x)/(cos 4x + cos 2x) = tan 3x

Exercise 3.6 | Q 9 | Page 121

Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x

Exercise 3.6 | Q 10 | Page 122

Prove that sin  theta/2 sin  (7theta)/2 + sin  (3theta)/2 sin  (11theta)/2 =  sin 2θ sin 5θ

Exercise 3.6 | Q 11 | Page 122

Prove that cos(30° – A) cos(30° + A) + cos(45° – A) cos(45° + A) = cos 2"A" + 1/4

Exercise 3.6 | Q 12 | Page 122

Prove that (sin x + sin 3x + sin 5x + sin 7x)/(cos x + cos x + cos 5x  cos 7x) = tan 4x

Exercise 3.6 | Q 13 | Page 122

Prove that (sin(4"A" - 2"B") + sin(4"B" - 2"A"))/(cos(4"A" - 2"B") + cos(4"B" - 2"A")) = tan(A + B)

Exercise 3.6 | Q 14 | Page 122

Show that cot(A + 15°) – tan(A – 15°) = (4cos2"A")/(1 + 2 sin2"A")

Exercise 3.7 [Page 124]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.7 [Page 124]

Exercise 3.7 | Q 1. (i) | Page 124

If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C

Exercise 3.7 | Q 1. (ii) | Page 124

If A + B + C = 180°, prove that cos A + cos B − cos C = - 1 + 4cos  "A"/2 cos  "B"/2 sin  "C"/2

Exercise 3.7 | Q 1. (iii) | Page 124

If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C

Exercise 3.7 | Q 1. (iv) | Page 124

If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C

Exercise 3.7 | Q 1. (v) | Page 124

If A + B + C = 180°, prove that tan  "A"/2  tan  "B"/2 + tan  "B"/2 tan  "C"/2 + tan  "C"/2 tan  "A"/2 = 1

Exercise 3.7 | Q 1. (vi) | Page 124

If A + B + C = 180°, prove that sin A + sin B + sin C = 4 cos  "A"/2 cos  "B"/2 cos  "C"/2

Exercise 3.7 | Q 1. (vii) | Page 124

If A + B + C = 180°, prove that sin(B + C − A) + sin(C + A − B) + sin(A + B − C) = 4 sin A sin B sin C

Exercise 3.7 | Q 2 | Page 124

If A + B + C = 2s, then prove that sin(s – A) sin(s – B)+ sin s  sin(s – C) = sin A sin B

Exercise 3.7 | Q 3 | Page 124

If x + y + z = xyz, then prove that (2x)/(1 - x^2) + (2y)/(1 - y^2) + (2z)/(1 - z^2) = (2x)/(1 - x^2) (2y)/(1 - y^2) (2z)/(1 - z^2)

Exercise 3.7 | Q 4. (i) | Page 124

If A + B + C = pi/2, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C

Exercise 3.7 | Q 4. (ii) | Page 124

If A + B + C = pi/2, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C

Exercise 3.7 | Q 5. (i) | Page 124

If ∆ABC is a right triangle and if ∠A = pi/2 then prove that cosB + cosC = 1

Exercise 3.7 | Q 5. (ii) | Page 124

If ∆ABC is a right triangle and if ∠A = pi/2 then prove that sinB + sinC = 1

Exercise 3.7 | Q 5. (iii) | Page 124

If ∆ABC is a right triangle and if ∠A = pi/2 then prove that cos B – cos C = - 1 + 2sqrt(2) cos  "B"/2  sin  "C"/2

Exercise 3.8 [Page 133]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.8 [Page 133]

Exercise 3.8 | Q 1. (i) | Page 133

Find the principal solution and general solution of the following:
sin θ = -1/sqrt(2)

Exercise 3.8 | Q 1. (ii) | Page 133

Find the principal solution and general solution of the following:
cot θ = sqrt(3)

Exercise 3.8 | Q 1. (iii) | Page 133

Find the principal solution and general solution of the following:
tan θ = - 1/sqrt(3)

