# NCERT solutions for Mathematics Exemplar Class 11 chapter 3 - Trigonometric Functions [Latest edition]

## Chapter 3: Trigonometric Functions

Solved ExamplesExercise
Solved Examples [Pages 39 - 51]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 3 Trigonometric Functions Solved Examples [Pages 39 - 51]

Solved Examples | Q 1 | Page 39

A circular wire of radius 3 cm is cut and bent so as to lie along the circumference of a hoop whose radius is 48 cm. Find the angle in degrees which is subtended at the centre of hoop.

Solved Examples | Q 2 | Page 40

If A = cos2θ + sin4θ for all values of θ, then prove that 3/4 ≤ A ≤ 1.

Solved Examples | Q 3 | Page 40

Find the value of sqrt(3) cosec 20° – sec 20°

Solved Examples | Q 4 | Page 41

If θ lies in the second quadrant, then show that sqrt((1 - sin theta)/(1 + sin theta)) + sqrt((1 + sin theta)/(1 - sin theta)) = −2sec θ

Solved Examples | Q 5 | Page 41

Find the value of tan 9° – tan 27° – tan 63° + tan 81°

Solved Examples | Q 6 | Page 41

Prove that (sec8 theta - 1)/(sec4 theta - 1) = (tan8 theta)/(tan2 theta)

Solved Examples | Q 7 | Page 42

Solve the equation sin θ + sin 3θ + sin 5θ = 0

Solved Examples | Q 8 | Page 42

Solve 2 tan2x + sec2x = 2 for 0 ≤ x ≤ 2π.

Solved Examples | Q 9 | Page 43

Find the value of (1 + cos  pi/8)(1 + cos  (3pi)/8)(1 + cos  (5pi)/8)(1 + cos  (7pi)/8)

Solved Examples | Q 10 | Page 43

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

Solved Examples | Q 11 | Page 44

If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = (2ac)/(a^2 - c^2).

Solved Examples | Q 12 | Page 46

Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α

Solved Examples | Q 13 | Page 46

If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = (k + 1)/(k - 1) sin Φ

Solved Examples | Q 14 | Page 47

Solve sqrt(3) cos θ + sin θ = sqrt(2)

#### Objective Type Questions from 15 to 19

Solved Examples | Q 15 | Page 47

If tan θ = (-4)/3, then sin θ is ______.

• (-4)/5 but not 4/5

• (-4)/5 or 4/5

• 4/5 but not - 4/5

• None of these

Solved Examples | Q 16 | Page 48

If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.

• a2 + b2 + 2ac = 0

• a2 – b2 + 2ac = 0

• a2 + c2 + 2ab = 0

• a2 – b2 – 2ac = 0

Solved Examples | Q 17 | Page 48

The greatest value of sin x cos x is ______.

• 1

• 2

• sqrt(2)

• 1/2

Solved Examples | Q 18 | Page 48

The value of sin 20° sin 40° sin 60° sin 80° is ______.

• (-3)/16

• 5/16

• 3/16

• 1/16

Solved Examples | Q 19 | Page 49

The value of cos  pi/5 cos  (2pi)/5 cos  (4pi)/5 cos  (8pi)/5  is ______.

• 1/16

• 0

• (-1)/8

• (-1)/16

#### Fill in the blank:

Solved Examples | Q 20 | Page 50

If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.

#### State whether the following statement is True or False:

Solved Examples | Q 21 | Page 50

“The inequality 2^sintheta + 2^costheta ≥ 2^(1/sqrt(2)) holds for all real values of θ”

• True

• False

Solved Examples | Q 22 | Page 51

Match each item given under column C1 to its correct answer given under column C2.

