# NCERT solutions for Mathematics Exemplar Class 10 chapter 8 - Introduction to Trignometry & its Equation [Latest edition]

## Chapter 8: Introduction to Trignometry & its Equation

Exercise 8.1Exercise 8.2Exercise 8.3Exercise 8.4
Exercise 8.1 [Pages 89 - 91]

### NCERT solutions for Mathematics Exemplar Class 10 Chapter 8 Introduction to Trignometry & its Equation Exercise 8.1 [Pages 89 - 91]

#### Choose the correct alternative:

Exercise 8.1 | Q 1 | Page 89

If cos A = 4/5, then the value of tan A is ______.

• 3/5

• 3/4

• 4/3

• 5/3

Exercise 8.1 | Q 2 | Page 90

If sin A = 1/2, then the value of cot A is ______.

• sqrt(3)

• 1/sqrt(3)

• sqrt(3)/2

• 1

Exercise 8.1 | Q 3 | Page 90

The value of the expression [cosec(75° + θ) – sec(15° – θ) – tan(55° + θ) + cot(35° – θ)] is ______.

• – 1

• 0

• 1

• 3/2

Exercise 8.1 | Q 4 | Page 90

Given that sinθ = a/b, then cosθ is equal to ______.

• b/sqrt(b^2 - a^2)

• b/a

• sqrt(b^2 - a^2)/b

• a/sqrt(b^2 - a^2)

Exercise 8.1 | Q 5 | Page 90

If cos(α + β) = 0, then sin(α – β) can be reduced to ______.

• cos β

• cos 2β

• sin α

• sin 2α

Exercise 8.1 | Q 6 | Page 90

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

• 0

• 1

• 2

• 1/2

Exercise 8.1 | Q 7 | Page 90

If cos 9α = sinα and 9α < 90°, then the value of tan5α is ______.

• 1/sqrt(3)

• sqrt(3)

• 1

• 0

Exercise 8.1 | Q 8 | Page 90

If ∆ABC is right-angled at C, then the value of cos(A + B) is ______.

• 0

• 1

• 1/2

• sqrt(3)/2

Exercise 8.1 | Q 9 | Page 90

If sinA + sin2A = 1, then the value of the expression (cos2A + cos4A) is ______.

• 1

• 1/2

• 2

• 3

Exercise 8.1 | Q 10 | Page 90

Given that sinα = 1/2 and cosβ = 1/2, then the value of (α + β) is ______.

• 30°

• 60°

• 90°

Exercise 8.1 | Q 11 | Page 91

The value of the expression [(sin^2 22^circ + sin^2 68^circ)/(cos^2 22^circ + cos^2 68^circ) + sin^2  63^circ + cos 63^circ sin 27^circ] is ______.

• 3

• 2

• 1

• 0

Exercise 8.1 | Q 12 | Page 91

If 4 tanθ = 3, then ((4 sintheta - costheta)/(4sintheta + costheta)) is equal to ______.

• 2/3

• 1/3

• 1/2

• 3/4

Exercise 8.1 | Q 13 | Page 91

If sinθ – cosθ = 0, then the value of (sin4θ + cos4θ) is ______.

• 1

• 3/4

• 1/2

• 1/4

Exercise 8.1 | Q 14 | Page 91

sin(45° + θ) – cos(45° – θ) is equal to ______.

• 2 cosθ

• 0

• 2 sinθ

• 1

Exercise 8.1 | Q 15 | Page 91

A pole 6 m high casts a shadow 2sqrt(3) m long on the ground, then the Sun’s elevation is ______.

• 60°

• 45°

• 30°

• 90°

Exercise 8.2 [Page 93]

### NCERT solutions for Mathematics Exemplar Class 10 Chapter 8 Introduction to Trignometry & its Equation Exercise 8.2 [Page 93]

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

Exercise 8.2 | Q 1 | Page 93

tan 47^circ/cot 43^circ = 1

• True

• False

Exercise 8.2 | Q 2 | Page 93

The value of the expression (cos2 23° – sin2 67°) is positive.

• True

• False

Exercise 8.2 | Q 3 | Page 93

The value of the expression (sin 80° – cos 80°) is negative.

• True

• False

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

Exercise 8.2 | Q 4 | Page 93

sqrt((1 - cos^2theta) sec^2 theta) = tan theta

• True

• False

Exercise 8.2 | Q 5 | Page 93

If cosA + cos2A = 1, then sin2A + sin4A= 1.

• True

• False

Exercise 8.2 | Q 6 | Page 93

(tan θ + 2)(2 tan θ + 1) = 5 tan θ + sec2θ.

• True

• False

Exercise 8.2 | Q 7 | Page 93

If the length of the shadow of a tower is increasing, then the angle of elevation of the sun is also increasing.

• True

• False

Exercise 8.2 | Q 8 | Page 93

If a man standing on a platform 3 metres above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.

• False

• True

Exercise 8.2 | Q 9 | Page 93

The value of 2sinθ can be a + 1/a, where a is a positive number, and a ≠ 1.

• True

• False

Exercise 8.2 | Q 10 | Page 93

$\cos \theta = \frac{a^2 + b^2}{2ab}$where a and b are two distinct numbers such that ab > 0.

• True

• False

Exercise 8.2 | Q 11 | Page 93

The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

• True

• False

Exercise 8.2 | Q 12 | Page 93

If the height of a tower and the distance of the point of observation from its foot, both, are increased by 10%, then the angle of elevation of its top remains unchanged.

