Some Applications of Trigonometry is a chapter in the CBSE Class 10 Mathematics syllabus from Mathematics. This chapter hub brings together revision notes, practice questions, worksheets, flashcards, formula sheet to help students learn, practice, and revise Some Applications of Trigonometry effectively.

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Some Applications of Trigonometry

NCERT Class 10 Mathematics Chapter 9: Some Applications of Trigonometry (Pages 133–143)

Class 10 CBSE hub Mathematics chapters Mathematics

Summary of Some Applications of Trigonometry

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Some Applications of Trigonometry at a Glance

Board

CBSE

Class

Class 10

Subject

Mathematics

Book

Mathematics

Chapter

Pages

133–143

Resources

7 study resources

Some Applications of Trigonometry Summary

In this chapter, we delve into the fascinating applications of trigonometry, particularly in measuring heights and distances without the need for direct measurements. You begin by reviewing the concepts of angles of elevation and depression. An angle of elevation occurs when an observer looks upward from a horizontal line, while an angle of depression arises when the observer looks downward. These angles are essential in everyday situations, such as determining the height of a building or a tree from a distance. The chapter presents various scenarios to illustrate how to apply trigonometric ratios to find unknown heights and distances. For instance, when trying to determine the height of an object like a tower, you will be given the distance from the observer to the base of the tower, along with the angle of elevation from the observer's eye level to the top of the tower. Using the tangent ratio, which relates the opposite side to the adjacent side of a right triangle, you can compute the height based on these measurements. Practical examples are provided, such as a tower with a known distance and angle of elevation, enabling you to calculate its height step-by-step. The process typically involves drawing a right triangle, identifying the appropriate trigonometric ratio (like tangent or sine), and solving for the height. Additionally, the chapter covers more complex problems, including multiple steps and varying angles. For instance, when a flag is hoisted on a building, you can apply your knowledge of trigonometric ratios to determine not only the height of the flagstaff but also to derive distances such as how far away one should stand to view the object clearly. The importance of these applications is highlighted when completing real-world tasks like assessing the safety of ladders when working at heights or determining the distance across a river using angles of depression. Each example carefully guides you through the necessary calculations, reinforcing how mathematics is applied to solve real-life challenges. By the end of this chapter, you should be comfortable identifying the right triangles formed by angles of elevation and depression, using trigonometric ratios to calculate heights and distances, and understanding how these principles are integrated into practical situations. These skills will not only prepare you for further studies in mathematics but also equip you with valuable tools for everyday problem-solving.

Some Applications of Trigonometry Revision Guide

Download the Some Applications of Trigonometry revision guide with key points, summaries, and quick revision notes for CBSE Class 10 Mathematics.

Key Points

Angle of Elevation: Define and use it.

The angle formed by the line of sight and horizontal when an observer looks upwards.

Angle of Depression: Define and apply.

The angle formed by the line of sight and horizontal when an observer looks downwards.

Trigonometric Ratios: Core concepts.

Primary ratios are sin, cos, and tan used to calculate angles and heights in right triangles.

Height from Angle: Use tan θ for height.

tan θ = opposite/adjacent helps find heights using known distances and angles of elevation.

Height of a Tower Example: Calculate using elevation.

From 15 m away and 60° elevation, height = 15√3 m using tan 60° = height/distance.

Electric Pole Example: Angle use for ladder length.

Length of a ladder needed is calculated using sin θ and height differences from pole top.

Chimney Height Example: Multiple parameters.

Combine observer height and angle using tan θ to derive total chimney height.

Shadow Length: Relation with sun's altitude.

Shadow length varies with sun's angle, applying tan θ reveals object height from shadow lengths.

Flagstaff Height Example: Combine two triangles.

Analyze two triangles to find the height of a flagstaff above a building using angles.

Width of River: Multi-angles application.

Use different angles of depression from a height to derive total width of a river.

Angles in Triangle: 180° rule.

The sum of angles in a triangle is always 180°, essential for calculations and proofs.

Optical Effects: Real-world applications.

Understanding angles of elevation/depression applies in architecture, navigation, and photography.

Right Angle Triangle: Fundamental structure.

Essential for all trigonometric calculations, basic properties crucial for stability and measurements.

Misconceptions: Elevation vs. Depression.

Students often confuse these angles; remember elevation is upwards, depression is downwards.

Real Heights Calculation: Use context.

Know the distance from the object and angle to calculate heights without direct measurement.

Common Ratios: Know your values.

Be familiar with basic angle values: sin, cos, tan for 30°, 45°, and 60° degrees.

Ladder Placement Example: Geometry in action.

Calculating distance from a wall using cosine functions helps with ladder safety and positioning.

Height Determination: Practical purpose.

Angles of elevation help engineers and architects estimate building heights more efficiently.

Elevation to Foot Distances: Solve for x.

Use back calculations with angles to find unknown distances in various examples and tasks.

Diagrams Importance: Visual learning.

Sketching triangles and angles helps clarify relationships and solving equations with trigonometry.

Review Key Formulas: Essential to memorize.

Understanding formulas like h = d * tan(θ) can assist transformations of problems into equations.

Some Applications of Trigonometry Practice Questions & Answers

Practice important questions and exam-style problems from Some Applications of Trigonometry. These questions cover key topics from the CBSE Class 10 Mathematics syllabus.

How to practice: Start with the questions below to test your understanding of Some Applications of Trigonometry. Use the revision guide to review concepts you find difficult, then come back and retry the questions for better retention.

What is the angle of elevation when a person looks at the top of a tree from a point 30 m away at an angle of 45°?

Single Answer MCQ

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If the height of a building is 24 m and the angle of elevation from a point on the ground 10 m away from its base is 60°, what would be the height of the observer's eyes if the height of the building is added?

Single Answer MCQ

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A person standing on the ground observes a balloon at an angle of elevation of 30°. If the balloon rises to a height where the angle of elevation is now 60°, how will this affect the height calculated?

