Applications of Trigonometry

• Fundamental Concepts
  • Definition of trigonometry
  • Role in measuring heights and distances
• Basic Trigonometric Ratios
  • Sine (sin θ) = opposite / hypotenuse
  • Cosine (cos θ) = adjacent / hypotenuse
  • Tangent (tan θ) = opposite / adjacent
• Angle of Elevation & Depression
  • Angle of Elevation – measured upward from the horizontal
  • Angle of Depression – measured downward from the horizontal
  • Common set‑ups: observer on ground, object above/below line of sight
• Height and Distance Problems
  • Using tan θ to find unknown height or distance
  • Using sin θ or cos θ when the hypotenuse is known
  • Typical steps:
    1. Draw a right‑angled diagram
    2. Identify known angles and sides
    3. Apply the appropriate ratio
    4. Solve for the unknown
• Key Trigonometric Identities
  • Pythagorean Identity: sin²θ + cos²θ = 1
  • Co‑function Relations: sin(90°‑θ)=cosθ, cos(90°‑θ)=sinθ
  • Sum & Difference Formulas (used rarely in Class 10):
    • sin(A ± B), cos(A ± B)
• Real‑World Applications
  • Navigation – determining bearing and distance between points
  • Surveying – measuring land elevations, building heights
  • Engineering & Physics – analyzing forces, inclines, and trajectories
• Solved Examples
  • Example 1: Find the height of a tree when the angle of elevation from a point 30 m away is 35°.
  • Example 2: Two towers are 50 m apart. From a point between them, the angles of elevation to the tops are 40° and 30°. Find the heights of the towers.
• Practice Problems
  • Compute the distance to a hilltop given an angle of elevation of 22° from a point 120 m away.
  • A lighthouse is observed from a ship; the angle of depression is 12°. If the ship is 5 km from the base, find the lighthouse’s height.
  • Multiple‑choice questions on selecting the correct trigonometric ratio for a given diagram.