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

#### Bywisdomira

(Last Updated On: April 1, 2023)

Some Applications of Trigonometry in Class 10 is an exercise consisting of 16 questions. In this chapter, you will study about the real life applications of trigonometry and the questions are based on practical applications of trigonometry.

Topics and Sub Topics in Class 10 Maths Chapter 9 Some Applications of Trigonometry:

 Section Name Topic Name 9 Some Applications of Trigonometry 9.1 Introduction 9.2 Heights And Distances 9.3 Summary

## NCERT Solutions For Class 10 Maths Chapter 9 Some Applications of Trigonometry Ex 9.1

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 Board CBSE Textbook NCERT Class Class 10 Subject Maths Chapter Chapter 9 Chapter Name Some Applications of Trigonometry Exercise Ex 9.1 Number of Questions Solved 16 Category NCERT Solutions

Hi all, We have also solved 68 questions of Chapter 12 – Some Applications of Trigonometry from RD Sharma Class 10 Maths textbook. You can download these solutions in PDF from the above link.

Ex 9.1 Class 10 Maths Question 1.
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°.
Solution:

Ex 9.1 Class 10 Maths Question 2.
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 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.
Solution:

Ex 9.1 Class 10 Maths Question 4.
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.
Solution:

Let’s draw a diagram to better understand the situation:

In the diagram, we can see that the angle of elevation of the top of the tower (T) from a point on the ground (F), which is 30 m away from the foot of the tower is 30°. Let h be the height of the tower.

We can use the tangent function to find the height of the tower:

tan(30°) = h/30 m [tan(theta) = opposite/adjacent = height/30 m]
h = 30 m * tan(30°) = 30 m * 1/√3 = 30/√3 = 17.32 m (approx.)

Therefore, the height of the tower is approximately 17.32 meters.

Ex 9.1 Class 10 Maths Question 5.
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.
Solution:

In the diagram, we can see that a kite (K) 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 (P). The inclination of the string with the ground is 60°. Let’s assume that the length of the string is ‘x’.

We can use the sine function to find the length of the string:

sin(60°) = opposite/hypotenuse = 60 m/x
x = 60 m/sin(60°) = 60 m/(√3/2) = 40√3 m ≈ 69.28 m

Therefore, the length of the string is approximately 69.28 meters.

Ex 9.1 Class 10 Maths Question 6.
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.
Solution:

Let’s draw a diagram to better understand the situation:In the diagram, we can see that a 1.5 m tall boy is standing at point A at some distance ‘x’ from a 30 m tall building (B). The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building and reaches point B. Let’s assume that the height of the boy’s eyes from the ground is ‘h’.

Let’s first find the initial distance of the boy from the building when the angle of elevation is 30°. We can use the tangent function to find this distance:

```tan(30°) = opposite/adjacent = (30 - h)/x x = (30 - h)/tan(30°) ```

Now, let’s find the final distance of the boy from the building when the angle of elevation is 60°. We can use the same formula to find this distance:

```tan(60°) = opposite/adjacent = (30 - h)/(x - d) x - d = (30 - h)/tan(60°) d = x - (30 - h)/tan(60°) ```

We can subtract the initial distance from the final distance to find the distance the boy walked towards the building:

```distance walked = d - x = (30 - h)/tan(60°) - (30 - h)/tan(30°) distance walked = (30 - h)(1/tan(60°) - 1/tan(30°)) ```

Now, we just need to substitute the given values and simplify the expression:

```distance walked = (30 - 1.5)(√3 - 1)/(√3 + 1) distance walked = 16.5(√3 - 1) distance walked ≈ 8.91 meters ```

Therefore, the boy walked approximately 8.91 meters towards the building.

## Angle of Elevation in Trigonometry

The angle of elevation is a trigonometric concept that refers to the angle formed between a horizontal line and a line of sight (or line of vision) pointing upwards to an object. It is typically used in scenarios where an observer is looking at an object above them, such as a tower, mountain, or airplane.

The angle of elevation is calculated using basic trigonometric functions such as sine, cosine, and tangent. To find the angle of elevation, you need to know the vertical height of the object above the horizontal line, as well as the horizontal distance between the observer and the object.