Single Answer MCQ

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From a point 50 m away from the base of a tower, the angle of elevation to the top of the tower is 45°. What is the height of the tower?

Single Answer MCQ

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If a student is 1.5 m tall and stands 20 m away from a tree, forming an angle of elevation of 60°, how tall is the tree?

Single Answer MCQ

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When a girl looks down at a flower pot from an angle of depression of 30° from a height of 10 m, what is the horizontal distance to the pot?

Single Answer MCQ

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What happens to the perceived height of an object when the observer moves further away?

Single Answer MCQ

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If a collector views an object from 45 m away at an angle of elevation of 30°, how do you calculate the height of the object?

Single Answer MCQ

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View all 200 Some Applications of Trigonometry questions

A hill is observed from the ground at a distance of 100 m with an angle of elevation of 45°. What is the vertical height of the hill?

Which of the following best describes the angle of depression?

For a man 1.75 m tall looking up at a lamppost with an angle of elevation of 60° standing 25 m away, how can you find the height of the lamppost?

What effect does increasing the distance from the object have on the angle of elevation if the height remains unchanged?

If a tower is 40 m high and an observer sees it from 30 m away with an angle of elevation of 60°, what is the relationship between the height of the tower and the observed height?

A kite flying at an angle of elevation of 45° from a point 20 m away from the base of a tree makes which geometry in sighting?

From a point 10 m away from the base of a tower, the angle of elevation to the top of the tower is 60°. What is the height of the tower?

An observer 1.2 m tall is standing 25 m away from a building. If the angle of elevation to the top of the building from the observer's eyes is 30°, what is the height of the building?

A tree casts a shadow of 15 m when the angle of elevation of the sun is 45°. What is the height of the tree?

Find the height of a flagpole if a person standing 20 m away observes it with an angle of elevation of 60°.

A lighthouse is 50 m tall. If a boat is 100 m away from the base of the lighthouse, what is the angle of elevation to the top of the lighthouse?

An observer is standing at the ground level and looks at the top of a tree with an angle of elevation of 30°. If the tree is 10 m tall, how far is the observer from the base of the tree?

A flagstaff on a hill has a height of 12 m. From a point on the same level as the base of the hill, the angle of elevation to its top is 45°. What is the distance from this point to the base of the hill?

A tower stands vertically on the ground. If a person is standing 10 m away from the base of the tower and the angle of elevation to the top of the tower is 30°, what is the height of the tower?

The angle of elevation of the sun when the length of the shadow of a 20 m tall tree is 10 m is what?

From a point 50 m away from a building, the angle of elevation to the top of the building is 60°. What is the height of the building?

If the angle of elevation from the foot of a hill to a tower is 60° and the height of the tower is 30 m, find the distance to the foot of the hill.

An observer 1.8 m tall is standing 30 m away from a tree. If the angle of elevation to the top of the tree is 45°, what is the height of the tree?

A person standing 50 m away from a monument sees it at an angle of elevation of 55°. Find the height of the monument.

If a ladder makes an angle of elevation of 60° with the ground and reaches a point 4 m up on the wall, how long is the ladder?

What is the height of a tower whose angle of elevation at a point 30 m away is 45°?

The height of a building is 12 m. From the top of the building, the angle of depression to a point on the ground is 30°. What is the distance of the point from the base of the building?

A tree is observed at an angle of elevation of 30°. If the distance to the tree from the observer is 25 m, what height can be expected?

A flagstaff of height x meters is erected on a building of height 15 m. From a point 30 m away from the building, the angle of elevation to the top of the flagstaff is 60°. What is the value of x?

If the angle of elevation changes from 30° to 60°, how does the height of a building remain constant but change the observed distance?

The shadow of a pole is found to be 10 m long when the angle of elevation of the sun is 60°. How tall is the pole?

A man 1.8 m tall observes a building from a distance of 10 m, noticing it's 2 m taller than him. What is the angle of elevation from where he stands?

An observer is standing 20 m away from a vertical pole. If the angle of elevation from the observer's eyes (at a height of 1.5 m) to the top of the pole is 30°, what is the height of the pole?

The angles of elevation from the ground to the top and bottom of a 10 m tall building are 30° and 10°, respectively. Find the distance of the observer from the base of the building.

From a point 45 m away from a building, the angle of elevation to the top of the building is 60° and the angle of depression to the bottom is 30°. How tall is the building?

From the top of a multi-storeyed building, the angles of depression of two points on the ground are 30° and 45°. If the height of the building is H, what is the distance between the two points on the ground?

An observer's height is 1.75 m. If they stand 15 m from a tree and observe it at an angle of elevation of 45°, what is the height of the tree?

A tower stands vertically on the ground. From a point on the ground, which is 20 m away from the foot of the tower, the angle of elevation of the top of the tower is 30°. What is the height of the tower?

What is the angle of depression?

An observer 1.2 m tall is standing 15 m away from a lamp post. The angle of elevation of the top of the lamp post from her eyes is 60°. What is the height of the lamp post?

A person is standing on a hill and looking down at a point on the ground that is 30 m below. If the angle of depression is 45°, what is the horizontal distance from the person to the point?

From a point on the ground, the angle of elevation of a quayside crane is 45° and it is 10 m away. What is the height of the crane?

From a tower, the angle of depression to the foot of a hill is 30°. If the height of the tower is 20 m, how far is the foot of the hill from the base of the tower?

The shadow of a building is 50 m long when the angle of elevation of the sun is 30°. Find the height of the building.

If the angle of depression from a point A on a building is 60° to a point B on the ground, and the distance from A to B vertically is 10 m, what is the horizontal distance from the base of the building to point B?

From the top of a 50 m high tower, the angle of depression to a point on the ground is 30°. How far away is the point from the base of the tower?

When observing a ship at sea, if the angle of depression from the top of a lighthouse is 30° and the lighthouse is 50 m high, how far is the ship from the base of the lighthouse?

Two towers are 100 m apart. The angles of elevation to the top of each tower from the foot of the other tower are 45° and 30°, respectively. Compute the heights of the towers.

