#### Online Mock Tests

#### Chapters

Chapter 2: Polynomials

Chapter 3: Pair of Liner Equation in Two Variable

Chapter 4: Quadatric Euation

Chapter 5: Arithematic Progressions

Chapter 6: Triangles

Chapter 7: Coordinate Geometry

Chapter 8: Introduction to Trignometry & its Equation

Chapter 9: Circles

Chapter 10: Construction

Chapter 11: Area Related To Circles

Chapter 12: Surface Areas and Volumes

Chapter 13: Statistics and Probability

## Chapter 8: Introduction to Trignometry & its Equation

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

#### Choose the correct alternative:

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

`3/5`

`3/4`

`4/3`

`5/3`

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

`sqrt(3)`

`1/sqrt(3)`

`sqrt(3)/2`

1

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

– 1

0

1

`3/2`

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)`

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

cos β

cos 2β

sin α

sin 2α

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

0

1

2

`1/2`

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

`1/sqrt(3)`

`sqrt(3)`

1

0

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

0

1

`1/2`

`sqrt(3)/2`

If sinA + sin^{2}A = 1, then the value of the expression (cos^{2}A + cos^{4}A) is ______.

1

`1/2`

2

3

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

0°

30°

60°

90°

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

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

`2/3`

`1/3`

`1/2`

`3/4`

If sinθ – cosθ = 0, then the value of (sin^{4}θ + cos^{4}θ) is ______.

1

`3/4`

`1/2`

`1/4`

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

2 cosθ

0

2 sinθ

1

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°

### 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:

`tan 47^circ/cot 43^circ` = 1

True

False

The value of the expression (cos^{2} 23° – sin^{2} 67°) is positive.

True

False

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

True

False

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

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

True

False

If cosA + cos^{2}A = 1, then sin^{2}A + sin^{4}A= 1.

True

False

(tan θ + 2)(2 tan θ + 1) = 5 tan θ + sec^{2}θ.

True

False

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

True

False

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

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

True

False

Write True' or False' and justify your answer the following :

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

True

False

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

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

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

`(sqrt(3) + 1) (3 - cot 30^circ)` = tan^{3} 60° – 2 sin 60°

Prove the following

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

Prove the following:

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

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.

Prove the following:

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

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.

Prove the following:

Simplify (1 + tan^{2}θ) (1 – sinθ)(1 + sinθ)

Prove the following:

If 2sin^{2}θ – cos^{2}θ = 2, then find the value of θ.

Prove the following:

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

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.

Show that tan^{4}θ + tan^{2}θ = sec^{4}θ – sec^{2}θ.

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

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

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

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

If 1 + sin^{2}θ = 3sinθ cosθ , then prove that tanθ = 1 or `1/2`.

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

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)`

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.

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))`

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

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

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

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

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.

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.

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)`

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.

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.

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

## 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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