Draw the graph of `y=sin x ` and `y=sin . (x)/(2)`.

Answer

To draw the graphs of \( y = \sin x \) and \( y = \sin \frac{x}{2} \), we will follow these steps:

### Step 1: Understand the basic properties of the sine function
The sine function oscillates between -1 and 1, with a period of \( 2\pi \). The key points for the graph of \( y = \sin x \) are:
- At \( x = 0 \), \( y = \sin(0) = 0 \)
- At \( x = \frac{\pi}{2} \), \( y = \sin\left(\frac{\pi}{2}\right) = 1 \)
- At \( x = \pi \), \( y = \sin(\pi) = 0 \)
- At \( x = \frac{3\pi}{2} \), \( y = \sin\left(\frac{3\pi}{2}\right) = -1 \)
- At \( x = 2\pi \), \( y = \sin(2\pi) = 0 \)

### Step 2: Plot the graph of \( y = \sin x \)
Using the points identified:
- Plot the points: \( (0, 0) \), \( \left(\frac{\pi}{2}, 1\right) \), \( (\pi, 0) \), \( \left(\frac{3\pi}{2}, -1\right) \), and \( (2\pi, 0) \).
- Connect these points smoothly to form the sine wave, which will repeat every \( 2\pi \).

### Step 3: Understand the properties of \( y = \sin \frac{x}{2} \)
The function \( y = \sin \frac{x}{2} \) has a different period. The period of \( \sin \frac{x}{2} \) is \( 4\pi \) because:
- The sine function completes one full cycle when its argument changes by \( 2\pi \), so \( \frac{x}{2} = 2\pi \) gives \( x = 4\pi \).

Key points for the graph of \( y = \sin \frac{x}{2} \) are:
- At \( x = 0 \), \( y = \sin(0) = 0 \)
- At \( x = 2\pi \), \( y = \sin\left(\frac{2\pi}{2}\right) = \sin(\pi) = 0 \)
- At \( x = 4\pi \), \( y = \sin\left(\frac{4\pi}{2}\right) = \sin(2\pi) = 0 \)
- At \( x = \pi \), \( y = \sin\left(\frac{\pi}{2}\right) = 1 \)
- At \( x = 3\pi \), \( y = \sin\left(\frac{3\pi}{2}\right) = -1 \)

### Step 4: Plot the graph of \( y = \sin \frac{x}{2} \)
Using the points identified:
- Plot the points: \( (0, 0) \), \( (\pi, 1) \), \( (2\pi, 0) \), \( (3\pi, -1) \), and \( (4\pi, 0) \).
- Connect these points smoothly to form the sine wave, which will repeat every \( 4\pi \).

### Step 5: Combine the graphs
- Draw both graphs on the same coordinate system to compare them.
- The graph of \( y = \sin x \) will oscillate more frequently than \( y = \sin \frac{x}{2} \).