If a person looks down at an angle of depression of 40° to the foot of a tree and the tree is 10 m tall, what is the distance of the person from the base of the tree?

When the angle of elevation of the sun is 60°, the shadow of a tree is 5 m long. What is the height of the tree?

From the top of a hill, the angle of depression to a point on the ground is measured to be 55°. If the distance from the hill to the point is 200 m, what is the height of the hill?

An observer standing 10 m away from a vertical cliff measures the angle of elevation to the top of the cliff to be 45°. Determine the height of the cliff.

A lookout tower has a height of 80 m. If the angle of depression to a ship is 30°, how far is the ship from the base of the tower?

A building casts a shadow of 60 m when the angle of elevation of the sun is at 30°. Find the height of the building.

If the angle of elevation to the top of a building is 70° and the angle of depression to the base of the building is 40°, find the height of the building if the observer is 20 m away.

From a point on a bridge 5 m above the ground, the angles of depression to two points of the banks of a river are 30° and 60°. Calculate the width of the river.

A drone is flying at a height of 50 m. If the angle of depression to a car on the ground is 25°, how far is the car from the point directly below the drone?

A ladder leans against a wall forming an angle of elevation of 60° with the ground. If the ladder is 10 m long, how high does the ladder reach on the wall?

An observer sees a tower with an angle of elevation of 30° standing on a 10 m hill. If the observer is 40 m away from the foot of the hill, what is the height of the tower?

What is the angle of elevation from a point 10 m away from a tree that is 8 m tall?

If the angle of depression from the top of a tower is 30°, what is the relationship between the height of the tower and the horizontal distance to the observer at its foot?

From a point on the ground, the angle of elevation to the top of a building is 45°. If the observer is 20 meters away from the building, what is the height of the building?

If the angle of elevation to the top of a hill is 60° and the distance from the observer to the base is 30 m, what is the height of the hill?

A ladder 10 m long leans against a wall, forming an angle of 60° with the ground. How high up the wall does the ladder reach?

What is the angle of depression if the height of an object is 15 m and the distance from the observer to the base of the object is 20 m?

If a person is standing 40 m away from a building and the angle of elevation to the top is 30°, what is the height of the building?

In a right triangle, if the height is 24 m and the base is 10 m, what is the angle of elevation from the base to the top?

What will be the angle of elevation if a person measures the height of a pole to be 25 m standing 30 m away from its base?

A drone is flying at an angle of 45° above the ground and is 20 m away horizontally from the observer. How high is the drone?

If the angle of depression to the bottom of a well from a point 25 m above the ground is 53°, what is the depth of the well?

A person standing on the ground sees the top of a building at an angle of elevation of 30°. If the building is 10 m tall, how far is the person from the base?

Two poles of equal heights are standing opposite to each other on either side of the road which is 90 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 30° and 60° respectively. Find the height of the poles and the distances of the point from the poles. Use √3 = 1.732.

A straight road leads to foot of a tower whose shadow is found to be 40 m longer when sun’s altitude is 30° than when it is 60°. Find the height of the tower and length of shadow in both the situations. Use √3 = 1.73.

A car is moving away from the base of a 30 m high tower. The angle of elevation of the top of the tower from the car at an instant, when the car is 10√3 m away from the base of the tower, is:

Radio towers are used for transmitting a range of communication services. In the given tower, the distance between the base of the tower and point O is 6 m. From point O, the angle of elevation of the top of section B is 30°. Find the length of the wire from point O to the top of section B.

In the given tower, the distance between the base of the tower and point O is 6 m. From point O, the angle of elevation of the top of section A is 60°. Find the length of the wire from point O to the top of section A.

In the given tower case study, find the distance AB.

In the given tower case study, find the area of △OPB.

In the given figure, which of the following angles represents the angle of depression?

Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building.

An injured bird was found on the roof of a building. The building is 15 m high. A fireman used an adjustable ladder to reach the roof. He placed the ladder in such a way that the ladder makes an angle of 60° with the ground. Find the length of the ladder used by the fireman to reach the roof.

In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall opposite to the building, making an angle of 30° with the ground. Draw a neat diagram to represent the situation and hence find the width of the road between the building and the wall.

In the same case, find the length of the ladder used by the fireman.

A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as 60°. If AB = 5√3 m, then length of the wire is

Case Study: Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window W of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°. Determine the value of h.

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be 45°. The height (in metres) of the tower is:

Tejas is standing at the top of a building and observes a car at an angle of depression of 30° as it approaches the base of the building at a uniform speed. 6 seconds later, the angle of depression increases to 60°, and at that moment, the car is 25 m away from the building. What is the height of the building?

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be 45°. The height (in metres) of the tower is:

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be 45°. The height (in metres) of the tower is:

What is the height of the building?

What is the distance between the two positions of the car?

What would be the total time taken by the car to reach the foot of the building from the starting point?

What is the distance of the observer from the car when it makes an angle of 60°?

If the length of the shadow of a tower is √3 times that of its height, then altitude of the Sun is:

A boy standing on a horizontal plane is flying a kite with a string of length 60 m, at an angle of elevation of 30°. Another boy standing on the roof of a 20 m high building, finds the angle of elevation of same kite to be 45°. If both the boys are on opposite sides of the kite, find the distance of the first boy from the base of the building. Also, find the height of the kite from the ground. (Use √3 = 1.73)

If the length of the shadow of a tower is √3 times that of its height, then altitude of the Sun is:

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff. From a point on the ground 30 m away from the tower, angles of elevation of the top and foot of the flagstaff are 60° and 30° respectively. Find the height of the tower and the length of the flagstaff. Take √3 = 1.73.

If the length of the shadow of a tower is √3 times that of its height, then the altitude of the Sun is:

The angle of elevation of the top of a building from a point A, on the ground, is 30°. On moving a distance of 24 m towards its base to the point B, the angle of elevation changes to 60°. Find the height of the building and distance of point A from the base of the building. Take √3 = 1.73.

