Class 10 Notes - Some Applications of Trigonometry

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Some Applications of Trigonometry

Trigonometry is not just about triangles and angles—it is widely used to solve real-life problems involving heights and distances. In this chapter, you will learn how to use trigonometric ratios to find unknown heights and distances without actually measuring them.

Key Concepts

  • Line of Sight: The imaginary line drawn from the eye of an observer to the point being viewed.
  • Angle of Elevation: The angle formed by the line of sight with the horizontal when the object is above the horizontal level.
  • Angle of Depression: The angle formed by the line of sight with the horizontal when the object is below the horizontal level.

Trigonometric Ratios Used

  • Sine (sin): Opposite side / Hypotenuse
  • Cosine (cos): Adjacent side / Hypotenuse
  • Tangent (tan): Opposite side / Adjacent side

Steps to Solve Height and Distance Problems

  1. Draw a diagram based on the problem statement.
  2. Mark all known and unknown values (heights, distances, angles).
  3. Identify the right triangle(s) involved.
  4. Use the appropriate trigonometric ratio (sin, cos, tan) based on the given information.
  5. Solve for the unknown value.

Common Scenarios

  • Finding the height of a building or a tree using the angle of elevation.
  • Finding the distance between two objects using angles of elevation or depression.
  • Finding the height of an object when the shadow length and angle of elevation of the sun are known.

Example Problems

Example 1: The angle of elevation of the top of a tower from a point on the ground 30 m away from its base is 45°. Find the height of the tower.

Solution: Let the height of the tower be h.
tan 45° = h / 30
1 = h / 30 ⇒ h = 30 m

Example 2: A ladder 10 m long rests against a wall. The foot of the ladder is 6 m from the wall. Find the angle the ladder makes with the ground.

Solution: Let θ be the angle.
cos θ = 6 / 10 = 0.6
θ = cos-1(0.6) ≈ 53.13°

Tips

  • Always draw a clear diagram for the problem.
  • Label all points, angles, and lengths clearly.
  • Use the correct trigonometric ratio based on the information given.
  • Check if the angle is of elevation or depression.
  • Keep units consistent throughout the calculation.

Practice Questions

  1. The angle of elevation of the top of a building from a point 50 m away from its base is 30°. Find the height of the building.
  2. A man observes the top of a tower at an angle of elevation of 60°. If he is standing 20 m from the base, find the height of the tower.
  3. The angle of depression from the top of a lighthouse to a boat is 45°. If the lighthouse is 40 m high, how far is the boat from the base of the lighthouse?
  4. A tree casts a shadow 15 m long when the angle of elevation of the sun is 30°. Find the height of the tree.

Summary

  • Trigonometry helps solve real-world problems involving heights and distances.
  • Angles of elevation and depression are key concepts.
  • Draw diagrams and use trigonometric ratios to find unknown values.