Exercise 3.8 | Q 2. (i) | Page 133

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

sin4x = sin2x

Exercise 3.8 | Q 2. (ii) | Page 133

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

2 cos2x + 1 = – 3 cos x

Exercise 3.8 | Q 2. (iii) | Page 133

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

2 sin2x + 1 = 3 sin x

Exercise 3.8 | Q 2. (iv) | Page 133

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

cos 2x = 1 − 3 sin x

Exercise 3.8 | Q 3. (i) | Page 133

Solve the following equations:
sin 5x − sin x = cos 3

Exercise 3.8 | Q 3. (ii) | Page 133

Solve the following equations:
2 cos2θ + 3 sin θ – 3 = θ

Exercise 3.8 | Q 3. (iii) | Page 133

Solve the following equations:
cos θ + cos 3θ = 2 cos 2θ

Exercise 3.8 | Q 3. (iv) | Page 133

Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0

Exercise 3.8 | Q 3. (v) | Page 133

Solve the following equations:
sin 2θ – cos 2θ – sin θ + cos θ = θ

Exercise 3.8 | Q 3. (vi) | Page 133

Solve the following equations:
sin θ + cos θ = sqrt(2)

Exercise 3.8 | Q 3. (vii) | Page 133

Solve the following equations:
sin theta + sqrt(3) cos theta = 1

Exercise 3.8 | Q 3. (viii) | Page 133

Solve the following equations:
cot θ + cosec θ = sqrt(3)

Exercise 3.8 | Q 3. (ix) | Page 133

Solve the following equations:
tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)

Exercise 3.8 | Q 3. (x) | Page 133

Solve the following equations:
cos 2θ = (sqrt(5) + 1)/4

Exercise 3.8 | Q 3. (xi) | Page 133

Solve the following equations:
2cos 2x – 7 cos x + 3 = 0

Exercise 3.9 [Pages 142 - 143]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.9 [Pages 142 - 143]

Exercise 3.9 | Q 1 | Page 142

In a ∆ABC, if sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C")) prove that a2, b2, C2 are in Arithmetic Progression

Exercise 3.9 | Q 2 | Page 142

The angles of a triangle ABC, are in Arithmetic Progression and if b : c = sqrt(3) : sqrt(2), find ∠A

Exercise 3.9 | Q 3 | Page 143

In a ∆ABC, if cos C = sin "A"/(2sin"B") show that the triangle is isosceles

Exercise 3.9 | Q 4 | Page 143

In a ∆ABC, prove that sin "B"/sin "C" = ("c" - "a"cos "B")/("b" - "a" cos"C")

Exercise 3.9 | Q 5 | Page 143

In an ∆ABC, prove that a cos A + b cos B + c cos C = 2a sin B sin C

Exercise 3.9 | Q 6 | Page 143

In a ∆ABC, ∠A = 60°. Prove that b + c = 2"a" cos (("B" - "C")/2)

Exercise 3.9 | Q 7. (i) | Page 143

In an ∆ABC, prove the following, "a"sin ("A"/2 + "B") = ("b" + "c") sin  "A"/2

Exercise 3.9 | Q 7. (ii) | Page 143

In a ∆ABC, prove the following, a(cos B + cos C) = 2("b" + "c") sin^2  "A"/2

Exercise 3.9 | Q 7. (iii) | Page 143

In a ∆ABC, prove the following, ("a"^2 - "c"^2)/"b"^2 = (sin ("A" - "C"))/(sin("A" + "C"))

Exercise 3.9 | Q 7. (iv) | Page 143

In a ∆ABC, prove the following, ("a"sin("B" - "C"))/("b"^2 - "c"^2) = ("b"sin("C" - "A"))/("c"^2 - "a"^2) = ("c"sin("A" - "B"))/("a"^2 - "b"^2)

Exercise 3.9 | Q 7. (v) | Page 143

In a ∆ABC, prove the following, ("a"+ "b")/("a" - "b") = tan(("A" + "B")/2) cot(("A" - "B")/2)