A tower stands vertically on the ground. A man standing at the top of the tower observes his friend at an angle of depression of 30°, who is approaching the foot of the tower with a uniform speed. 30 seconds later, the angle of depression changes to 60°. Find the time taken by his friend to reach the foot of the tower from this point.

A car is moving away from the base of a 30 m high tower. The angle of elevation of the top of the tower from the car at an instant, when the car is at 10√3 m away from the base of the tower, is:

A radio station tower is built in two sections A and B. Tower is supported by wires from point O. Distance between the base of the tower and point O is 6 m. From point O, the angle of elevation of the top of section B is 30° and the angle of elevation of the top of section A is 60°. Find the length of the wire from point O to the top of section B.

In the same tower figure, find the length of the wire from point O to the top of section A.

In the same tower figure, find the distance AB.

In the same tower figure, find the area of ΔOPB.

The string of a flying kite is tied to a point on the ground. The length of the string between the kite and the point on the ground is 80 m. The string makes an angle of 30° with the ground. The height of the kite above the ground is:

Radio towers are used for transmission range of communication services. Distance between the base of the tower and point O is 6 m. From point O, the angle of elevation of the top of section B is 30°. Find the length of the wire from point O to the top of section B.

From point O, the angle of elevation of the top of section A is 60°. Find the length of the wire from point O to the top of section A.

Using the radio tower figure, find the distance AB.

Using the radio tower figure, find the area of ΔOPB.

In the given figure, which of the following angles represents the angle of depression?

An injured bird was found on the roof of a building. The building is 15 m high. A fireman used a ladder making an angle of 60° with the ground to reach the roof. Find the length of the ladder used by the fireman to reach the roof.

Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building, when the ladder makes an angle of 60° with the ground and the building is 15 m high.

In the same situation, find the length of the ladder used by the fireman when it makes an angle of 30° with the ground and reaches the roof of a 15 m high building.

In the given figure, which of the following angles represents the angle of depression?

An injured bird was found on the roof of a building. The building is 15 m high. A fireman uses a ladder making an angle of 60° with the ground to reach the roof. Find the length of the ladder used by the fireman to reach the roof.

An injured bird was found on the roof of a building. The building is 15 m high. A fireman uses a ladder making an angle of 60° with the ground to reach the roof. Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building.

In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of 30° with the ground. Draw a neat diagram to represent the above situation and hence find the width of the road between the building and the wall.

In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of 30° with the ground. Find the length of the ladder used by the fireman in this case.

The length of the shadow of a tower when the sun’s altitude changes from 30° to 60° will:

Kite festival is a popular festival in India. Reena's kite is 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground and the inclination of the string with the ground is 30°. Find the length of string used by Reena.

Ravi is flying a kite from a 10 m high building. His kite is also flying 60 m above the ground and the length of the string used by Ravi is same as that of Reena. θ is the angle of elevation of Ravi’s kite from a point on the rooftop. Find the value of sin θ.

If θ changes to 60°, without changing the length of the string, what will be the height of Ravi’s kite above the ground? Use √3 = 1.7.

What would have been the height of Ravi’s kite above the ground, if the string had an inclination of 30° with the ground, assuming that the length of the string does not change?

The length of the shadow of a tower when the sun’s altitude changes from 30° to 60° will:

Reena's kite is 60 m above the ground. The string attached to the kite is tied to a point on the ground and the inclination of the string with the ground is 30°. Find the length of string used by Reena.

Ravi is flying a kite from a 10 m high building. His kite is flying 60 m above the ground and the length of the string used by Ravi is same as that of Reena. If θ is the angle of elevation of Ravi's kite from a point on the rooftop, find sin θ.

If θ changes to 60°, without changing the length of the string, what will be the height of Ravi’s kite above the ground? Use √3 = 1.7.

What would have been the height of Ravi’s kite above the ground, if the string had an inclination of 30° with the ground, assuming that the length of the string does not change?

The length of the shadow of a tower when the sun’s altitude changes from 30° to 60° will:

In a kite festival, Reena's kite is flying 60 m above the ground and the string to the kite is temporarily tied to a point on the ground, making an inclination of 30° with the ground. Find the length of string used by Reena.

Ravi is flying a kite from a 10 m high building. His kite is also flying 60 m above the ground and the length of the string used by Ravi is same as that of Reena. If θ is the angle of elevation of Ravi's kite from a point on the rooftop, find the value of sin θ.

If θ changes to 60°, without changing the length of the string, what will be the height of Ravi's kite above the ground? Use √3 = 1.7.

What would have been the height of Ravi's kite above the ground, if the string had an inclination of 30° with the ground, assuming that the length of the string does not change?

In the given figure, the angle of elevation of point A from point C is:

A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a shadow of 45 m on the ground. Find the height of the tower.

What is the width of the river?

Both the boats leave at the same time at the speeds of 10 m/s and 5 m/s respectively. Which boat will reach point D first, and how much earlier than the other boat?

Two motorboats A and B are waiting at the opposite banks of a river. From a point P on the bridge, 20 m above the river, the angles of depression of the boats are 30° and 45° respectively. Find the distance travelled by boat A to reach point D in the river, vertically below point P. Use √3 = 1.73.

What is the distance between the two boats after 3 seconds?

In the given figure, the angle of elevation of point A from point C is:

A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a shadow of 45 m on the ground. Find the height of the tower.

What is the width of the river?

Both the boats leave at the same time at the speeds of 10 m/s and 5 m/s respectively. Which boat will reach point D first, and how much earlier than the other boat?

What is the distance between the two boats after 3 seconds?

In the given figure, the angle of elevation of point A from point C is:

A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a shadow of 45 m on the ground. Find the height of the tower.

Two motorboats A and B are waiting at the opposite banks of a river. From a point P on the bridge, 20 m above the river, the angles of depression of the boats are 30° and 45° respectively. Both boats leave at the same time at speeds of 10 m/s and 5 m/s respectively. Find the distance travelled by boat A to reach point D in the river, vertically below point P. Use √3 = 1.73.

