Some Applications of Trigonometry Chapter 9 | Mathematics

Complete NCERT notes, line of sight, angle of elevation, angle of depression, heights and distances problems, and solved examples for CBSE Class 10 Board Exam 2025-26.

📖 Introduction 📏 Basic Terms 📐 Angle of Elevation 📐 Angle of Depression 📐 Important Formulas 📝 Solved Examples ⭐ Key Points

📖 Introduction

Trigonometry is widely used in real-life applications to measure heights and distances indirectly. It is used in navigation, surveying, astronomy, architecture, and engineering.

💡 Theodolite is an instrument used to measure angles of elevation and depression.

📏 Basic Terms

Line of Sight: The line drawn from the eye of the observer to the point being viewed.
Horizontal Line: A line parallel to the ground through the observer's eye.
Angle of Elevation: Angle between line of sight and horizontal when looking up.
Angle of Depression: Angle between line of sight and horizontal when looking down.

📐 Angle of Elevation

📌 Definition

When we look up at an object, the angle formed between the line of sight and the horizontal line is called the angle of elevation.

📌 Formula

tan θ = Height / Distance

Height = Distance × tan θ

Distance = Height / tan θ

📐 Angle of Depression

📌 Definition

When we look down at an object, the angle formed between the line of sight and the horizontal line is called the angle of depression.

💡 Angle of elevation from one point = Angle of depression from the other point when both points are at different heights.

📐 Important Formulas

1️⃣ Height from angle and distance

h = d × tan θ

2️⃣ Distance from height and angle

d = h / tan θ

3️⃣ Two angles of elevation

If angle changes from α to β, distance moved = h(cotα - cotβ)

4️⃣ Height of cloud/balloon

Height = d × tanθ

📌 Standard Values Table

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞

📝 NCERT Solved Examples

Example 1: Angle of Elevation

A tower stands vertically on the ground. From a point on the ground 15 m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. Find the height of the tower.

tan 60° = Height / 15 → √3 = h/15 → h = 15√3 = 15 × 1.732 = 25.98 m
∴ Height of tower = 15√3 m (≈ 25.98 m)

Example 2: Angle of Depression

From the top of a 7 m high building, the angle of depression of the foot of a tower is 45°. Find the distance between the building and the tower.

tan 45° = Height of building / Distance → 1 = 7 / d → d = 7 m
∴ Distance = 7 m

Example 3: Two Angles of Elevation

The angle of elevation of the top of a tower from a point on the ground is 30°. After moving 50 m towards the tower, the angle becomes 60°. Find the height of the tower.

Let height = h. From first point: tan30° = h/x → 1/√3 = h/x → x = h√3
From second point: tan60° = h/(x-50) → √3 = h/(x-50) → x-50 = h/√3
Substitute: h√3 - 50 = h/√3 → Multiply by √3: 3h - 50√3 = h → 2h = 50√3 → h = 25√3 m

Example 4: Two Objects on Same Side

From the top of a 60 m high lighthouse, the angles of depression of two ships on the same side are 30° and 45°. Find the distance between the ships.

tan45° = 60/d₁ → 1 = 60/d₁ → d₁ = 60 m
tan30° = 60/d₂ → 1/√3 = 60/d₂ → d₂ = 60√3 m
Distance between ships = d₂ - d₁ = 60√3 - 60 = 60(√3 - 1) m ≈ 60 × 0.732 = 43.92 m

Example 5: Airplane Problem

An airplane flying at a height of 3000 m passes vertically above another airplane at an instant when the angles of elevation of the two airplanes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between them.

Let distance of point from ground = d. For lower plane: tan45° = h₁/d → 1 = h₁/d → h₁ = d
For higher plane: tan60° = h₂/d → √3 = h₂/d → h₂ = d√3
But h₂ = 3000 m → d = 3000/√3 = 1000√3 m
Then h₁ = 1000√3 m
Vertical distance = h₂ - h₁ = 3000 - 1000√3 = 1000(3 - √3) m

⭐ Key Points for Board Exam

Angle of Elevation: Measured upward from horizontal.
Angle of Depression: Measured downward from horizontal.
Angle of elevation = Angle of depression when looking from one point to another at different heights.
Always draw a clear figure showing the horizontal line, line of sight, and known distances.
Use tan ratio most frequently for height-distance problems.
When an object moves towards/away from the observer, the angle of elevation changes.
For two points on the same side of the observer, subtract distances.
For two points on opposite sides, add distances.
Memorize values of tan for 0°, 30°, 45°, 60°, 90°.
Rationalize denominators in final answers.

📌 Quick Reference Table

Situation	Formula
Height given distance and angle	h = d × tanθ
Distance given height and angle	d = h / tanθ
Distance moved (angle changes from α to β)	d = h(cotα - cotβ)
Height difference (two objects)	Δh = d(tanθ₁ - tanθ₂)

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