Trigonometry Real-World Applications 3

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?

Accepted Answer

67.4°

Answer

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?

Accepted Answer

89.6°

Answer

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?

Accepted Answer

S 53.1° W

Answer

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?

Accepted Answer

440 Hz

Answer

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?

Accepted Answer

427 feet

Answer

Horizontal distance = 60 / tan(8°) ≈ 60 / 0.1405 ≈ 427 feet.

Trigonometry Practice Test

Trigonometry Practice Test

Trigonometry Real-World Applications 3