Which boat will reach point D first, and how much earlier than the other boat?

What is the distance between the two boats after 3 seconds?

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 60 m high, find the height of the building and distance between the building and the tower. (Use √3 = 1.73)

A statue, 2 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 bottom of the statue is 45°. Find the height of the pedestal and its distance from the point of observation on the ground. Use sqrt(3) = 1.73.

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 distance of the point from the poles.

The angle of elevation of the top of a tower as observed from a point in a horizontal plane through the foot of the tower is 30°. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 60°. Find the height of the tower and the distance of first position from the tower.

Two pillars of equal heights stand on either side of a roadway 150 m wide. From a point on the roadway between the pillars, the angles of elevation of the top of the pillars are 60° and 30° respectively. Find the height of the pillars and the position of the point from the pillars.

A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as 60°. If AB = 5√3 m, then length of the wire is

In the water tank case-study diagram, AB is an elevated water tank and CD is a nearby multistorey building 54 metres away. From window W, the angle of elevation of the top of the tank is 45° and angle of depression of its foot is 30°. Write a relation between d, the height of window, and y.

In the water tank case-study diagram, determine the value of h.

In the water tank case-study diagram, determine height of the water tank.

In the water tank case-study diagram, find the value of x and height of the window above ground level.

A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as 60°. If AB = 5√3 m, then length of the wire is

Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window W of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°. Write a relation between d, the height of window, and y.

Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window W of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°. Find the value of x and height of the window above ground level.

A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as 60°. If AB = 5√3 m, then length of the wire is

Determine the value of h.

Find the value of x and height of the window above ground level.

Determine height of the water tank.

A kite is flying at a height of 60 m above the ground level. Ravi, standing at the roof of the house is holding the string straight and observes the angle of elevation of kite as 30°. From the bottom of the same building, the angle of elevation of kite is 45°. Find the length of the string and height of roof from the ground. Use √3 = 1.73.

A kite is flying at a height of 60 m above the ground level. Ravi, standing at the roof of the house, is holding the string straight and observes the angle of elevation of kite as 30°. From the bottom of the same building, the angle of elevation of kite is 45°. Find the length of the string and height of roof from the ground. Use √3 = 1.73.

A man standing on the deck of a ship, which is 10 m above the water level, observes that the angle of elevation of the top of a hill is 60° and the angle of depression of the base of the hill is 30°. Find the height of the hill.

The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. From a point Y, 40 m vertically above X, the angle of elevation of Q is 45°. Find the height of the tower PQ and the distance XP. Use √3 = 1.732.

A ladder 14 m long leans against a wall. If the foot of the ladder is 7 m from the wall, then the angle of elevation of the top of the wall is:

Case Study 2: Amrita measured the angle of elevation to the top of a lighthouse as 60°. From an observation deck 40 m higher, the angle of elevation to the top was 45°. If CD is h metres, find the distance BD in terms of h.

Case Study 2: Amrita measured the angle of elevation to the top of a lighthouse as 60°. From an observation deck 40 m higher, the angle of elevation to the top was 45°. Find distance BC in terms of h.

Case Study 2: Amrita measured the angle of elevation to the top of a lighthouse as 60°. From an observation deck 40 m higher, the angle of elevation to the top was 45°. Find the height CE of the lighthouse. Use √3 = 1.73.

Case Study 2: Amrita measured the angle of elevation to the top of a lighthouse as 60°. From an observation deck 40 m higher, the angle of elevation to the top was 45°. Find distance AE, if AC = 100 m.

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Some Applications of Trigonometry Practice Worksheets

Download and practice Some Applications of Trigonometry worksheets to improve problem-solving accuracy and speed for CBSE Class 10 Mathematics exams.

Some Applications of Trigonometry - Practice Worksheet

This worksheet covers essential long-answer questions to help you build confidence in Some Applications of Trigonometry from Mathematic for Class 10 (Mathematics).

Practice

Questions

Explain the concept of angle of elevation and provide an example with a diagram illustrating how it is applied in real life.

The angle of elevation is the angle formed by the line of sight from an observer to a point above the horizontal level. For example, if a person is standing 10 meters away from a tree and looking up at the top of the tree, the angle formed at the observer's eye level and the line of sight to the top of the tree is the angle of elevation. Using trigonometric ratios, we can calculate the height of the tree based on this angle and the distance from the base. Draw a right triangle to visualize this relationship. The formula tan(θ) = opposite/adjacent can be used where θ is the angle of elevation.

What is the angle of depression? Illustrate its significance using a relevant example.

The angle of depression is the angle formed by the line of sight from an observer to a point below the horizontal level. For example, if someone standing on a hill looks down at a car parked at the base, the angle between their line of sight and the horizontal line is the angle of depression. This concept helps determine the height of the hill or any elevation by using trigonometric ratios. To solve problems, often tan(θ) = opposite/adjacent is employed. Diagrams can clarify the observer’s position and the point below them.

Calculate the height of a tower if the angle of elevation from a point 20 m away from its base is 45°.

To find the height of the tower, label the tower's height as h. You can use the tan function, considering tan(45°) = 1. The distance from the base is 20 m, so tan(45°) = h/20. This simplifies to h = 20 m. Thus, the tower's height is 20 m. Key points include the right triangle formed and understanding that the opposite side is the height and the adjacent is the distance from the tower.

Describe how to determine the height of a building using the angle of elevation and distance from the building.

To determine the height of the building (let's say it is h), identify the distance (d) from the observer to the building and the angle of elevation (θ). Using the formula tan(θ) = h/d, you can rearrange it to solve for h: h = d × tan(θ). For example, if the distance is 30 m and the angle of elevation is 30°, then h = 30 × tan(30°), which is 30 × (1/√3) = 10√3 m.

A ladder needs to be propped against a wall at an angle of 60°. If the foot of the ladder is 5 m away from the wall, how long is the ladder?

