📐 Some Applications of Trigonometry — Class 10

Heights and distances using angles of elevation and depression

1. Key Concepts

📖 Angle of Elevation

When we look upward from a horizontal line to observe an object above us, the angle formed between the horizontal line and the line of sight is called the angle of elevation.

Example: Looking up at the top of a tower from the ground.

📖 Angle of Depression

When we look downward from a horizontal line to observe an object below us, the angle formed between the horizontal line and the line of sight is called the angle of depression.

Example: Looking down at a boat in the sea from a cliff.

⚡ Key Relationship

The angle of elevation from point A to point B equals the angle of depression from point B to point A.

This is because they are alternate interior angles formed when a transversal (line of sight) cuts two parallel lines (horizontal lines at A and B).

🌟 Real-Life Applications

• Surveying land and measuring heights of mountains
• Navigation in ships and aircraft
• Finding distance of stars and planets in astronomy
• Construction — finding height of buildings
• Military — finding range of targets

2. Problem-Solving Strategy

🔑 Steps to Solve Height & Distance Problems

Draw a neat diagram based on the problem description.
Label all known and unknown quantities clearly.
Identify the right triangle(s) formed.
Choose the appropriate trig ratio (sin/cos/tan) based on what's given and required.
Set up the equation and solve.
State the answer with proper units.

3. Solved Examples

💡 Example 1: Finding Height of Tower

Q: A tower stands vertically on the ground. From a point on the ground 20 m away from the foot, the angle of elevation of the top is 60°. Find the height of the tower.

Let height of tower = h metres

Distance from point to foot = 20 m, Angle of elevation = 60°

In right triangle: tan60° = h/20

√3 = h/20 → h = 20√3 m

Answer: Height = 20√3 ≈ 34.64 m

💡 Example 2: Angle of Depression

Q: From the top of a 75 m high lighthouse, the angles of depression of two ships are 30° and 45°. If the ships are on the same side of the lighthouse, find the distance between the ships.

Let height of lighthouse = 75 m.

Let d₁ = distance of nearer ship from base, d₂ = distance of farther ship from base

For nearer ship (angle 45°): tan45° = 75/d₁ → 1 = 75/d₁ → d₁ = 75 m

For farther ship (angle 30°): tan30° = 75/d₂ → 1/√3 = 75/d₂ → d₂ = 75√3 m

Distance between ships = d₂ – d₁ = 75√3 – 75 = 75(√3 – 1)

Answer: 75(√3 – 1) ≈ 75 × 0.732 ≈ 54.9 m

💡 Example 3: Two Angles of Elevation

Q: The angle of elevation of the top of a building from a point on the ground is 30°. Moving 20 m towards the building, the angle becomes 60°. Find the height of the building.

Let height = h, and initial distance = d

tan30° = h/d → h = d/√3 ...(i)

tan60° = h/(d–20) → h = √3(d–20) ...(ii)

From (i) and (ii): d/√3 = √3(d–20) → d = 3(d–20) → d = 3d – 60 → 2d = 60 → d = 30 m

h = 30/√3 = 10√3 m

Answer: Height = 10√3 ≈ 17.32 m

💡 Example 4: Width of River

Q: A person standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he retreats 20 m from the bank, the angle becomes 30°. Find the width of the river and height of the tree.

Let width of river = d, height of tree = h

From bank: tan60° = h/d → h = d√3 ...(i)

After retreating: tan30° = h/(d+20) → h = (d+20)/√3 ...(ii)

From (i) and (ii): d√3 = (d+20)/√3 → 3d = d+20 → 2d = 20 → d = 10 m

h = 10√3 m

Answer: Width of river = 10 m, Height of tree = 10√3 ≈ 17.32 m

4. Important Points

Angle of elevation is measured from horizontal upward to the object.
Angle of depression is measured from horizontal downward to the object.
The line of sight, horizontal line, and the vertical distance form a right triangle.
tan θ is most commonly used as it directly relates height and horizontal distance.
Always check if the diagram is drawn correctly before solving.

📋 Key Relationships

Elevation (looking up)

tan(angle) = height/distance

height = distance × tan(angle)

Depression (looking down)

Angle of depression = Angle of elevation from below

(Alternate interior angles)

Common Values

tan30° = 1/√3 ≈ 0.577

tan45° = 1

tan60° = √3 ≈ 1.732