Exercise 3.9 | Q 8 | Page 143

In a ∆ABC, prove that (a2 – b2 + c2) tan B = (a2 + b2 – c2) tan C

Exercise 3.9 | Q 9 | Page 143

An Engineer has to develop a triangular shaped park with a perimeter 120 m in a village. The park to be developed must be of maximum area. Find out the dimensions of the park

Exercise 3.9 | Q 10 | Page 143

A rope of length 42 m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so formed

Exercise 3.9 | Q 11. (i) | Page 143

Derive Projection formula from Law of sines

Exercise 3.9 | Q 11. (ii) | Page 143

Derive Projection formula from Law of cosines

Exercise 3.10 [Pages 146 - 147]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.10 [Pages 146 - 147]

Exercise 3.10 | Q 1 | Page 146

Determine whether the following measurements produce one triangle, two triangles or no triangle:
∠B = 88°, a = 23, b = 2. Solve if solution exists

Exercise 3.10 | Q 2 | Page 146

If the sides of a ∆ABC are a = 4, b = 6 and c = 8, then show that 4 cos B + 3 cos C = 2

Exercise 3.10 | Q 3 | Page 146

In a ∆ABC, if a = sqrt(3) - 1, b = sqrt(3) + 1 and C = 60° find the other side and other two angles

Exercise 3.10 | Q 4 | Page 146

In any ∆ABC, prove that the area ∆ = ("b"^2 + "c"^2 - "a"^2)/(4 cot "A")

Exercise 3.10 | Q 5 | Page 146

In a ∆ABC, if a = 12 cm, b = 8 cm and C = 30°, then show that its area is 24 sq.cm

Exercise 3.10 | Q 6 | Page 146

In a ∆ABC, if a = 18 cm, b = 24 cm and c = 30 cm, then show that its area is 216 sq.cm

Exercise 3.10 | Q 7 | Page 146

Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the eastern direction are 30° and 45° respectively. If A and B stand 5km apart, find the distance of the intruder from B

Exercise 3.10 | Q 8 | Page 146

A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point P, he finds the distance to the eastern-most point of the pond to be 8 km, while the distance to the westernmost point from P to be 6 km. If the angle between the two lines of sight is 60°, find the width of the pond

Exercise 3.10 | Q 9 | Page 147

Two Navy helicopters A and B are flying over the Bay of Bengal at saine altitude from sea level to search a missing boat. Pilots of both the helicopters sight the boat at the same time while they are apart 10 km from each other. If the distance of the boat from A is 6 km and if the line segment AB subtends 60° at the boat, find the distance of the boat from B

Exercise 3.10 | Q 10 | Page 147

A straight tunnel is to be made through a mountain. A surveyor observes the two extremities A and B of the tunnel to be built from a point P in front of the mountain. If AP = 3 km, BP = 5 km, and ∠APB = 120°, then find the length of the tunnel to be built

Exercise 3.10 | Q 11 | Page 147

A farmer wants to purchase a triangular-shaped land with sides 120 feet and 60 feet and the angle included between these two sides is 60°. If the land costs Rs.500 per square feet, find the amount he needed to purchase the land. Also, find the perimeter of the land

Exercise 3.10 | Q 12 | Page 147

A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 30°. If after 100 km, the target has an angle of depression of 45°, how far is the target from the fighter jet at that instant?

Exercise 3.10 | Q 13 | Page 147

A plane is 1 km from one landmark and 2 km from another. From the plane’s point of view, the land between them subtends an angle of 45°. How far apart are the landmarks?

