Solutions — Angles of Elevation and Depression

1. Find the height. Fluency

   • (a) 40 m, 30°: h = 40 × tan 30° = 40 × 0.5774 ≈ 23.09 m.
   • (b) 25 m, 55°: h = 25 × tan 55° = 25 × 1.4281 ≈ 35.70 m.
   • (c) 80 m, 12°: h = 80 × tan 12° = 80 × 0.2126 ≈ 17.01 m.
   • (d) 15 m, 45°: h = 15 × tan 45° = 15 × 1 = 15.00 m. (Exact: at 45°, height = horizontal distance.)

2. Find the horizontal distance. Fluency

   • (a) 50 m, 40°: d = 50/tan 40° = 50/0.8391 ≈ 59.59 m.
   • (b) 20 m, 65°: d = 20/tan 65° = 20/2.1445 ≈ 9.33 m.
   • (c) 100 m, 30°: d = 100/tan 30° = 100/(1/√3) = 100√3 ≈ 173.21 m.
   • (d) 35 m, 50°: d = 35/tan 50° = 35/1.1918 ≈ 29.37 m.

3. Find the angle of elevation. Fluency

   • (a) h=30, d=40: tan θ = 30/40 = 0.75 ⇒ θ = tan²¹(0.75) ≈ 36.9°.
   • (b) h=10, d=10: tan θ = 1 ⇒ θ = 45.0°.
   • (c) h=15, slant=20: sin θ = 15/20 = 0.75 ⇒ θ = sin²¹(0.75) ≈ 48.6°.
   • (d) h=8, d=12: tan θ = 8/12 = 0.667 ⇒ θ = tan²¹(0.667) ≈ 33.7°.

4. Angle of depression problems. Fluency

   • (a) 60 m cliff, boat 100 m away: tan θ = 60/100 = 0.6 ⇒ θ = tan²¹(0.6) ≈ 31.0°.
   • (b) Plane 2000 m up, tower 1500 m away: tan θ = 2000/1500 = 1.333 ⇒ θ = tan²¹(1.333) ≈ 53.1°.
   • (c) Window 8 m up, depression 20°: tan 20° = 8/d ⇒ d = 8/tan 20° = 8/0.364 ≈ 21.98 m.
   • (d) Cliff 45 m, depression 35°: tan 35° = 45/d ⇒ d = 45/tan 35° = 45/0.7002 ≈ 64.27 m.

5. Building height from diagram. Understanding

   • (a) Equation for height above eye: tan 55° = x / 20, where x is the height above the observer’s eye level.
   • (b) Height above eye level: x = 20 × tan 55° = 20 × 1.4281 ≈ 28.56 m.
   • (c) Total building height: h = 28.56 + 1.6 = 30.16 m.
   • (d) Why add eye height?: The angle of elevation was measured from eye level, not from ground level. The trig calculation gives the height above the observer’s eye; the full building height includes the section below eye level (from the ground to eye height). Always add the observer’s eye height to get the full building height.

6. Two angles from the same point. Understanding

   • (a) Shorter building (30°, 60 m): h = 60 × tan 30° = 60 × 0.5774 ≈ 34.64 m.
   • (b) Taller building (50°, 60 m): h = 60 × tan 50° = 60 × 1.1918 ≈ 71.51 m.
   • (c) Height difference: 71.51 − 34.64 = 36.87 m.
   • (d) From shorter to taller: Elevation. The top of the shorter building (34.64 m) is below the top of the taller building (71.51 m), so you look up — an angle of elevation.

7. Elevation = Depression. Understanding

   • (a) Horizontal distance to boat: tan 22° = 50/d ⇒ d = 50/tan 22° = 50/0.4040 ≈ 123.8 m.
   • (b) Angle of elevation from boat: 22° — the same as the angle of depression.
   • (c) Why equal?: The horizontal line from the lighthouse top and the horizontal ground form parallel lines. The line of sight is a transversal cutting these parallels. The angle of depression (at the lighthouse) and the angle of elevation (at the boat) are alternate interior angles, which are equal.
   • (d) As boat approaches: The angle of depression increases. As the horizontal distance decreases, the ratio height/distance grows, so the angle gets larger.

8. Ramp and slope angles. Understanding

   • (a) Angle of ramp: tan θ = 0.6/4 = 0.15 ⇒ θ = tan²¹(0.15) ≈ 8.5°.
   • (b) Slant length: slant = √(0.6² + 4²) = √(0.36+16) = √16.36 ≈ 4.04 m.
   • (c) Compliance check: Ramp gradient = rise:run = 0.6:4 = 1:6.67. Since 6.67 < 8, the gradient is steeper than 1:8 — this ramp does not comply with the standard.
   • (d) Maximum angle at 1:8: tan θ = 1/8 = 0.125 ⇒ θ = tan²¹(0.125) ≈ 7.1°.

9. Lighthouse and ship. Problem Solving

   • (a) Distance to Ship A (depression 18°): d_A = 45/tan 18° = 45/0.3249 ≈ 138.5 m.
   • (b) Distance to Ship B (depression 32°): d_B = 45/tan 32° = 45/0.6249 ≈ 72.0 m.
   • (c) Distance between ships: 138.5 − 72.0 = 66.5 m.
   • (d) Which is closer?: Ship B (72.0 m < 138.5 m). The larger angle of depression corresponds to the closer ship.

10. Inaccessible height. Problem Solving

    • (a) Expression from A: h = d × tan 52° where d is horizontal distance from A to cliff base.
    • (b) Expression from B: h = (d + 80) × tan 28° since B is 80 m further from the cliff than A.
    • (c) Solve for d: d×tan52° = (d+80)×tan28°
      1.2799d = 0.5317d + 42.536
      0.7482d = 42.536
      d ≈ 56.85 m.
    • (d) Height of cliff: h = 56.85 × tan 52° = 56.85 × 1.2799 ≈ 72.76 m. (Check: h = (56.85+80)×tan28° = 136.85×0.5317≈72.77 m ✓)