Page 1: Introduction

This chapter applies trigonometric ratios to real-life problems involving heights and distances.

Main concepts: angle of elevation, angle of depression, line of sight.

Problems usually involve right triangles.

Page 2: Angle of Elevation

Angle between line of sight and horizontal when object is above observer.

Used for height of tower, tree, building, cloud.

Page 3: Angle of Depression

Angle between line of sight and horizontal when object is below observer.

Used for depth, distance of ship from lighthouse.

Page 4: Key Assumptions

• Observer’s eye at ground level (unless stated)
• Distances measured horizontally
• Objects vertical (tower, pole)

Page 5: Basic Problem Type 1

Find height of object.

tan θ = height / distance → height = distance × tan θ

Page 6: Basic Problem Type 2

Find distance from object.

tan θ = height / distance → distance = height / tan θ

Page 7: Two Angles of Elevation

Observer moves towards/away from object.

Form two equations → solve for height and distance.

Page 8: Angle of Depression Problems

From top of lighthouse/tower to ship.

Same as elevation but angle below horizontal.

Page 9: Multiple Objects

Two towers, buildings, or objects.

Use common angles or difference.

Page 10: Key Tips for Solving

• Draw clear labelled diagram
• Identify right triangle
• Choose correct ratio (tan most common)
• Use values from Chapter 8 table

Page 11: Practice Questions - Easy (1-10)

1. Define angle of elevation.
2. tan 30° value.
3. Height = ? if distance 100 m, elevation 30°.
4. Distance if height 50 m, elevation 45°.
5. Angle of depression from 60 m tower to ship 100 m away.
6. tan 60° = √3 use.
7. Simple height numerical.
8. Simple distance numerical.
9. Diagram description for elevation.
10. Line of sight meaning.

Page 12: Practice Questions - Medium (11-20)

11. Height of tower if elevation 30° at 50 m, 60° at 20 m.
12. Distance of ship from lighthouse 100 m high, depression 30°.
13. Two angles of elevation 30° and 45° from point to tower.
14. Width of river using elevation from bank.
15. Height of cloud above lake using reflection.
16. Man 1.7 m tall, shadow problem type.
17. Angle changes when moving towards tower.
18. Find height using two positions.
19. Depression to boat from cliff.
20. Tower height with broken top type hint.

Page 13: Practice Questions - Hard (21-30)

21. Two towers on either side of road.
22. Complex elevation from two points.
23. Board level word problem with multiple steps.
24. Height of building with flagstaff.
25. Angle to top and bottom.
26. Shadow of two poles.
27. Airplane height problem.
28. River width with angles from both banks.
29. Advanced depression numerical.
30. Combined elevation and depression.

Page 14: Common Problem Types

Single angle, two angles, depression, width, shadow.

Page 15: Common Mistakes

• Wrong diagram
• Using wrong ratio
• Forgetting tan for height/distance
• Wrong angle identification
• Calculation error in values

Page 16: Previous Year Questions

Height of tower, width of river, angle changes.

Page 17: Exam Tips

• Draw diagram always
• Label angles clearly
• Write tan θ = opposite/adjacent
• Use exact values (√3, 1/√3)
• Show all steps

Page 18: Quick Revision Sheet

Key concepts, common setups.

Page 19: Final Motivation

Chapter 9 complete! Applications is scoring with practice.

Master diagram and tan usage.

Class 10 Maths unstoppable 🦖

Page 20: Extra Numericals

More solved problems.

Page 21: Diagram Tips

How to draw correctly.

Page 22: Common Setups

Tower, river, ship.

Page 23: Board Pattern Questions

Typical word problems.

Page 24: Extra Practice

More questions.

Page 25: Real-Life Uses

Surveying, navigation, architecture.

Page 26: Trigonometric Values Recap

30°, 45°, 60° table.