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NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry

NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry

Edited By Komal Miglani | Updated on Mar 31, 2025 07:43 PM IST | #CBSE Class 10th

Trigonometry is used in real life situations where we want to find angles or distances. For example, in measuring the height of a tall building or a mountain, we can use trigonometric ratios like sine, cosine, or tangent. It’s also used in navigation, which includes locating the location of a ship or a plane. Concepts like line of sight, angle of elevation, and angle of depression help us measure things that are far away.

This Story also Contains
  1. NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry PDF Free Download
  2. Applications Of Trigonometry Class 10 NCERT Solution - Important Notes
  3. NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry (Exercise)
  4. NCERT Solutions for Class 10 Maths: Chapter Wise
  5. NCERT Exemplar solutions - Subject-wise
  6. NCERT Solutions for Class 10 for other subjects
  7. NCERT Books and NCERT Syllabus
NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry
NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry

This article on NCERT Class 10 Maths Chapter 9 Solutions of Some Applications of Trigonometry provides clear and step-by-step solutions for exercise problems in NCERT Class 10 maths book. These NCERT Solutions are created by subject experts at Careers360, considering the latest CBSE syllabus, 2025-26. For further reference, students can check NCERT Notes For Class 10 Mathematics Chapter Some Applications of Trigonometry and NCERT Exemplar Solutions For Class 10 Mathematics Chapter Trigonometry.

NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry PDF Free Download

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Applications Of Trigonometry Class 10 NCERT Solution - Important Notes

Line of Sight - The Line of Sight is the line formed by our vision as it passes through an item when we look at it.

Horizontal Line - The distance between the observer and the object is measured by a horizontal line.

The angle of Elevation:

  • The angle formed by the line of sight to the top of the item and the horizontal line is called an angle of elevation.
  • It is above the horizontal line, i.e., when we gaze up at the item, we make an angle of elevation.
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JEE Main Important Mathematics Formulas

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

1

The Angle of Depression:

  • The angle of depression is formed when you have to look down to see something.
  • The angle of depression is created between the horizontal line (above the angle) and the line of sight to the object.

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NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry (Exercise)

Class 10 Maths Chapter 9 Solutions Exercise: 9.1
Page Number: 141-143
Total Questions: 15

Q1 A circus artist is climbing a 20 m-long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).

2

Answer:

Given that,
The length of the rope (AC) = 20 m. and ∠ACB =30o
Let the height of the pole (AB) be h

So, in the right triangle ΔABC

3

By using the Sin rule
sin⁡θ=PH=ABAC
sin⁡30o=h20
h=10 m.

Hence, the answer is 10 m.

Q2 A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Answer:

4

Suppose DB is a tree and the AD is the broken height of the tree which touches the ground at C.
Given that,
∠ACB=30o , BC = 8 m
Let AB = x m and AD = y m
So, AD+AB = DB = x+y

In right angle triangle ΔABC ,
tan⁡θ=PB=x8
tan⁡30o=x8=13
So, the value of x = 8/3

Similarly,
cos⁡30o=BCAC=8y
The value of y is 16/3

So, the total height of the tree is-

x+y=243=83

= 8 (1.732) = 13.856 m (approx)

Hence, the answer is 13.856 m (approx)

Q3 A contractor plans to install two slides for the children to play in a park. For children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

Answer:

1742387416956

Suppose x m is the length of slides for children below 5 years and the length of slides for elderly children is y m.

Given that,
AF = 1.5 m, BC = 3 m, ∠AEF=30o and ∠BDC=60o

In triangle Δ EAF,
sin⁡θ=AFEF=1.5x
sin⁡30o=1.5x
The value of x is 3 m.

Similarly in Δ CDB,
sin⁡60o=3y
32=3y
The value of y is 23 = 2(1.732) = 3.468

Hence the length of the slide for children below 5 years. is 3 m and for the elder children is 3.468 m.

