Radians and Wave Oscillation

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Date:

Circle with radian measurement and wave diagram

📐 Part 1: Understanding Radians

Key Information: A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. One complete circle = 2π radians = 360°.

1. How many radians are there in a complete circle?

π radians

2π radians

4π radians

6π radians

2. Convert 90° to radians:

π/4 radians

π/2 radians

π radians

3π/2 radians

3. Which of the following are equivalent to π/3 radians? (Select all that apply)

60°

120°

One-sixth of a complete rotation

One-third of a semicircle

4. Draw a circle and mark an angle of π/4 radians from the positive x-axis:

🌊 Part 2: Wave Oscillation and Radians

Key Information: Waves are periodic functions that repeat every 2π radians. The general form is y = A sin(ωt + φ), where ω is angular frequency in radians per second.

5. A wave completes one full oscillation. How many radians has it travelled through?

Answer: _____________ radians

6. If a wave has a frequency of 5 Hz, what is its angular frequency in radians per second?

7. For the wave equation y = 3 sin(2πt), what is the amplitude and period?

Amplitude: _________ units

Period: _________ seconds

8. Explain why radians are particularly useful when working with wave equations rather than degrees:

🔧 Part 3: Problem Solving

9. A pendulum swings through an angle of 2π/3 radians from its starting position. If the pendulum arm is 1.5 metres long, calculate the arc length travelled by the pendulum bob.

10. A sound wave has the equation y = 4 sin(6πt). At what times (in seconds) during the first second will the wave reach its maximum positive displacement?

11. Convert the following angles to radians, leaving your answers in terms of π:

a) 45° = _________ radians

b) 150° = _________ radians

c) 270° = _________ radians

12. A Ferris wheel rotates at 0.5 radians per second. How long does it take to complete one full rotation?