 C1 C2 (a) (1 - cosx)/sinx (i) cot^2  x/2 (b) (1 + cosx)/(1 - cosx) (ii) cot  x/2 (c) (1 + cosx)/sinx (iii) |cos x + sin x| (d) sqrt(1 + sin 2x) (iv) tan  x/2
Exercise [Pages 52 - 60]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 3 Trigonometric Functions Exercise [Pages 52 - 60]

Exercise | Q 1 | Page 52

Prove that (tanA + secA  - 1)/(tanA - secA + 1) = (1 + sinA)/cosA

Exercise | Q 2 | Page 52

If (2sinalpha)/(1 + cosalpha + sinalpha) = y, then prove that (1 - cosalpha + sinalpha)/(1 + sinalpha) is also equal to y.
["Hint": "Express" (1 - cosalpha + sinalpha)/(1 + sinalpha) = (1 - cosalpha + sinalpha)/(1 + sinalpha) * (1 + cosalpha + sinalpha)/(1 + cosalpha + sinalpha)]

Exercise | Q 3 | Page 52

If m sinθ = n sin(θ + 2α), then prove that tan(θ + α)cotα = (m + n)/(m - n)

[Hint: Express (sin(theta + 2alpha))/sintheta = m/n and apply componendo and dividendo]

Exercise | Q 4 | Page 52

If cos(α + β) = 4/5 and sin(α – β) = 5/13, where α lie between 0 and pi/4, find the value of tan2α.
[Hint: Express tan2α as tan(α + β + α – β)]

Exercise | Q 5 | Page 53

If tanx = b/a, then find the value of sqrt((a + b)/(a - b)) + sqrt((a - b)/(a + b))

Exercise | Q 6 | Page 53

Prove that cosθ cos  theta/2 - cos 3theta cos  (9theta)/2 = sin 7θ sin 8θ.

[Hint: Express L.H.S. = 1/2[2costheta cos  theta/2 - 2 cos 3theta cos  (9theta)/2]

Exercise | Q 7 | Page 53

If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2

Exercise | Q 8 | Page 53

Find the value of tan22°30′. ["Hint:"  "Let" θ = 45°, "use" tan  theta/2 = (sin  theta/2)/(cos  theta/2) = (2sin  theta/2 cos  theta/2)/(2cos^2  theta/2) = sintheta/(1 + costheta)]

Exercise | Q 9 | Page 53

Prove that sin 4A = 4sinA cos3A – 4 cosA sin3A

Exercise | Q 10 | Page 53

If tanθ + sinθ = m and tanθ – sinθ = n, then prove that m2 – n2 = 4sinθ tanθ
[Hint: m + n = 2tanθ, m – n = 2sinθ, then use m2 – n2 = (m + n)(m – n)]

Exercise | Q 11 | Page 53

If tan(A + B) = p, tan(A – B) = q, then show that tan 2A = (p + q)/(1 - pq)

Exercise | Q 12 | Page 53

If cosα + cosβ = 0 = sinα + sinβ, then prove that cos2α + cos2β = -2cos(α + β).
[Hint: (cosα + cosβ)2 - (sinα + sinβ)2 = 0]

Exercise | Q 13 | Page 53

If (sin(x + y))/(sin(x - y)) = (a + b)/(a - b), then show that tanx/tany = a/b [Hint: Use Componendo and Dividendo].

Exercise | Q 14 | Page 53

If tanθ = (sinalpha - cosalpha)/(sinalpha + cosalpha), then show that sinα + cosα = sqrt(2) cosθ.

[Hint: Express tanθ = tan (alpha - pi/4) theta = alpha - pi/4]

Exercise | Q 15 | Page 53

If sinθ + cosθ = 1, then find the general value of θ.

Exercise | Q 16 | Page 53

Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = 1/sqrt(2).

Exercise | Q 17 | Page 54

If cotθ + tanθ = 2cosecθ, then find the general value of θ.

Exercise | Q 18 | Page 54

If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.

Exercise | Q 19 | Page 54

If secx cos5x + 1 = 0, where 0 < x ≤ pi/2, then find the value of x.