To find the ladder's length (hypotenuse), use the sine relation. In this case, h is the height opposite to the angle of elevation. The equation sin(60°) = opposite/hypotenuse applies. The height can be defined as h, and based on trigonometry, h = 5 × tan(60°). Given tan(60°) = √3, we find the height is 5√3. Using the Pythagorean theorem, the length of the ladder can then be calculated as length = √(5² + (5√3)²). This simplifies to find the accurate ladder length.

Explain how to find the width of a river if angles of depression from a point are known.

From a certain height, if the angles of depression to banks on either side are given, you can denote the height as h. By creating two right triangles on either side, you can use tan(θ) = h/distance. Calculate both sides' distances using tan(30°) and tan(45°). For example, if height is 3 m, use distance calculations for each angle, then add these distances for the total width of the river. This approach highlights principles of parallel lines and transversals in triangles.

Given a tower's shadow is longer at 30° than at 60°, how can you find the height of the tower?

Given two angles, you know the length of shadows will differ. Use tan(60°) = height/shadow length to derive equations for both angles. For instance, at 60°, if shadow x → height = x√3. At 30°, shadow extends to x + 40m → height = (x + 40)/√3. Equate both height formulas to establish an equation. Solve for x to find the height. Utilize the relationship between different triangle angles to guide calculations.

How do you calculate the total height of a flagpole given the building height and the angle of elevation?

If you know the building height (h) and the angle of elevation from a distance (d) to the top of the flagpole, add the height of the building to the additional height gained through tan(θ) application. Use the formula: Total Height = Building Height + (d tan(θ)). This method illustrates how angles of elevation gain height additively as distances are calculated forward from a point.

Describe how to find the height of a multi-storey building using angles of depression.

To determine the height of a building (let's name it PC), observe angles of depression to a shorter building (AB). By applying tan(θ), derive equations for both height relationships. Use properties of transversal angles as you relate angles to identified heights. Essentially, you will calculate PD using tangents, relating ratios of heights to base distances as you simplify ratios downward towards the observer’s point.

Some Applications of Trigonometry - Mastery Worksheet

This worksheet challenges you with deeper, multi-concept long-answer questions from Some Applications of Trigonometry to prepare for higher-weightage questions in Class 10.

Mastery

Questions

1. A tower stands vertically on the ground. From a point 20 m away from its foot, the angle of elevation of the top of the tower is 30°. Calculate the height of the tower. Additionally, if a flagstaff of height x is placed on top of the tower, what is the total height of the tower (including the flagstaff) if the angle of elevation increases to 45° when viewed from the same point?

Using tan(30°) = height/20. Calculate height = 20√3 m. Second part: tan(45°) = (height + x)/20, leading to x = 20 m. Total height = height + x = 20√3 + 20 m.

2. An observer 1.6 m tall is standing 25 m away from a building. The angle of elevation to the top of the building is 60°. Find the height of the building. If an adjacent building is 10 m shorter, calculate its height.

Height = AE + BE where AE = 1.6 m. Using tan(60°) = height/25. Solve to find the height of the taller building to be approximately 43.3 m, hence the shorter building is 33.3 m.

3. A flagstaff is placed atop a 12 m high building. If an observer at a distance of 15 m measures the angle of elevation to the top of the flagstaff as 45°, determine the height of the flagstaff.

Use tan(45°) = (height of building + height of flagstaff) / 15. This gives height of flagstaff = 15 m - 12 m = 3 m.

4. A person is standing at the top of a hill. The angle of depression to a point on the ground is 30°. If they know the height of the hill is 100 m, determine the horizontal distance from the base of the hill to the point on the ground.

Using tan(30°) = 100/distance, solve to find distance = 100√3 m, approximately 173.2 m.

5. From the top of a 50 m high building, the angle of depression to the foot of another building is 60°. Calculate the height of the second building if the angle of elevation to the top of the first building is 45° from the ground.

Use tan(60°) = 50/(distance). Then, height of second building = height of first building - (distance * tan(45°)). After calculations, get height = 50 - (50/√3) m.

6. A ladder leans against a wall. The foot of the ladder is 2 m away from the wall and makes a 60° angle with the ground. Find the height at which the ladder touches the wall.

Using sin(60°) = height/2. Solve for height = 2√3 m, approximately 3.46 m.

7. Two buildings are 100 m apart. From the top of one building, the angle of depression to the foot of the other is 30°. If the height of the first building is 80 m, determine the height of the second building.

Use h = 80 - 100*(tan 30°). Calculate to find the second building is approximately 72.3 m tall.

8. The shadow of a 9 m tall pole is found to be 15 m long when the angle of elevation of the sun is θ. If the angle of elevation increases to 45°, what will be the new height of light intensity from the same pole's base using trigonometric ratios?

Using tan(θ) = 9/15 gives you θ. When it’s 45°, height remains 9 m. This is more of a reasoning question to derive tan implications.

9. A train is moving at a mountain whose height is 300 m. From a distance of 1 km, it views the mountain at an angle of 45°. Find what fraction of the mountain is viewed?

Using tan(45°) = 300/1000. This represents full view, hence an accessible height of 300 m.

10. An observer looks at two points on a horizontal plane from an elevated point 50 m above. The angles of depression to the points are 30° and 45°. Calculate the distance between the two points.

Using tan(30°) = 50/x1 and tan(45°) = 50/x2. Solve these for x1, x2, and find the difference in distances.

Some Applications of Trigonometry - Challenge Worksheet

The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Some Applications of Trigonometry in Class 10.

Challenge

Questions

Using the concept of angles of elevation, analyze how the height of a lighthouse can change the navigational safety of ships. Consider variations in ship distance and height.

Evaluate the role of lighthouse height relative to distance. Explore the implications for navigation safety using trigonometric principles.

Discuss how trigonometry can be applied to optimize the design of a ramp for accessibility at different angles. Evaluate economic factors involved.

Assess various angle configurations and their practicality. Back your evaluation with example designs and cost considerations.

Explore the scenario where a basketball player attempts a shot from various angles. How does trigonometric analysis improve shot accuracy?

