Chapter Summary: Some Applications of Trigonometry

A one-page recap of every key idea in this chapter. Read this the night before the exam.

1. Key Terms

  • Line of sight: the line from the observer's eye to the object.
  • Horizontal: the level line through the eye, parallel to the ground.
  • Angle of elevation: the upward angle (object above the horizontal).
  • Angle of depression: the downward angle (object below the horizontal).
  • Crucial fact: angle of depression (from the top) = angle of elevation (from the bottom), by alternate angles.

2. The Core Relation

For a right triangle with vertical (height) hh and horizontal (distance) dd at angle θ\theta: tan⁡θ=hd.\tan\theta = \frac{h}{d}. Use sin⁡\sin when the slant (line of sight) is involved, cos⁡\cos for the adjacent with the slant.

3. Standard Angle Shortcuts

  • tan⁡30°=13\tan 30° = \tfrac{1}{\sqrt3}, tan⁡45°=1\tan 45° = 1, tan⁡60°=3\tan 60° = \sqrt3.
  • At 45°: height = distance.
  • At 60°: height = 3×\sqrt3 \times distance.
  • At 30°: distance = 3×\sqrt3 \times height.

4. Solving Strategy

  1. Draw and label the right triangle(s).
  2. Transfer any depression angle to the alternate (elevation) position.
  3. Write a tan⁡\tan equation per triangle.
  4. For two-triangle problems, solve the pair together (substitution / subtraction).
  5. Add the observer's eye height when the question asks for height above ground.

5. Common Set-Ups

  • Two points, same side: distances subtract; the nearer point has the larger angle.
  • Object between two points (opposite sides): distances add to the separation.
  • Moving object: distance moved = difference of horizontal distances; speed = distance ÷ time.

Common mistakes to avoid

  • Placing the depression angle inside the triangle at the top — move it to the bottom (alternate angle).
  • Confusing the slant (line of sight) with the horizontal distance — they need different ratios.
  • Forgetting to add the observer's eye height for 'height above ground'.
  • Not rationalising surd denominators like 3−1\sqrt3 - 1 at the end.

Handy results

  • 30°–60° two-point: separation d=2h3d = \dfrac{2h}{\sqrt3}.
  • Complementary angles from distances aa and bb: height =ab= \sqrt{ab}.

Final tip: A correct labelled diagram is half the solution. Draw it, pick tan⁡\tan (height vs distance), and keep your surds exact until the final step.