Here’s how to calculate the angle of elevation:

1. Draw a right-angled triangle, where the horizontal line is the base of the triangle, and the line of sight pointing upwards to the object is the hypotenuse.
2. Label the angle of elevation as θ (theta), which is the angle formed between the horizontal line and the line of sight.
3. Measure the vertical height of the object above the horizontal line, which is the opposite side of the triangle. Label this as h.
4. Measure the horizontal distance between the observer and the object, which is the adjacent side of the triangle. Label this as d.
5. Use the trigonometric function tangent (tan) to calculate the angle of elevation. tan(θ) = opposite/adjacent = h/d.
6. Solve for θ by taking the inverse tangent (tan^-1) of the ratio h/d. θ = tan^-1(h/d).

The angle of elevation is often used in navigation, surveying, and engineering applications, where it is important to calculate the height or distance of an object above ground level.

### Angle of Depression trigonometry

The angle of depression is a trigonometric concept that refers to the angle formed between a horizontal line and a line of sight (or line of vision) pointing downwards to an object. It is typically used in scenarios where an observer is looking at an object below them, such as a boat in the water or a building from a higher floor.

The angle of depression is calculated using basic trigonometric functions such as sine, cosine, and tangent. To find the angle of depression, you need to know the vertical depth of the object below the horizontal line, as well as the horizontal distance between the observer and the object.

Here’s how to calculate the angle of depression:

1. Draw a right-angled triangle, where the horizontal line is the base of the triangle, and the line of sight pointing downwards to the object is the hypotenuse.
2. Label the angle of depression as θ (theta), which is the angle formed between the horizontal line and the line of sight.
3. Measure the vertical depth of the object below the horizontal line, which is the opposite side of the triangle. Label this as h.
4. Measure the horizontal distance between the observer and the object, which is the adjacent side of the triangle. Label this as d.
5. Use the trigonometric function tangent (tan) to calculate the angle of depression. tan(θ) = opposite/adjacent = h/d.
6. Solve for θ by taking the inverse tangent (tan^-1) of the ratio h/d. θ = tan^-1(h/d).

The angle of depression is often used in navigation, surveying, and engineering applications, where it is important to calculate the depth or distance of an object below ground level or sea level.

## Line of Sight and Angle of Elevation Trigonometry

The line of sight and the angle of elevation are both trigonometric concepts that are related to each other. The line of sight refers to an imaginary straight line that extends from an observer’s eye to a target or object that the observer is looking at. The angle of elevation, on the other hand, is the angle formed between a horizontal line and a line of sight pointing upwards to an object.

The line of sight and the angle of elevation are used together to determine the height or distance of an object above the observer’s position. To do this, the observer needs to measure both the angle of elevation and the horizontal distance between the observer and the object.

Here’s how to use the line of sight and the angle of elevation to calculate the height or distance of an object:

1. Draw a right-angled triangle, where the horizontal line is the base of the triangle, and the line of sight pointing upwards to the object is the hypotenuse. Label the angle of elevation as θ.
2. Measure the horizontal distance between the observer and the object, which is the adjacent side of the triangle. Label this as d.
3. Use the trigonometric function tangent (tan) to calculate the height of the object. tan(θ) = opposite/adjacent.
4. Solve for the height (opposite) by multiplying the tangent of the angle of elevation by the horizontal distance. opposite = tan(θ) x d.

By using the line of sight and the angle of elevation, you can calculate the height or distance of an object from your position. This is a useful tool in fields such as surveying, engineering, and navigation, where it is important to determine the exact location and size of objects in relation to the observer.

## How to Download NCERT Solutions for Class 10 Maths Chapter 9

I can guide you on how to find the solutions.

You can access the NCERT Solutions for Class 10 Maths Chapter 9 “Some Applications of Trigonometry” on the official website of NCERT or by using the NCERT Book of Class 10 Maths.

To access the NCERT Solutions on the official website, follow the steps below:

1. Go to the official website of NCERT (https://ncert.nic.in/).
2. Click on the “Textbooks” tab.
3. Select “PDF (I-XII)” under the “Publications” section.
4. Select “Textbooks” and choose Class 10.
5. Select “Mathematics” and choose the book title.
6. Select Chapter 9 “Some Applications of Trigonometry” and click on “View/Download”.
7. The solutions will be available in the PDF file.

Alternatively, you can use the NCERT Book of Class 10 Maths, which contains the solutions for Chapter 9. You can either buy the physical book or access the e-book version on the official website of NCERT.

I hope this helps you find the solutions for Class 10 Maths Chapter 9.

