YOUR STUDY SPACE · CBSE

Some Applications of Trigonometry — Class 10 Mathematics

Mathematics · Chapter 9

Download revision card →

CBSE Class 10 community

View all

All available subjects

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Mathematics
Science
+ 10 more

~7 pages per kitPage 1 free

Get all 12 kits · ₹299

Mathematics (Standard and Basic)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

70+ formulas and theorems
Board-paper chapter counts
24 traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹69

Social Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

History
Economics
+ 2 more

~7 pages per kitPage 1 free

Get all 4 kits · ₹69

English

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

First Flight
Footprints without Feet

~7 pages per kitPage 1 free

Get both kits · ₹69

Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Equations and formulas
Book tables as charts
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹69

Hindi A · Kshitij 2 + Kritika 2

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

15 पाठ
व्याकरण
लेखन प्रारूप

Only 7 pagesPage 1 free

Get the kit · ₹69

Computer Applications 2027

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

HTML & CSS examples
Protocol tables
Cyber ethics

Only 7 pagesPage 1 free

Get the kit · ₹69

Hindi B · Sparsh 2 + Sanchayan 2

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

17 पाठ
व्याकरण
लेखन प्रारूप

Only 7 pagesPage 1 free

Get the kit · ₹69

Sanskrit · Shemushi 2 + Vyakaranavithi + Abhyaswaan Bhav

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

9 पाठ
सन्धि-समास
प्रत्यय

Only 7 pagesPage 1 free

Get the kit · ₹69

Browse all Class 10 kits →

Chapter summary

This chapter applies trigonometric ratios to practical problems of heights and distances that cannot be measured directly. The key ideas are the angle of elevation (looking up at an object) and the angle of depression (looking down). By modelling a situation as a right-angled triangle and using the appropriate ratio, students can find the height of a building, the width of a river or the distance of a ship. The chapter strengthens the ability to translate a real scene into a diagram and solve it with trigonometry.

Angle of elevationAngle of depressionHeights and distances problems

Key ideas & terms

Angle of elevation: The angle above the horizontal when looking up at an object.
Angle of depression: The angle below the horizontal when looking down at an object.
Line of sight: The straight line from the observer’s eye to the object.
Heights and distances: Problems solved using right-triangle trigonometry.

Extra questions & answers

Define the angle of elevation.

The angle formed by the line of sight with the horizontal when the object viewed is above the horizontal level.

Which trigonometric ratio links the opposite side and the hypotenuse?

The sine ratio, .

What is the line of sight?

The straight line drawn from the eye of an observer to the point being viewed on the object.

Define the angle of elevation.

The angle between the horizontal and the line of sight when the object viewed is above the horizontal level (refer to the figure in your NCERT textbook).

Define the angle of depression.

The angle between the horizontal and the line of sight when the object viewed is below the horizontal level.

Which trigonometric ratio links the height and the base in a height-and-distance problem?

The tangent ratio, since .

A tower casts a shadow such that the angle of elevation of the sun is . How are the height and shadow related?

Since , the height of the tower equals the length of its shadow.

The angle of elevation of the top of a 30 m tower from a point on the ground is . Find the distance of the point from the base.

Why does the angle of elevation increase as one moves towards a tower?

Because the horizontal distance decreases while the height stays the same, so the line of sight makes a larger angle with the horizontal.

What practical problems can be solved using trigonometry?

Finding the heights of towers, buildings, hills and trees, and the widths of rivers or distances of ships, without measuring them directly.

EXERCISE 9.1

1. A circus artist is climbing a 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 (see Fig. 9.11).

Let the height of the pole be . The rope is the hypotenuse of the right triangle. Therefore, Hence, Therefore, the height of the pole is .

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 with it. The distance between the foot of the tree to the point where the top touches the ground is . Find the height of the tree.

Let the unbroken vertical part be and the length of the broken part be . For the unbroken part, Hence, Also, so Therefore, the original height of the tree was

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

For the first slide, let its length be . Then Therefore, For the second slide, let its length be . Then Therefore, Thus, the required slide lengths are and , respectively.

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

Let the height of the tower be . Then Hence, Therefore, the height of the tower is .

5. A kite is flying at a height of 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 . Find the length of the string, assuming that there is no slack in the string.

Let the length of the string be . Then Hence, Therefore, the length of the string is .

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

The vertical distance from the boy's eyes to the top of the building is Let his initial distance from the building be . Then so Let his distance after walking be . Then so Therefore, the distance walked is

7. 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 high building are and respectively. Find the height of the tower.

Let the horizontal distance of the observation point from the building be . Since the angle of elevation of the top of the building is , Hence, . Let the total height of the building and tower be . Then so Therefore, the height of the transmission tower is

8. A statue, 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 and from the same point the angle of elevation of the top of the pedestal is . Find the height of the pedestal.

Let the height of the pedestal be and the horizontal distance of the observation point from it be . Since we get . For the top of the statue, Therefore, Thus, the pedestal is high.

9. The angle of elevation of the top of a building from the foot of the tower is and the angle of elevation of the top of the tower from the foot of the building is . If the tower is high, find the height of the building.

Let the distance between the tower and the building be , and let the height of the building be . From the foot of the building, Hence, From the foot of the tower, Therefore, Thus, the height of the building is .

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

Let the distance from the point to the pole whose angle of elevation is be . The distance to the other pole is then . If the common height is , then and Therefore, The other distance is , and Thus, the poles are high, and the point is and from them.

11. A TV tower stands vertically on a 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 . From another point away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is (see Fig. 9.12). Find the height of the tower and the width of the canal.

Let the width of the canal be and the height of the tower be . From the point directly opposite the tower, so From the point farther away, so Equating the two values of , Hence, Therefore, the width of the canal is and the height of the tower is .

12. From the top of a high building, the angle of elevation of the top of a cable tower is and the angle of depression of its foot is . Determine the height of the tower.

Let the horizontal distance between the building and the tower be . Since the angle of depression of the tower's foot is , Hence, . Let the tower's height above the top of the building be . Then so Therefore, the total height of the tower is

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

Let the distances of the ships from the foot of the lighthouse be and , where the farther ship has an angle of depression of . Then so For the nearer ship, so Hence, the distance between the ships is

14. A tall girl spots a balloon moving with the wind in a horizontal line at a height of from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is . After some time, the angle of elevation reduces to (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

The vertical distance between the girl's eyes and the balloon is Let the initial horizontal distance be . Then so Let the later horizontal distance be . Then so Therefore, the distance travelled by the balloon is

15. 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 , 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 . Find the time taken by the car to reach the foot of the tower from this point.

Let the height of the tower be . When the angle of depression is , let the car's distance from the tower be . Then so Six seconds later, let its distance be . Then so The distance travelled in seconds is The remaining distance is half the distance covered in seconds. Therefore, at the same uniform speed, the car will cover it in