Q4 The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.

Answer:

1742387432125

Let the height of the tower AB be h and the angle of elevation from the ground at point C is ∠ACB=30o
According to the question,
In the right triangle ΔABC ,
tan⁡θ=ABBC=h30
tan⁡30o=13=h30
The value of h is 103 = 10(1.732) = 17.32 m

Hence, the answer is 17.32 m.

Q5 A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.

Answer:

1742387442979

Given that,

The length of AB = 60 m and the inclination of the string with the ground at point C is ∠ACB=60o.

Let the length of the string AC be l.

According to the question,

In right triangle Δ CBA,

sin⁡60o=ABAC=60l

32=60l

The value of length of the string ( l ) is 403 = 40(1.732) = 69.28 m

Hence, the answer is 69.28 m

Q6 A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

Answer:

1742387454193

Given that,
The height of the tallboy (DC) is 1.5 m and the height of the building (AB) is 30 m.
∠ADF=30o and ∠AEF=60o

According to the question,
In right triangle AFD,
⇒tan⁡30o=AFDF=28.5DF⇒13=28.5DF
So, DF = (28.5)3

In right angle triangle ΔAFE ,
tan⁡60o=AFFE=28.5EF
3=28.5EF
EF = 9.53

So, distance walked by the boy towards the building = DF - EF = 193

Q7 From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

Answer:

1742387465243

Suppose BC = h is the height of the transmission tower AB is

the height of the building and AD is the distance between the building and the observer point (D).
We have,
AB = 20 m, BC = h m and AD = x m
∠CDA=60o and ∠BDA=45o

According to the question,
In triangle Δ BDA,
tan⁡45o=ABAD=20x
So, x = 20 m

Again,
In triangle Δ CAD,

⇒tan⁡60o=AB+BCAD=20+h20⇒3=1+h20⇒h=20(3−1)⇒20(0.732)=14.64m

Hence, the answer is 14.64m

Q8 A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Answer:

1742387484682

Let the height of the pedestal be h m. and the height of the statue is 1.6 m.
The angle of elevation of the top of the statue and top of the pedestal is( ∠DCB= 60o )and( ∠ACB= 45o ) respectively.

Now,
In triangle ΔABC ,
tan⁡45o=1=ABBC=hBC
therefore, BC = h m

In triangle ΔCBD ,
⇒tan⁡60o=BDBC=h+1.6h⇒3=1+1.6h
the value of h is 0.8(3+1) m
Hence the height of the pedestal is 0.8(3+1) m

Q9 The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

Answer:

1742387493372

It is given that, the height of the tower (AB) is 50 m. ∠AQB=30o and ∠PBQ=60o
Let the height of the building be h m

According to the question,
In triangle PBQ,
tan⁡60o=PQBQ=50BQ
3=50BQBQ=503 .......................(i)

In triangle ABQ,

tan⁡30o=hBQ
BQ=h3 .........................(ii)
On equating the eq(i) and (ii) we get,

503=h3
Therefore, h = 50/3 = 16.66 m = height of the building.

Q10 Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the points from the poles.

Answer:

1742387504017

Given that,
The height of both poles is equal to DC = AB. The angle of elevation of the top of the poles is ∠DEC=30o and ∠AEB=60o resp.
Let the height of the poles be h m and CE = x and BE = 80 - x

According to the question,
In triangle DEC,

⇒tan⁡30o=DCCE=hx⇒13=hx⇒x=h3 ..............(i)

In triangle AEB,
⇒tan⁡60o=ABBE=h80−x⇒3=h80−x⇒x=80−h3 ..................(ii)
On equating eq (i) and eq (ii), we get

3h=80−h3

h3=20 So, x = 60 m

Hence the height of both poles is ( h=203 )m and the position of the point is at 60 m from the pole CD and 20 m from the pole AB.

Q11 A TV tower stands vertically on the bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30° (see Fig. 9.12). Find the height of the tower and the width of the canal.