• True

• False

Exercise 8.3 [Page 95]

### NCERT solutions for Mathematics Exemplar Class 10 Chapter 8 Introduction to Trignometry & its Equation Exercise 8.3 [Page 95]

Exercise 8.3 | Q 1 | Page 95

Prove the following:

sintheta/(1 + cos theta) + (1 + cos theta)/sintheta = 2cosecθ

Exercise 8.3 | Q 2 | Page 95

Prove the following:

tanA/(1 + sec A) - tanA/(1 - sec A) = 2cosec A

Exercise 8.3 | Q 3 | Page 95

Prove the following:

If tan A = 3/4, then sinA cosA = 12/25

Exercise 8.3 | Q 4 | Page 95

Prove the following:

(sin α + cos α)(tan α + cot α) = sec α + cosec α

Exercise 8.3 | Q 5 | Page 95

Prove the following:

(sqrt(3) + 1) (3 - cot 30^circ) = tan3 60° – 2 sin 60°

Exercise 8.3 | Q 6 | Page 95

Prove the following

1 + (cot^2 alpha)/(1 + cosec alpha) = cosec α

Exercise 8.3 | Q 7 | Page 95

Prove the following:

tanθ + tan (90° – θ) = sec θ sec(90° – θ)

Exercise 8.3 | Q 8 | Page 95

Prove the following:

Find the angle of elevation of the sun when the shadow of a pole h metres high is sqrt(3) h metres long.

Exercise 8.3 | Q 9 | Page 95

Prove the following:

If sqrt(3) tan theta = 1, then find the value of sin^2 theta - cos^2 theta.

Exercise 8.3 | Q 10 | Page 95

Prove the following:

A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

Exercise 8.3 | Q 11 | Page 95

Prove the following:

Simplify (1 + tan2θ) (1 – sinθ)(1 + sinθ)

Exercise 8.3 | Q 12 | Page 95

Prove the following:

If 2sin2θ – cos2θ = 2, then find the value of θ.

Exercise 8.3 | Q 13 | Page 95

Prove the following:

Show that (cos^2 (45^circ + theta) + cos^2 (45^circ - theta))/(tan(60^circ + theta) tan(30^circ - theta)) = 1

Exercise 8.3 | Q 14 | Page 95

An observer 1.5 metres tall is 20.5 metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from the eye of the observer.

Exercise 8.3 | Q 15 | Page 95

Show that tan4θ + tan2θ = sec4θ – sec2θ.

Exercise 8.4 [Pages 99 - 100]

### NCERT solutions for Mathematics Exemplar Class 10 Chapter 8 Introduction to Trignometry & its Equation Exercise 8.4 [Pages 99 - 100]

Exercise 8.4 | Q 1 | Page 99

If cosecθ + cotθ = p, then prove that cosθ = (P^2 - 1)/(p^2 + 1)

Exercise 8.4 | Q 2 | Page 99

Prove that sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta

Exercise 8.4 | Q 3 | Page 99

The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower.

#### Choose the correct option. Justify your choice

Exercise 8.4 | Q 4 | Page 99

If 1 + sin2θ = 3sinθ cosθ , then prove that tanθ = 1 or 1/2.

Exercise 8.4 | Q 5 | Page 99

Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.

Exercise 8.4 | Q 6 | Page 99

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is sqrt(st)

Exercise 8.4 | Q 7 | Page 99

The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.

Exercise 8.4 | Q 8 | Page 99

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is ((h tan alpha)/(tan beta - tan alpha))

Exercise 8.4 | Q 9 | Page 99

If tan θ + sec θ = l, then prove that secθ = (l^2 + 1)/(2l)

Exercise 8.4 | Q 10 | Page 99

If sinθ + cosθ = p and sec θ + cosec θ = q, then prove that q(p2 – 1) = 2p.

Exercise 8.4 | Q 11 | Page 99

If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = sqrt(a^2 + b^2 - c^2)

Exercise 8.4 | Q 12 | Page 99

Prove that (1 + sec theta - tan theta)/(1 + sec theta + tan theta) = (1 - sin theta)/cos theta

Exercise 8.4 | Q 13 | Page 99

The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower.

Exercise 8.4 | Q 14 | Page 100

From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot of the tower are α and β (β > α). Find the distance between the two objects.

Exercise 8.4 | Q 15 | Page 100

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal. Show that p/q = (cos beta - cos alpha)/(sin alpha - sin beta)

Exercise 8.4 | Q 16 | Page 100

The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower.

Exercise 8.4 | Q 17 | Page 100

A window of a house is h metre above the ground . From the window , the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be α and β respectively. Prove that the height of the house is h(1+ tan α tan β) metres.

Exercise 8.4 | Q 18 | Page 100

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be 60° and 30°, respectively. Find the height of the balloon above the ground.

## Chapter 8: Introduction to Trignometry & its Equation

Exercise 8.1Exercise 8.2Exercise 8.3Exercise 8.4

## NCERT solutions for Mathematics Exemplar Class 10 chapter 8 - Introduction to Trignometry & its Equation

NCERT solutions for Mathematics Exemplar Class 10 chapter 8 (Introduction to Trignometry & its Equation) 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 10 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 10 chapter 8 Introduction to Trignometry & its Equation are Trigonometry, Trigonometric Ratios, Trigonometric Ratios of Some Special Angles, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Proof of Existence, Relationships Between the Ratios, Trigonometry, Trigonometric Ratios and Its Reciprocal.

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