Exercise 3.10 | Q 14 | Page 147

A man starts his morning walk at a point A reaches two points B and C and finally back to A such that ∠A = 60° and ∠B = 45°, AC = 4 km in the ∆ABC. Find the total distance he covered during his morning walk

Exercise 3.10 | Q 15 | Page 147

Two vehicles leave the same place P at the same time moving along two different roads. One vehicle moves at an average speed of 60 km/hr and the other vehicle moves at an average speed of 80 km/hr. After half an hour the vehicle reaches destinations A and B. If AB subtends 60° at the initial point P, then find AB

Exercise 3.10 | Q 16 | Page 147

Suppose that a satellite in space, an earth station, and the centre of earth all lie in the same plane. Let r be the radius of earth and R he the distance from the centre of earth to the satellite. Let d be the distance from the earth station to the satellite. Let 30° be the angle of elevation from the earth station to the satellite, If the line segment connecting the earth station and satellite subtends angle α at the centre of earth then prove that d = "R"sqrt(1 + ("r"/"R")^2 - 2 ("r"/"R") cos alpha)

Exercise 3.11 [Page 149]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.11 [Page 149]

Exercise 3.11 | Q 1. (i) | Page 149

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

Exercise 3.11 | Q 1. (ii) | Page 149

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

Exercise 3.11 | Q 1. (iii) | Page 149

Find the principal value of cosec–1(– 1)

Exercise 3.11 | Q 1. (iv) | Page 149

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

Exercise 3.11 | Q 1. (v) | Page 149

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

Exercise 3.11 | Q 2 | Page 149

A man standing directly opposite to one side of a road of width x meter views a circular shaped traffic green signal of diameter ‘a’ meter on the other side of the road. The bottom of the green signal Is ‘b’ meter height from the horizontal level of viewer’s eye. If ‘a’ denotes the angle subtended by the diameter of the green signal at the viewer’s eye, then prove that α = tan^-1 (("a" + "b")/x) - tan^-1 ("b"/x)

Exercise 3.12 [Pages 150 - 151]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 3 TrigonometryExercise 3.12 [Pages 150 - 151]

#### MCQ

Exercise 3.12 | Q 1 | Page 150

Choose the correct alternative:
1/(cos 80^circ) - sqrt(3)/(sin 80^circ) =

• sqrt(2)

• sqrt(3)

• 2

• 4

Exercise 3.12 | Q 2 | Page 150

Choose the correct alternative:
If cos 28° + sin 28° = k3, then cos 17° is equal to

• "k"^3/sqrt(2)

• - "k"^3/sqrt(2)

•  +- "k"^3/sqrt(2)

• - "k"^3/sqrt(3)

Exercise 3.12 | Q 3 | Page 150

Choose the correct alternative:
The maximum value of 4sin^2x + 3cos^2x + sin   x/2 + cos  x/2 is

• 4 + sqrt(2)

• 3 + sqrt(2)

• 9

• 4

Exercise 3.12 | Q 4 | Page 150

Choose the correct alternative:
(1 + cos  pi/8) (1 + cos  (3pi)/8) (1 + cos  (5pi)/8) (1 + cos  (7pi)/8) =

• 1/8

• 1/2

• 1/sqrt(3)

• 1/sqrt(2)

Exercise 3.12 | Q 5 | Page 150

Choose the correct alternative:
If pi < 2theta < (3pi)/2, then sqrt(2 + sqrt(2 + 2cos4theta) equals to

• −2 cos θ

• −2 sin θ

• 2 cos θ

• 2 sin θ

Exercise 3.12 | Q 6 | Page 150

Choose the correct alternative:
If tan 40° = λ, then (tan 140^circ - tan 130^circ)/(1 + tan 140^circ *  tan 130^circ) =

• (1 - lambda^2)/lambda

• (1 + lambda^2)/lambda

• (1 + lambda^2)/(2lambda)

• (1 - lambda^2)/(2lambda)

Exercise 3.12 | Q 7 | Page 150

Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =

• 0

• 1

• −1

• 89

Exercise 3.12 | Q 8 | Page 150

Choose the correct alternative:
Let fk(x) = 1/"k" [sin^"k" x + cos^"k" x] where x ∈ R and k ≥ 1. Then f4(x) − f6(x) =

• 1/4

• 1/12

• 1/6

• 1/3

Exercise 3.12 | Q 9 | Page 150

Choose the correct alternative:
Which of the following is not true?