1742387514600

Answer:

Suppose the h is the height of the tower AB and BC = x m
It is given that, the width of the CD is 20 m,
According to the question,

In triangle ΔADB ,

tan⁡30∘=AB20+x=h20+x⇒13=h20+x⇒20+x=h3⇒x=h3−20

In triangle ACB,
⇒tan⁡60o=hx=3⇒x=h3 .............(ii)

On equating eq (i) and (ii) we get:

h3−20=h3
from here we can calculate the value of h=103=10(1.732)=17.32m and the width of the canal is 10 m.

Q12 From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

Answer:

1742387534387

Let the height of the cable tower be (AB = h+7 )m
Given,
The height of the building is 7 m and the angle of elevation of the top of the tower ∠ACE=60o, angle of depression of its foot ∠BCE=45o

According to the question,

In triangle ΔDBC ,
tan⁡45o=CDBD=7BD=1BD=7 m
since DB = CE = 7 m

In triangle ΔACE ,

tan⁡60o=hCE=h7=3∴h=73 m

Thus, the total height of the tower equal to h+7 =7(1+3) m

Q13 As observed from the top of a 75 m high lighthouse from the sea level, the depression angles of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Answer:

Given that,
The height of the lighthouse (AB) is 75 m from the sea level. And the angle of depression of two different ships are ∠ADB=300 and ∠ACB=450 respectively
1742387544339

Let the distance between both the ships be x m.
According to the question,

In triangle ΔADB ,

tan⁡300=ABBD=75x+y=13
∴x+y=753 .............(i)

In triangle ΔACB ,

tan⁡450=1=75BC=75y
∴y=75 m .............(ii)

From equation (i) and (ii) we get;
x=75(3−1)=75(0.732)
x=54.9≃55 m

Hence, the distance between the two ships is approx 55 m.

Q14 A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30° (see Fig. 9.13). Find the distance traveled by the balloon during the interval.

1742387556311

Answer:

Given that,
The height of the girl is 1.2 m. The height of the balloon from the ground is 88.2 m and the angle of elevation of the balloon from the eye of the girl at any instant is ( ∠ACB=600 ) and after some time ∠DCE=300.
1742387565300

Let the x distance traveled by the balloon from position A to position D during the interval.
AB = ED = 88.2 - 1.2 =87 m

Now, In triangle ΔBCA ,

tan⁡600=3=ABBC=87BC∴BC=293

In triangle ΔDCE ,

tan⁡300=13=DECE=87CE∴CE=873

Thus, the distance traveled by the balloon from position A to D

=CE−BC=873−293=583 m

Q15 A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

Answer:

1742387575715

Let h be the height of the tower (DC) and the speed of the car be x ms−1. Therefore, the distance (AB)covered by the car in 6 seconds is 6 x m. Let t time required to reach the foot of the tower. So, BC = x t

According to the question,
In triangle ΔDAC ,
tan⁡300=13=h6x+xtx(6+t)=h3 ..........................(i)

In triangle ΔBCD ,

tan⁡600=3=hxt∴h=3.xt ...................(ii)

Put the value of h in equation (i) we get,
x(6+t)=(3.3)xt6x+xt=3xt6x=2xt
t=3

Hence, from point B car takes 3 sec to reach the foot of the tower.

NCERT Exemplar solutions - Subject-wise

To get subject wise exemplars, students can click the following links.

NCERT Solutions for Class 10 for other subjects

To get subject wise NCERT solutions, students can click the following links.

NCERT Books and NCERT Syllabus

Here is the latest NCERT syllabus which is very useful for students before strategizing their study plan. Also, links to some reference books which are important for further studies.