Delve into the relationship between angle of elevation and shot success, supported by statistical shooting data.

A farmer is planning an irrigation system using water from a well. How does understanding the angle of elevation help them optimize water usage?

Evaluate the relationship between height and distance. Discuss various irrigation designs based on trigonometric calculations.

Critically analyze how the angles of depression from a drone can be used in surveying land. Discuss the precision and accuracy in measurement.

Examine different drone heights and their relation to measurement errors. Support your argument with case studies.

Determine how shadow length change can affect a solar panel's efficiency throughout the day using trigonometry.

Use trigonometric ratios to analyze the relationship between angle of elevation of the sun and shadow length. Propose solutions to maximize efficiency.

How can architects use trigonometric concepts to design buildings that improve natural lighting? Evaluate the effectiveness of angles of elevation.

Use examples of known buildings and their light intake. Assess angles that optimize natural light while considering thermal efficiency.

Examine a scenario where emergency services need to rescue a person on a cliff. How do angles of elevation and depression play a critical role in strategizing the rescue?

Evaluate rescue techniques and assess the role of trigonometry in determining distances and safety measures.

Discuss the implications of trigonometric ratios in the construction of bridges over varying terrains. How does this influence structural integrity?

Analyze different slope angles and their effects on material choice and load capacity. Use examples from civil engineering.

Consider the case of a race car on a banked track. How does the angle of the banking affect the vehicle's speed and safety?

Discuss the trigonometric principles behind forces acting on the car and calculate optimal banking angles.

Some Applications of Trigonometry Formula Sheet

Use this Class 10 Mathematics Some Applications of Trigonometry Formula Sheet for quick revision before school exams and CBSE exams. It brings together the important formulas, key concepts, and worked examples in one place so students can revise faster and download a printable PDF for offline study.

Important Formulas

Height of an Object: h = d × tan(θ)

h is the height of the object, d is the distance from the object, and θ is the angle of elevation. This formula is used to determine the height of an inaccessible object by measuring the angle from a known distance.

Distance from Object: d = h / tan(θ)

d is the distance to the object, h is the height of the object, and θ is the angle of elevation. Used to calculate how far one should stand to measure an object's height.

Length of Ladder: L = h / sin(θ)

L is the length of the ladder, h is the vertical height to be reached, and θ is the angle of elevation of the ladder to the horizontal ground. This is used for determining ladder length needed to reach a specific height at a given angle.

Shadow Length: L = h / tan(α)

L is the length of the shadow, h is the height of the object, and α is the angle of elevation of the sunlight. This helps to find the length of the shadow cast by an object.

Width between two points: W = tan(θ1) × h1 + tan(θ2) × h2

W is the width between two observation points, θ1 and θ2 are the angles of depression to the base of the observations, and h1 and h2 are the heights of the observation points.

Height from Angle of Depression: h = d × tan(θ)

h is the height of an object viewed from an angle of depression θ, and d is the horizontal distance. This is used when assessing heights from above.

Angle of Elevation: tan(θ) = opposite / adjacent

This foundational relationship ties trigonometric functions to right triangles and assists in solving for unknown angles.

Angle of Depression: tan(θ) = opposite / adjacent

Similar to angle of elevation, this applies to angles measured looking downward. Useful for calculating heights or distances from above objects.

Height of Chimney: h = d × tan(θ) + observer height

Calculates the total height of structures by adding observer height at the point of measurement to the result of the tangent function.

Height of Tower: h = d × tan(θ)

Finds the height of a tower using distance from the base (d) and angle of elevation (θ).

Worked Examples

tan(60°) = h / 15

Used in calculating the height of a tower where 15 m is the distance from the base, deriving the equation for h will give h = 15√3.

sin(60°) = 3.7 / L

From the angle measurement on a ladder, this equation is used to calculate the length (L) required when the height is 3.7 m.

tan(45°) = AE / 28.5

This equation helps derive the height of a chimney where AE needs to be solved; results in AE = 28.5.

tan(30°) = 10 / PA

This relates the distance PA to the height of a 10 m building, and solving gives PA = 10√3.

tan(60°) = h / x

This is used in triangles for tower shadow problems where h is the unknown height in terms of shadow length (x).

AB = AE + 1.5

Defines the total height of the chimney in terms of the observer's height and the distance observed.

BC = 40 + x

This describes the relation where distances to shadows switch due to angle variations.

DC / BD = cot(60°)

This equation relates vertical distance to the angle and is key in calculating the foot distance from the ladder to its base.

tan(30°) = PD / AD

This equation helps in solving for the width of a river, using heights and angles observed above.

h = 20√3

This equation denotes the height of a physical object, like a tower, derived from trigonometric ratios and distances.

Explore More Some Applications of Trigonometry Resources

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Some Applications of Trigonometry Frequently Asked Questions

Explore the applications of trigonometry focusing on heights and distances in real-life scenarios in the Class 10 Mathematics curriculum. Learn key concepts and solve practical problems.

QWhat is the angle of elevation?

The angle of elevation is the angle formed by the line of sight of an observer looking upwards from a horizontal surface to an object above that surface. It helps in determining heights without direct measurement.

QWhat is the angle of depression?

The angle of depression is the angle formed by the line of sight of an observer looking downwards from a horizontal line. This angle is critical for calculating distances and heights of objects below the observer's level.

QHow can I calculate the height of a tower using trigonometry?

To calculate the height of a tower, you can measure the distance from the foot of the tower and determine the angle of elevation using a protractor. Then, apply the tangent function: height = distance * tan(angle).

QWhat are trigonometric ratios?

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. The primary ratios include sine, cosine, and tangent, which help calculate unknown side lengths or angles in triangles.

QWhat role do trigonometric ratios play in real-life applications?

Trigonometric ratios are vital in fields such as architecture, engineering, navigation, and physics. They help in solving practical problems related to heights, distances, and angles occurring in real-world scenarios.

QCan I find the height of a building just by observing its angle of elevation?

