Some Applications of Trigonometry — Class 10 Mathematics

"Trigonometry's true power is measuring what we cannot directly reach — towers, mountains, stars."

1. About the Chapter

This chapter APPLIES trigonometric ratios (Chapter 8) to REAL-WORLD problems of heights and distances.

Key Concepts

• Angle of Elevation
• Angle of Depression
• Line of sight, horizontal line
• Solving problems systematically

2. Key Terms

Line of Sight

The line drawn from the OBSERVER's eye to the OBJECT being seen.

Horizontal Line

The horizontal level passing through the observer's eye.

Angle of Elevation

The angle between LINE OF SIGHT (going UP) and HORIZONTAL LINE.

• Used when object is ABOVE observer
• Example: looking up at a tower

Angle of Depression

The angle between LINE OF SIGHT (going DOWN) and HORIZONTAL LINE.

• Used when object is BELOW observer
• Example: looking down from a balcony

Important Note

Angle of elevation = Angle of depression (alternate interior angles when looking from one to the other).

3. Solving Height-Distance Problems

General Procedure

1. DRAW DIAGRAM: sketch the situation with observer, object, angle
2. IDENTIFY known and unknown quantities
3. SET UP trigonometric ratio (usually tan, sometimes sin/cos)
4. SOLVE for unknown
5. VERIFY answer

Standard Setup

For a tower of height h, observer at distance d, angle θ: tan θ = h / d

So:

• If d and θ known: h = d × tan θ
• If h and θ known: d = h / tan θ
• If h and d known: tan θ = h/d → θ = arctan(h/d)

4. Worked Examples

Example 1: Find Height

A man 1.8 m tall stands 30 m from a tower. Angle of elevation of top of tower is 60°. Find tower height.

Setup:

• Man's height = 1.8 m
• Horizontal distance = 30 m
• Angle of elevation = 60°

Solve:

• Height of tower above man's eye = 30 × tan 60° = 30√3 m
• Total tower height = 30√3 + 1.8 ≈ 30(1.732) + 1.8 = 51.96 + 1.8 = 53.76 m

(For Class 10, often ignore observer's height for simplicity.)

Example 2: Find Distance

A boy looks at the top of a 30 m tall building at an angle of elevation of 30°. Find the distance from the building.

• tan 30° = 30 / d
• 1/√3 = 30 / d
• d = 30√3 ≈ 52 m

Example 3: Angle of Depression

From the top of a 100 m tall tower, a car on the ground is observed at an angle of depression of 45°. Find the distance of the car from the foot of the tower.

• tan 45° = 100 / d
• 1 = 100 / d
• d = 100 m

Example 4: Two Observation Points

The angle of elevation of the top of a tower from a point on the ground is 30°. After walking 100 m toward the tower, the angle becomes 60°. Find tower height.

Let height = h, initial distance = d.

• tan 30° = h/d → h = d/√3
• After walking 100m toward: tan 60° = h/(d−100) → h = √3(d−100)

Set equal: d/√3 = √3(d−100) d = 3(d−100) d = 3d − 300 −2d = −300 d = 150 m

h = 150/√3 = 50√3 ≈ 86.6 m

Example 5: Cliff and Boat

From a cliff 100 m high, the angles of depression of two boats are 30° and 45°. Find the distance between them.

Let distances from cliff base be d₁ (closer) and d₂ (farther).

• tan 45° = 100/d₁ → d₁ = 100 m
• tan 30° = 100/d₂ → d₂ = 100√3 ≈ 173 m

Distance between boats = d₂ − d₁ = 173 − 100 = 73 m (approx)

5. Tips for Word Problems

Always Draw a Diagram

• Mark observer (eye level)
• Mark object (with height labelled)
• Mark angle clearly (elevation or depression)
• Mark known/unknown distances

Identify Sides

For angle θ in right triangle:

• Opposite (perpendicular)
• Adjacent (base)
• Hypotenuse

Choose Right Ratio

• tan = opposite/adjacent (most common for heights/distances)
• sin = opposite/hypotenuse (when hypotenuse mentioned)
• cos = adjacent/hypotenuse (when hypotenuse mentioned)

Common Pitfalls

• Don't confuse 'angle of elevation' (up) and 'angle of depression' (down)
• Distance is usually HORIZONTAL distance (along ground)
• Be careful with units

6. Real-World Applications

Civil Engineering

• Building heights from ground
• Bridge angles
• Slope of roads

Surveying

• Land surveys use theodolite (measures angles)
• Indian Survey of India uses trigonometric methods

Aviation

• Plane altitude estimation
• Distance from runway

Astronomy

• Star altitudes
• Sun's angular elevation throughout day

Indian Context

• Trigonometric Survey of India (1802-1871) mapped India
• Famous: George Everest (mountain named after him)
• Modern: GPS uses trigonometry

7. Worked Example with Multiple Heights

Example: Pole and Tower

A pole 6 m high casts a shadow 8 m long. At the same time, a tower casts a shadow 24 m long. Find the tower's height.

Using similar triangles (sun's angle same):

• Pole: angle θ such that tan θ = 6/8 = 3/4
• Tower: tan θ = h/24
• So h/24 = 3/4
• h = 18 m

(Same angle, so ratios are equal.)

8. Common Mistakes

1. Elevation vs Depression confusion

   • LOOKING UP at object → elevation
   • LOOKING DOWN at object → depression

2. Wrong trigonometric ratio

   • Use TAN for most height-distance problems (no hypotenuse).

3. Ignoring observer height

   • In some problems, observer's height matters. Read carefully.

4. Direction of walking

   • 'Walking toward' decreases distance; 'walking away' increases.

5. Final answer should be POSITIVE

   • If you get negative distance, recheck setup.

9. Conclusion

Trigonometry's REAL POWER is in measuring what we cannot directly access:

• Tower and building heights
• Mountain altitudes
• Distances across rivers
• Aircraft elevation
• Sun and star positions

Master:

• Drawing clear diagrams
• Identifying angles correctly (elevation/depression)
• Choosing right ratios (usually tan)
• Solving with specific angle values

Practice 15+ problems to gain fluency.

Trigonometry: turning the unmeasurable into measurable.