Some Applications of Trigonometry — Practice

Question 1

Define angle of elevation.

Accepted Answer

✦ Answer: The angle between the line of sight (going UP) from observer to an object and the horizontal line. Used when object is ABOVE observer.

Question 2

From a point 30 m from the base of a tower, the angle of elevation of its top is 60°. Find the height of the tower.

Accepted Answer

Step 1 — Setup.
  Distance = 30 m, angle = 60°.

Step 2 — Apply tan.
  tan 60° = height / 30
  √3 = h / 30
  h = 30√3 m

Step 3 — Approximate.
  h ≈ 30 × 1.732 ≈ 51.96 m

✦ Answer: Height of tower = 30√3 m ≈ 52 m.

Question 3

The angle of elevation of a tower from a point on the ground is 30°. After walking 60 m toward the tower, it becomes 60°. Find the height of the tower.

Accepted Answer

Step 1 — Setup.
  Let height = h, initial distance = d (from first observation point).
  After walking 60 m toward tower, new distance = d − 60.

Step 2 — Form equations.
  From first point: tan 30° = h/d → 1/√3 = h/d → h = d/√3
  From second point: tan 60° = h/(d−60) → √3 = h/(d−60) → h = √3(d−60)

Step 3 — Equate.
  d/√3 = √3(d−60)
  d = 3(d − 60)
  d = 3d − 180
  −2d = −180
  d = 90 m

Step 4 — Find height.
  h = d/√3 = 90/√3 = 90√3/3 = 30√3 m
  h ≈ 51.96 m

Step 5 — Verify.
  At d = 90: tan 30° = 30√3/90 = √3/3 = 1/√3 ✓
  At d−60 = 30: tan 60° = 30√3/30 = √3 ✓

✦ Answer: Height of tower = 30√3 m ≈ 52 m.

Practice — Some Applications of Trigonometry

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