(c) Plot the graph of y = 0.5 sin 2x for 0 ≤ x ≤ 360. Use the graph to find the amplitude and the period of the graph. (6 marks)

To analyze the function \( y = 0.5 \sin(2x) \) over the interval \( 0 \leq x \leq 360 \), let's break down the steps to plot the graph and determine its amplitude and period.

### Step 1: Identify Key Characteristics

1. **Amplitude**: The amplitude of a sine function \( y = A \sin(Bx) \) is given by the absolute value of \( A \). Here, \( A = 0.5 \), so the amplitude is:
   \[
   \text{Amplitude} = |0.5| = 0.5
   \]

2. **Period**: The period of a sine function is calculated using the formula:
   \[
   \text{Period} = \frac{360^\circ}{B}
   \]
   where \( B = 2 \) in this case. Thus, the period is:
   \[
   \text{Period} = \frac{360^\circ}{2} = 180^\circ
   \]

### Step 2: Plot the Graph

1. **Create a Table of Values**: Calculate values of \( y \) at key points in the interval.

   | \( x \) (degrees) | \( y = 0.5 \sin(2x) \) |
   |--------------------|-----------------------|
   | 0                  | 0                     |
   | 30                 | 0.5                   |
   | 60                 | 0.866                 |
   | 90                 | 1                     |
   | 120                | 0.866                 |
   | 150                | 0.5                   |
   | 180                | 0                     |
   | 210                | -0.5                  |
   | 240                | -0.866                |
   | 270                | -1                    |
   | 300                | -0.866                |
   | 330                | -0.5                  |
   | 360                | 0                     |

2. **Plot Points**: Using these calculated values, plot the points on a coordinate system and connect them to form the sine wave.

### Step 3: Graphing Tool

To visualize, you can use graphing software or a graphing calculator to plot these points. The graph will oscillate between 0.5 and -0.5 with a repeating pattern every 180 degrees.

### Step 4: Conclusion

From the graph, you can confirm:
- **Amplitude**: 0.5
- **Period**: 180 degrees

If you need the graph plotted or further clarification, let me know!

### Additional Questions

1. How do you derive the sine function's properties from its equation?
2. What are other trigonometric functions and their characteristics?
3. How can you apply this knowledge to solve real-world problems?
4. What happens to the amplitude if the coefficient in front of the sine function changes?
5. How would the graph change if the function were \( y = -0.5 \sin(2x) \)?

**Tip**: When analyzing periodic functions, always note their amplitude and period as they significantly impact their shape and behavior.

Plot the graph of y = 0.5 sin 2x for 0 ≤ x ≤ 360. Use the graph to find the amplitude and the period of the graph.

Explore the graph of y = 0.5 sin 2x over the interval 0 ≤ x ≤ 360, and learn how to determine its amplitude (0.5) and period (180°). This step-by-step guide covers essential trigonometric concepts suitable for high school students.

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