## Applications of Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It has many applications in various fields such as engineering, physics, architecture, astronomy, surveying, navigation, and more. Here are some specific examples of how trigonometry is used:

1. Architecture and construction: Trigonometry is used in designing and constructing buildings and other structures. Architects and engineers use trigonometry to calculate angles and distances between various points in the design process.
2. Surveying: Surveyors use trigonometry to measure angles and distances between different points on the ground. This helps them create accurate maps and blueprints for construction projects.
3. Navigation: Trigonometry is used in navigation to calculate the distance and direction between two points. It is also used in aviation and marine navigation to determine the altitude and bearing of an aircraft or ship.
4. Physics: Trigonometry is used in physics to calculate the motion and trajectories of objects. For example, projectile motion can be analyzed using trigonometric functions.
5. Astronomy: Trigonometry is used in astronomy to calculate the positions and movements of celestial bodies. It is used to determine the distance between stars and planets, as well as the angles and distances between different points in the sky.
6. Electrical engineering: Trigonometry is used in electrical engineering to calculate the phase angles and frequencies of alternating currents and voltages.
7. Medical imaging: Trigonometry is used in medical imaging to calculate the angles and distances between different points in the human body. This helps doctors and surgeons to diagnose and treat medical conditions.

These are just a few examples of the many applications of trigonometry in different fields. Trigonometry is an essential tool for solving many real-world problems that involve angles, distances, and trajectories.

### Line of Sight

The line of sight refers to an imaginary straight line that extends from an observer’s eye to a target or object that the observer is looking at. It is used in various fields such as astronomy, surveying, navigation, and military targeting. Here are some specific examples of how the line of sight is used:

1. Astronomy: In astronomy, the line of sight is used to measure the distance between celestial objects such as stars and planets. By measuring the angle between the observer’s line of sight and the target object, astronomers can calculate the object’s distance from the Earth.
2. Surveying: Surveyors use the line of sight to measure the horizontal and vertical angles between two points on the ground. This allows them to create accurate maps and blueprints for construction projects.
3. Navigation: The line of sight is used in navigation to determine the bearing and direction of a ship or aircraft relative to a target or destination. It is also used in GPS systems to calculate the distance between two points on the ground.
4. Military targeting: In military operations, the line of sight is used to target enemy positions and calculate the trajectory of missiles and other projectiles. By measuring the angle and distance between the observer’s position and the target, military strategists can determine the best approach for attacking or defending a position.
5. Optics: In optics, the line of sight is used to calculate the focal length and magnification of lenses and other optical devices. By measuring the angle between the observer’s line of sight and the lens, opticians can determine the lens’s characteristics and performance.

Overall, the line of sight is an essential concept in many fields, allowing us to measure distances, angles, and trajectories accurately. It is a fundamental tool for solving real-world problems that involve observation, measurement, and analysis.

## FAQs About NCERT Solutions for Class 10 Maths Chapter 9

Here are some frequently asked questions (FAQs) about NCERT Solutions for Class 10 Maths Chapter 9 “Some Applications of Trigonometry”:

Q: What are NCERT Solutions for Class 10 Maths Chapter 9?

A: NCERT Solutions for Class 10 Maths Chapter 9 are the solutions to the exercises and questions given in the NCERT textbook for Class 10 Mathematics Chapter 9 “Some Applications of Trigonometry”. These solutions are designed to help students understand the concepts and solve the problems given in the textbook.

Q: Where can I find NCERT Solutions for Class 10 Maths Chapter 9?

A: NCERT Solutions for Class 10 Maths Chapter 9 can be found on the official website of NCERT or in the NCERT textbook for Class 10 Mathematics. You can also find them on various educational websites and platforms that provide study material and resources for students.

Q: Are NCERT Solutions for Class 10 Maths Chapter 9 helpful for board exams?

A: Yes, NCERT Solutions for Class 10 Maths Chapter 9 are helpful for board exams as they provide a comprehensive understanding of the concepts and help students solve the problems given in the textbook. These solutions are designed by subject matter experts and follow the latest CBSE guidelines, making them an ideal resource for board exam preparation.

Q: Can I download NCERT Solutions for Class 10 Maths Chapter 9?

A: Yes, you can download NCERT Solutions for Class 10 Maths Chapter 9 from the official website of NCERT or from various educational websites and platforms that provide study material and resources for students. These solutions are available in PDF format and can be downloaded and saved for offline use.

Q: Are NCERT Solutions for Class 10 Maths Chapter 9 free of cost?

A: Yes, NCERT Solutions for Class 10 Maths Chapter 9 are free of cost and can be accessed and downloaded from various educational websites and platforms that provide study material and resources for students. However, some websites may require you to create an account or sign up for a subscription to access these solutions.

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