Some Applications of Trigonometry – Class 10 Maths Chapter 9: Heights & Distances

Chapter 9: Some Applications of Trigonometry

9.1 Key Definitions

• Line of Sight: The line drawn from the eye of the observer to the point being observed.
• Angle of Elevation: The angle between the horizontal and the line of sight when looking upward at an object.
• Angle of Depression: The angle between the horizontal and the line of sight when looking downward at an object.

9.2 Problem-Solving Strategy

1. Draw a clear diagram showing all given information.
2. Identify the right triangle(s) in the figure.
3. Label the known and unknown sides/angles.
4. Choose the appropriate trigonometric ratio (sin, cos, or tan) based on which sides are involved.
5. Solve the equation and find the required quantity.

9.3 Solved Examples

Example 1: Finding the height of a tower

A person standing 30 m away from the base of a tower observes the top at an angle of elevation of 60°. Find the height of the tower.

Solution

Let the height of the tower = h. In the right triangle formed:

tan 60° = h/30

√3 = h/30

h = 30√3 = 30 × 1.732 = 51.96 m

Example 2: Finding the distance

From the top of a 45 m high lighthouse, the angle of depression of a ship is 30°. Find the distance of the ship from the base of the lighthouse.

Solution

Let the distance of the ship from the base = d.

Angle of depression = 30° = angle of elevation from ship to top (alternate angles)

tan 30° = 45/d → 1/√3 = 45/d → d = 45√3 = 77.94 m