Exercise | Q 20 | Page 54

If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2

[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]

Exercise | Q 21 | Page 54

If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = (1 - m)/(1 + m) cot phi

[Hint: Express (cos(theta + Φ))/(cos(theta - Φ)) = m/1 and apply Componendo and Dividendo]

Exercise | Q 22 | Page 54

Find the value of the expression 3[sin^4 ((3pi)/2 - alpha) + sin^4 (3pi + alpha)] - 2[sin^6 (pi/2 + alpha) + sin^6 (5pi - alpha)]

Exercise | Q 23 | Page 54

If acos2θ + bsin2θ = c has α and β as its roots, then prove that tanα + tanβ = (2b)/(a + c).

["Hint: Use the identities" cos2theta = (1 - tan^2theta)/(1 + tan^2theta) "and" sin2theta =  (2tantheta)/(1 + tan^2theta)].

Exercise | Q 24 | Page 54

If x = sec Φ – tan Φ and y = cosec Φ + cot Φ then show that xy + x – y + 1 = 0
[Hint: Find xy + 1 and then show that x – y = –(xy + 1)]

Exercise | Q 25 | Page 54

If θ lies in the first quadrant and cosθ = 8/17, then find the value of cos(30° + θ) + cos(45° – θ) + cos(120° – θ).

Exercise | Q 26 | Page 54

Find the value of the expression cos^4  pi/8 + cos^4  (3pi)/8 + cos^4  (5pi)/8 + cos^4  (7pi)/8

[Hint: Simplify the expression to 2(cos^4  pi/8 + cos^4  (3pi)/8) = 2[(cos^2  pi/8 + cos^2  (3pi)/8)^2 - 2cos^2  pi/8 cos^2  (3pi)/8]

Exercise | Q 27 | Page 55

Find the general solution of the equation 5cos2θ + 7sin2θ – 6 = 0

Exercise | Q 28 | Page 55

Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x

Exercise | Q 29 | Page 55

Find the general solution of the equation (sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta = 2

[Hint: Put sqrt(3) - 1 = r sinα, sqrt(3) + 1 = r cosα which gives tanα = tan(pi/4 - pi/6) α = pi/12]

#### Objective Type Questions from 30 to 59

Exercise | Q 30 | Page 55

If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.

• 1

• 4

• 2

• None of these

Exercise | Q 31 | Page 55

If f(x) = cos2x + sec2x, then ______.

[Hint: A.M ≥ G.M.]

• f(x) < 1

• f(x) = 1

• 2 < f(x) < 1

• f(x) ≥ 2

Exercise | Q 32 | Page 55

If tanθ = 1/2 and tanΦ = 1/3, then the value of θ + Φ is ______.

• pi/6

• pi

• 0

• pi/4

Exercise | Q 33 | Page 55

Which of the following is not correct?

• sinθ = -1/5

• cosθ = 1

• secθ = 1/2

• tanθ = 20

Exercise | Q 34 | Page 55

The value of tan1° tan2° tan3° ... tan89° is ______.

• 0

• 1

• 1/2

• Not defined

Exercise | Q 35 | Page 56

The value of (1 - tan^2 15^circ)/(1 + tan^2 15^circ) is ______.

• 1

• sqrt(3)

• sqrt(3)/2

• 2

Exercise | Q 36 | Page 56

The value of cos1° cos2° cos3° ... cos179° is ______.

• 1/sqrt(2)

• 0

• 1

• –1

Exercise | Q 37 | Page 56

If tan θ = 3 and θ lies in third quadrant, then the value of sin θ  ______.

• 1/sqrt(10)

• - 1/sqrt(10)

• (-3)/sqrt(10)

• 3/sqrt(10)

Exercise | Q 38 | Page 56

The value of tan 75° - cot 75° is equal to ______.

• 2sqrt(3)

• 2 + sqrt(3)

• 2 - sqrt(3)

• 1

Exercise | Q 39 | Page 56

Which of the following is correct?

[Hint: 1 radian = 180^circ/pi = 57^circ30^' approx]

• sin1° > sin1

• sin1° < sin1

• sin1° = sin1

• sin1° = pi/180^circ  sin1

Exercise | Q 40 | Page 56

If tanα = m/(m +  1), tanβ = 1/(2m + 1), then α + β is equal to ______.