Yes, by measuring the angle of elevation from a specific distance from the building and using trigonometric ratios, you can calculate the height of the building accurately.

QWhat is the significance of heights and distances in trigonometry?

Heights and distances are fundamental concepts in trigonometry that enable us to understand and calculate physical dimensions and layouts in various professional practices like surveying and architecture.

QWhat example illustrates the use of the angle of elevation?

An example involves a student standing a certain distance from a tower and measuring the angle of elevation to the top. This information can be used with trigonometric ratios to calculate the tower's height.

QHow do I calculate the length of a ladder needed to reach a height?

To find the ladder's length needed to reach a height, use the sine function: length = height/sin(angle), where the angle is the inclination of the ladder to the ground.

QWhat is one of the common uses of trigonometry in everyday life?

Trigonometry is commonly used in architecture to design buildings and bridges by calculating angles, heights, and distances to ensure structural integrity and aesthetics.

QHow does trigonometry help in determining the width of a river?

By measuring the angle of depression from a point above the river banks, and utilizing the heights and distances, you can apply trigonometric functions to calculate the river's width accurately.

QWhat is a practical example of using angles of depression?

A practical example includes an observer on a bridge measuring the angle of depression to points on opposite river banks. This helps determine the width of the river using trigonometric ratios.

QWhat are some common problems solved using trigonometry?

Common problems include finding the height of trees, buildings, or towers, determining distances across terrain, and calculating angles in various engineering projects.

QHow do you approach a problem involving angles of elevation?

To approach such problems, sketch the scenario, identify known values (distance and angle), and apply the appropriate trigonometric ratio, usually tangent, to find the unknown height.

QWhat should I remember when using trigonometry in calculations?

Always keep track of the angles and use the correct trigonometric ratios. Ensure your units match, and check your calculations carefully to avoid errors.

QWhat is the relationship between angles of elevation and angles of depression?

Angles of elevation and depression are complementary in many scenarios. If an angle of elevation is measured, the corresponding angle of depression from the same horizontal level can often be calculated.

QHow can trigonometry be applied in sports?

In sports, trigonometry can help analyze angles for shot trajectories in games like basketball or soccer, helping coaches and players improve performance based on these analyses.

QWhat is the height of a pole if the angle of elevation is known?

To find the pole's height, the distance from the observer to the pole and the angle of elevation are needed. Using tangent, height = distance * tan(angle), will provide the result.

QHow do you use trigonometric ratios in surveying?

Surveyors use trigonometric ratios to calculate distances and angles for land measurement. They can determine plot sizes, boundaries, and elevations without needing direct measurements.

QWhat techniques can be used for height calculation?

Height calculations can utilize direct measurement methods or trigonometric techniques, taking into account the distance from the object and the observed angle of elevation or depression.

QWhy is it important to understand angles in trigonometry?

Understanding angles in trigonometry is essential as it allows for accurate calculations in measurements and is fundamental in various applications across different scientific fields.

QCan trigonometry help in navigation?

Yes, trigonometry is crucial in navigation as it helps determine paths, distances, and angles required for plotting courses, especially in maritime and aerial navigation.

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Some Applications of Trigonometry Flashcards

Revise key terms and definitions from Some Applications of Trigonometry with interactive flashcards. Quick recall practice for CBSE Class 10 Mathematics.

These flash cards cover important concepts from Some Applications of Trigonometry in Mathematics for Class 10 (Mathematics).

1/20

What is the line of sight?

1/20

The line of sight is the line drawn from the eye of an observer to the point in the object viewed.

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2/20

Define the angle of elevation.

2/20

The angle of elevation is the angle formed by the line of sight with the horizontal when looking up at an object.

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3/20

Define the angle of depression.

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3/20

The angle of depression is the angle formed by the line of sight with the horizontal when looking down at an object.

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4/20

Formula for height using angle of elevation.

4/20

To find height (h), use: h = d * tan(θ), where d is the distance from the observer to the object and θ is the angle of elevation.

5/20

How to find height of a tower?

5/20

Use the tan function: tan(θ) = height of tower (h) / distance from base (d). Rearranging gives h = d * tan(θ).

6/20

Example of angle of elevation problem.

6/20

From 15 m away, if the angle of elevation is 60°, height = 15 * √3 m.

7/20

Length of a ladder problem.

7/20

For a 5 m pole and an angle of 60°, the ladder length = (3.7 * 2)/√3 ≈ 4.28 m.

8/20

How to calculate distance from a building?

8/20

Use the formula: Distance = height / tan(angle of elevation).

9/20

What is a common mistake with trigonometric ratios?

9/20

Mixing up sin, cos, and tan for the wrong triangle sides.

10/20

Difference between angle of elevation and angle of depression.

10/20

Angle of elevation looks upward, while angle of depression looks downward from a horizontal line.

11/20

What to include in a height problem?

11/20

Include distance to object, height of observer, and angle of elevation or depression.

12/20

How do you find the width of a river?

12/20

Use angles of depression from a height to calculate distances on either bank.

13/20

What determines the use of a trig ratio?

13/20

The known and unknown sides in relation to the angle determine whether to use sin, cos, or tan.

14/20

Calculate the height of a chimney.

14/20

If observer is 1.5 m tall and 28.5 m away with an angle of elevation of 45°, chimney height = 30 m.

15/20

What is the importance of using diagram?

15/20

Diagrams help visualize relationships and angles in trigonometric problems simplifying calculations.

16/20

What does a right triangle consist of?

16/20

A right triangle includes one angle measuring 90° and can be used to apply trigonometric ratios.

17/20

How do you approach an elevation problem?

17/20

Identify known distances and angles, choose the correct trigonometric function, and solve for the unknown.

18/20

What is the method to find a shadow's length?

18/20

Using tan(angle) = height of object / length of shadow aids in calculating shadow length.

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What can affect the calculated height?

19/20

Incorrect measurements of distance or angles can lead to errors in height calculations.

20/20

Define trigonometric ratio.

20/20

A trigonometric ratio compares the sides of a right triangle relative to its angles.

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