Trigonometry Real-World Applications 3 — Questions and Answers
Question 1: A security camera is mounted 12 feet high on a wall and needs to monitor a floor area starting 5 feet from the wall base. What is the depression angle the camera must tilt downward?
- 67.4° (Correct answer)
- 22.6°
- 45.0°
- 53.1°
Correct answer: 67.4°
Angle of depression = arctan(12/5) ≈ arctan(2.4) ≈ 67.4°.
Question 2: Sound waves traveling through air hit a glass window at an angle of incidence of 45°. If the speed of sound changes from 343 m/s in air to 5,100 m/s in glass, what is the refraction angle?
- 89.6° (Correct answer)
- 3.0°
- 45.0°
- 76.2°
Correct answer: 89.6°
By Snell's law: sin(θ₂) = (5100/343) × sin(45°) ≈ 10.51, which exceeds 1, indicating total internal reflection; however using glass-to-air: sin(θ) = (343/5100) × sin(45°) ≈ 0.0476, so θ ≈ 2.7° ≈ 3.0°.
Question 3: A cyclist travels 8 miles east and then 6 miles north. What bearing should the cyclist use to return directly to the starting point?
- S 53.1° W (Correct answer)
- S 36.9° W
- N 53.1° W
- N 36.9° W
Correct answer: S 53.1° W
The return bearing is south and west; the angle from south = arctan(8/6) ≈ 53.1°, so the bearing is S 53.1° W.
Question 4: A guitar string vibrates according to y = 0.003 sin(880πt) meters. What is the frequency of vibration?
- 440 Hz (Correct answer)
- 880 Hz
- 220 Hz
- 1760 Hz
Correct answer: 440 Hz
Frequency = ω/(2π) = 880π/(2π) = 440 Hz.
Question 5: A park ranger in a 60-foot tower spots a wildfire at an angle of depression of 8°. How far away (horizontal distance) is the fire from the base of the tower?
- 427 feet (Correct answer)
- 356 feet
- 510 feet
- 298 feet
Correct answer: 427 feet
Horizontal distance = 60 / tan(8°) ≈ 60 / 0.1405 ≈ 427 feet.
Question 6: An architect designs a triangular roof with sides of 18 ft, 18 ft, and 12 ft. What is the angle at the peak of the roof?
- 38.9° (Correct answer)
- 70.5°
- 45.0°
- 56.1°
Correct answer: 38.9°
Using the Law of Cosines: cos(C) = (18² + 18² − 12²) / (2×18×18) = (324+324−144)/648 = 504/648 ≈ 0.778, so C ≈ 38.9°.
Question 7: A horizontal beam extends 4 meters from a wall, supported by a cable attached to the wall 3 meters above the beam's end. What angle does the cable make with the wall?
- 53.1° (Correct answer)
- 36.9°
- 45.0°
- 63.4°
Correct answer: 53.1°
The angle with the wall = arctan(4/3) ≈ 53.1° since the opposite side is the beam (4 m) and adjacent is the wall height (3 m).
Trigonometry Real-World Applications 3
A security camera is mounted 12 feet high on a wall and needs to monitor a floor area starting 5 feet from the wall base.
What is the depression angle the camera must tilt downward?