• pi/2

• pi/3

• pi/6

• pi/4

Exercise | Q 41 | Page 56

The minimum value of 3cosx + 4sinx + 8 is ______.

• 5

• 9

• 7

• 3

Exercise | Q 42 | Page 56

The value of tan3A - tan2A - tanA is equal to ______.

• tan3A tan2A tanA

• -tan3A tan2A tanA

• tanA tan2A - tan2A tan3A - tan3A tanA

• None of these

Exercise | Q 43 | Page 57

The value of sin(45° + θ) - cos(45° - θ) is ______.

• 2cosθ

• 2sinθ

• 1

• 0

Exercise | Q 44 | Page 57

The value of cot(pi/4 + theta)cot(pi/4 - theta) is ______.

• -1

• 0

• 1

• Not defined

Exercise | Q 45 | Page 57

cos2θ cos2Φ + sin2(θ – Φ) – sin2(θ + Φ) is equal to ______.

• sin2(θ + Φ)

• cos2(θ + Φ)

• sin2(θ – Φ)

• cos2(θ – Φ)

Exercise | Q 46 | Page 57

The value of cos12° + cos84° + cos156° + cos132° is ______.

• 1/2

• 1

• -1/2

• 1/8

Exercise | Q 47 | Page 57

If tanA = 1/2, tanB = 1/3, then tan(2A + B) is equal to ______.

• 1

• 2

• 3

• 4

Exercise | Q 48 | Page 57

The value of sin  pi/10  sin  (13pi)/10 is ______.

["Hint: Use"  sin18^circ = (sqrt5 - 1)/4 "and"  cos36^circ = (sqrt5 + 1)/4]

• 1/2

• -1/2

• -1/4

• 1

Exercise | Q 49 | Page 57

The value of sin50° – sin70° + sin10° is equal to ______.

• 1

• 0

• 1/2

• 2

Exercise | Q 50 | Page 57

If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.

• 1

• 1/2

• 0

• –1

Exercise | Q 51 | Page 58

If α + β = pi/4, then the value of (1 + tan α)(1 + tan β) is ______.

• 1

• 2

• –2

• Not defined

Exercise | Q 52 | Page 58

If sinθ = (-4)/5 and θ lies in the third quadrant then the value of cos  theta/2 is ______.

• 1/5

• -1/sqrt(10)

• -1/sqrt(5)

• 1/sqrt(10)

Exercise | Q 53 | Page 58

Number of solutions of the equation tan x + sec x = 2 cosx lying in the interval [0, 2π] is ______.

• 0

• 1

• 2

• 3

Exercise | Q 54 | Page 58

The value of sin  pi/18 + sin  pi/9 + sin  (2pi)/9 + sin  (5pi)/18 is given by ______.

• sin  (7pi)/18 + sin  (4pi)/9

• 1

• cos  pi/6 + cos  (3pi)/7

• cos  pi/9 + sin  pi/9

Exercise | Q 55 | Page 58

If A lies in the second quadrant and 3tanA + 4 = 0, then the value of 2cotA – 5cosA + sinA is equal to ______.

• (-53)/10

• 23/10

• 37/10

• 7/10

Exercise | Q 56 | Page 58

The value of cos248° – sin212° is ______.

[Hint: Use cos2A – sin2 B = cos(A + B) cos(A – B)]

• (sqrt(5) + 1)/8

• (sqrt(5) - 1)/8

• (sqrt(5) + 1)/5

• (sqrt(5) + 1)/(2sqrt(2)

Exercise | Q 57 | Page 59

If tanα = 1/7, tanβ = 1/3, then cos2α is equal to ______.

• sin2β

• sin4β

• sin3β

• cos2β

Exercise | Q 58 | Page 59

If tanθ = a/b, then bcos2θ + asin2θ is equal to ______.

• a

• b

• a/b

• None

Exercise | Q 59 | Page 59

If for real values of x, cosθ = x + 1/x, then ______.

• θ is an acute angle.

• θ is a right angle.

• θ is an obtuse angle.

• No value of θ is possible.