• sin θ =  - 3/4

• cos θ = −1

• tan θ = 25

• sec θ = 1/4

Exercise 3.12 | Q 10 | Page 150

Choose the correct alternative:
cos 2θ cos 2ϕ+ sin2 (θ – ϕ) – sin2 (θ + ϕ) is equal to

• sin 2 (θ + Φ)

• cos 2 (8 + Φ)

• sin 2 (θ – Φ)

• cos 2(θ – Φ)

Exercise 3.12 | Q 11 | Page 150

Choose the correct alternative:
(sin("A" - "B"))/(cos"A" cos"B") + (sin("B" - "C"))/(cos"B" cos"C") + (sin("C" - "A"))/(cos"C" cos"A") is

• sin A + sin B + sin C

• 1

• 0

• cos A + cos B + cos C

Exercise 3.12 | Q 12 | Page 150

Choose the correct alternative:
If cos pθ + cos qθ = 0 and if p ≠ q, then θ is equal to (n is any integer)

• (pi(3"n" + 1))/("p" - "q")

• (pi(2"n" + 1))/("p" +- "q")

• (pi("n" +- 1))/("p" +- "q")

• (pi("n" + 2))/("p" + "q")

Exercise 3.12 | Q 13 | Page 151

Choose the correct alternative:
If tan α and tan β are the roots of x2 + ax + b = 0 then (sin(alpha + beta))/(sin alpha sin beta) is equal to

• "b"/"a"

• "a"/"b"

• - "a"/"b"

• - "b"/"a"

Exercise 3.12 | Q 14 | Page 151

Choose the correct alternative:
In a triangle ABC, sin2A + sin2B + sin2C = 2, then the triangle is

• equilateral triangle

• isosceles triangle

• right triangle

• scalene triangle

Exercise 3.12 | Q 15 | Page 151

Choose the correct alternative:
If f(θ) = |sin θ| + |cos θ| , θ ∈ R, then f(θ) is in the interval

• [0, 2]

• [1, sqrt(2)]

• [1, 2]

• [0, 1]

Exercise 3.12 | Q 16 | Page 151

Choose the correct alternative:
(cos 6x + 6 cos 4x + 15cos x + 10)/(cos 5x + 5cs 3x + 10 cos x) is equal to

• cos 2x

• cos x

• cos 3x

• 2 cos x

Exercise 3.12 | Q 17 | Page 151

Choose the correct alternative:
The triangle of maximum area with constant perimeter 12m

• is an equilateral triangle with side 4m

• is an isosceles triangle with sides 2m, 5m, 5m

• is a triangle with sides 3m, 4m, 5m

• Does not exist

Exercise 3.12 | Q 18 | Page 151

Choose the correct alternative:
A wheel is spinning at 2 radians/second. How many seconds will it take to make 10 complete rotations?

• 10π seconds

• 20π seconds

• 5π seconds

• 15π seconds

Exercise 3.12 | Q 19 | Page 151

Choose the correct alternative:
If sin α + cos α = b, then sin 2α is equal to

• b2 − 1, if "b" ≤ sqrt(2)

• b2 − 1, if "b" > sqrt(2)

• b2 − 1, if b ≥ 1

• b2 − 1, if "b" ≥ sqrt(2)

Exercise 3.12 | Q 20 | Page 151

Choose the correct alternative:
In a ∆ABC, if
(i) sin  "A"/2 sin  "B"/2 sin  "C"/2 > 0`
(ii) sin A sin B sin C > 0 then

• Both (i) and (ii) are true

• Only (i) is true

• Only (ii) is true

• Neither (i) nor (ii) is true

## Chapter 3: Trigonometry

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9Exercise 3.10Exercise 3.11Exercise 3.12

## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 3 - Trigonometry

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 3 (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 Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 3 Trigonometry are Introduction of Trigonometry, A Recall of Basic Results, Radian Measure, Trigonometric Functions and Their Properties, Trigonometric Equations, Properties of Triangle, Application to Triangle, Inverse Trigonometric Functions.

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