Frequently Asked Questions (FAQs)

1. What are the important topics covered in Class 10 Maths Chapter 9 in ncert?

In this chapter Trigonometry applications class 10 covered following topics-

1. Class 10 maths ch 9 introduction 

2. Line of Sight

3. Angle of Elevation

4. Angle of Depression

5. Trigonometric Ratios in Heights and Distances

6. Solving Problems Based on Heights and Distances

7. Assumptions in Height and Distance Problems

2. How can I download the NCERT Solutions for Class 10 Maths Chapter 9 PDF?

NCERT Solutions for Class 10 Maths Chapter 9 PDF is provided in this article. you can just click and download it. Also students can download maths class 10 chapter 9 using the official website fo careers360.

3. How to solve height and distance problems in Class 10 Maths?

Class 10 9th chapter maths include problems related to height and distance which can be solved using the following steps-

  • Draw a Diagram: Convert the given problem into right-angled triangle.
  • Find angles (elevation/depression), heights, or distances.
  • Choose the Right Trigonometric Ratio:
    • tan θ = Perpendicular / Base
    • sin θ = Perpendicular / Hypotenuse
    • cos θ = Base / Hypotenuse
  • Apply the Formula: Use the best trigonometric ratio.
  • Solve the Equation: Find unknown values grade by grade.
4. What is the easiest way to understand trigonometric ratios in Chapter 9?

The easiest way to understand trigonometric ratios in NCERT solutions class 10th maths chapter 9 is:

  1. Learn the Basic Ratios:
    • sin θ = Perpendicular / Hypotenuse
    • cos θ = Base / Hypotenuse
    • tan θ = Perpendicular / Base
  2. Use the Mnemonic: SOH-CAH-TOA
    • Sin = Opposite/Hypotenuse
    • Cos = Adjacent/Hypotenuse
    • Tan = Opposite/Adjacent
  3. Practice Right-Angled Triangle Problems
     
  4. Relate to Real-Life Applications (heights, distances, etc.)

Use these class 10 ncert maths chapter 9 solutions to command these formulas.

5. What are the important formulas to remember in Some Applications of Trigonometry?

In Applications of trigonometry class 10 questions with solutions, the following important formulas should be remembered:

Basic Trigonometric Ratios:

  1. sin θ = Perpendicular / Hypotenuse
  2. cos θ = Base / Hypotenuse
  3. tan θ = Perpendicular / Base
  4. cosec θ = 1 / sin θ
  5. sec θ = 1 / cos θ
  6. cot θ = 1 / tan θ

Height and Distance Formulas:

  1. Tan θ = Height / Distance (for the angle of elevation/depression)
  2. The angle of Elevation: The observer looks upward
  3. Angle of Depression: Observer looks downward

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Questions related to CBSE Class 10th

Have a question related to CBSE Class 10th ?

Hello

Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.

1. KCET (Karnataka Common Entrance Test): You must appear for the KCET exam, which is required for admission to undergraduate professional courses like engineering, medical, and other streams. Your exam score and rank will determine your eligibility for counseling.

2. Minority Income under 5 Lakh : If you are from a minority community and your family's income is below 5 lakh, you may be eligible for fee concessions or other benefits depending on the specific institution. Some colleges offer reservations or other advantages for students in this category.

3. Counseling and Seat Allocation:

After the KCET exam, you will need to participate in online counseling.

You need to select your preferred colleges and courses.

Seat allocation will be based on your rank , the availability of seats in your chosen colleges and your preferences.

4. Required Documents :

Domicile Certificate (proof that you are a resident of Karnataka).

Income Certificate (for minority category benefits).

Marksheets (11th and 12th from the Karnataka State Board).

KCET Admit Card and Scorecard.

This process will allow you to secure a seat based on your KCET performance and your category .

check link for more details

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Hope this helps you .

Hello Aspirant,  Hope your doing great,  your question was incomplete and regarding  what exam your asking.

Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.

hello Zaid,

Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.

best of luck!

According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.

You are not eligible for cbse board but you can still do 12th from nios which allow candidates to take admission in 12th class as a private student without completing 11th.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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