#### Fill in the blanks 60 to 67:

Exercise | Q 60 | Page 59

The value of (sin 50^circ)/(sin 130^circ) is ______.

Exercise | Q 61 | Page 59

If k = sin(pi/18) sin((5pi)/18) sin((7pi)/18), then the numerical value of k is ______.

Exercise | Q 62 | Page 59

If tanA = (1 - cos "B")/sin"B", then tan2A = ______.

Exercise | Q 63.(i) | Page 59

If sinx + cosx = a, then sin6x + cos6x = ______.

Exercise | Q 63.(ii) | Page 59

If sinx + cosx = a, then |sinx – cosx| = ______.

Exercise | Q 64 | Page 59

In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.

Exercise | Q 65 | Page 59

3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.

Exercise | Q 66 | Page 59

Given x > 0, the values of f(x) = -3cos sqrt(3 + x + x^2) lie in the interval ______.

Exercise | Q 67 | Page 60

The maximum distance of a point on the graph of the function y = sqrt(3) sinx + cosx from x-axis is ______.

Exercise | Q 68 | Page 60

State whether the statement is True or False? Also give justification.

If tanA = (1 - cos B)/sinB, then tan2A = tanB

• True

• False

Exercise | Q 69 | Page 60

State whether the statement is True or False? Also give justification.

The equality sinA + sin2A + sin3A = 3 holds for some real value of A.

• True

• False

Exercise | Q 70 | Page 60

State whether the statement is True or False? Also give justification.

Sin10° is greater than cos10°

• True

• False

Exercise | Q 71 | Page 60

State whether the statement is True or False? Also give justification.

cos  (2pi)/15 cos  (4pi)/15 cos  (8pi)/15 cos  (16pi)/15 = 1/16

• True

• False

Exercise | Q 72 | Page 60

State whether the statement is True or False? Also give justification.

One value of θ which satisfies the equation sin4θ - 2sin2θ - 1 lies between 0 and 2π.

• True

• False

Exercise | Q 73 | Page 60

State whether the statement is True or False? Also give justification.

If cosecx = 1 + cotx then x = 2nπ, 2nπ + pi/2

• True

• False

Exercise | Q 74 | Page 60

State whether the statement is True or False? Also give justification.

If tanθ + tan2θ + sqrt(3) tanθ tan2θ = sqrt(3), then θ = ("n"pi)/3 + pi/9

• True

• False

Exercise | Q 75 | Page 60

State whether the statement is True or False? Also give justification.

If tan(π cosθ) = cot(π sinθ), then cos(theta - pi/4) = +- 1/(2sqrt(2)).

• True

• False

Exercise | Q 76 | Page 60

In the following match each item given under the column C1 to its correct answer given under the column C2:

 Column A Column B (a) sin(x + y) sin(x – y) (i) cos2x – sin2y (b) cos (x + y) cos (x – y) (ii) (1 - tan theta)/(1 + tan theta) (c) cot(pi/4 + theta) (iii) (1 + tan theta)/(1 - tan theta) (d) tan(pi/4 + theta) (iv) sin2x – sin2y

## Chapter 3: Trigonometric Functions

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 11 chapter 3 - Trigonometric Functions

NCERT solutions for Mathematics Exemplar Class 11 chapter 3 (Trigonometric Functions) 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 CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 11 chapter 3 Trigonometric Functions are Transformation Formulae, Values of Trigonometric Functions at Multiples and Submultiples of an Angle, Sine and Cosine Formulae and Their Applications, 180 Degree Plusminus X Function, 2X Function, 3X Function, Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications, Concept of Angle, Introduction of Trigonometric Functions, Signs of Trigonometric Functions, Domain and Range of Trigonometric Functions, Trigonometric Functions of Sum and Difference of Two Angles, Trigonometric Equations, Trigonometric Functions, Truth of the Identity, Negative Function Or Trigonometric Functions of Negative Angles, 90 Degree Plusminus X Function, Conversion from One Measure to Another, Graphs of Trigonometric Functions.

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