Some Applications of Trigonometry

REVISION SUMMARY: Some Applications of Trigonometry (NCERT Class 10)

1. Chapter at a Glance

• The line of sight is drawn from the eye of an observer to the point viewed on the object.
• Angle of elevation is formed by the line of sight with the horizontal when the object is above eye level (observer raises head).
• Angle of depression is formed by the line of sight with the horizontal when the object is below eye level (observer lowers head).
• Heights and distances are found by drawing right triangles and applying trigonometric ratios (mainly tan, sin, cot) to the angle of elevation or depression.
• In all cases, the observer’s height is added or subtracted when the line of sight does not start from ground level.
• Problems may involve one angle, two different angles from the same point, or changing angles (e.g., shadows at 30° and 60°).
• The method always requires drawing a labelled diagram showing the line of sight, horizontal, and right angle before choosing the ratio.
• The chapter uses only the definitions of line of sight, elevation and depression together with standard trigonometric ratios in right triangles.

2. Definitions, Theorems and Results

• Line of sight: “the line drawn from the eye of an observer to the point in the object viewed by the observer.”
• Angle of elevation: “the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.”
• Angle of depression: “the angle formed by the line of sight with the horizontal when the point is below the horizontal level.”
• No theorems, lemmas or axioms are stated in the chapter. No proofs are required in the examination.

3. Formula Sheet

All other ratios (cos, sec, cosec) are not used in the chapter examples.

4. Solved-Example Patterns

• Single angle of elevation from ground level
  Draw right triangle with angle at observation point; apply tan θ = height/distance; solve for required height or distance.

• Angle of elevation with observer’s height given
  Subtract observer height from total height to get the vertical side opposite the angle; apply tan; add observer height back if total height is asked.

• Ladder or line inclined at given angle (sin or cot)
  Identify hypotenuse or adjacent side; choose sin θ or cot θ accordingly; calculate length and foot distance separately.

• Two angles of elevation from same point (flagstaff/building)
  First solve the known lower triangle for horizontal distance; then form second triangle with total height and same base; solve simultaneously.

• Shadow lengths at two different solar altitudes (30° & 60°)
  Let shadow at 60° be x; at 30° be x + 40 (or given difference). Write two tan equations, substitute and solve the linear equation for height.

• Angles of depression from top of one building to top and bottom of another
  Use alternate angles to convert depressions into angles of elevation in the lower triangles; equate the common horizontal distance and solve.

• Width of river or canal using angles of depression from a point above
  Form two right triangles on opposite sides of the point; apply tan on each; add the two base segments.

5. Common Mistakes and Exam Pitfalls

• Forgetting to add or subtract the observer’s/pedestal height, leading to incorrect vertical side.
• Confusing angle of elevation with angle of depression or drawing the angle on the wrong side of the horizontal.
• Choosing the wrong trigonometric ratio (e.g., using sin when tan is required).
• Not converting given approximate values (√3 ≈ 1.732) correctly or omitting units (m).
• Assuming the horizontal distance is the same in both triangles when it is not (e.g., different points of observation).
• Missing the “exactly behind” or “opposite sides” condition when two ships or two banks are involved.
• Omitting the diagram or failing to mark right angles and alternate angles formed by parallel lines.