Draw the graph of y = [sin x], where x ∈ [0, 2π] represents the greatest integer function.

To graph the function \( y = [\sin x] \), where \( x \) ranges from \( 0 \) to \( 2\pi \) and \( [\cdot] \) denotes the greatest integer function, we first need to understand the behavior of the sine function within this interval.

Understanding the Sine Function

The sine function oscillates between -1 and 1. Over the interval from \( 0 \) to \( 2\pi \), it has the following key points:

• \( \sin(0) = 0 \)
• \( \sin\left(\frac{\pi}{2}\right) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
• \( \sin(2\pi) = 0 \)

Applying the Greatest Integer Function

The greatest integer function, denoted as \( [x] \), rounds down to the nearest integer. Thus, we can determine the values of \( y = [\sin x] \) as follows:

• For \( 0 \leq x < \frac{\pi}{2} \): \( \sin x \) ranges from 0 to 1, so \( [\sin x] = 0 \).
• At \( x = \frac{\pi}{2} \): \( [\sin x] = 1 \).
• For \( \frac{\pi}{2} < x < \pi \): \( \sin x \) decreases from 1 to 0, so \( [\sin x] = 0 \).
• At \( x = \pi \): \( [\sin x] = 0 \).
• For \( \pi < x < \frac{3\pi}{2} \): \( \sin x \) decreases from 0 to -1, so \( [\sin x] = -1 \).
• At \( x = \frac{3\pi}{2} \): \( [\sin x] = -1 \).
• For \( \frac{3\pi}{2} < x < 2\pi \): \( \sin x \) increases from -1 to 0, so \( [\sin x] = -1 \).
• At \( x = 2\pi \): \( [\sin x] = 0 \).

Graphing the Function

Now, we can summarize the values of \( y \) over the interval:

• From \( 0 \) to \( \frac{\pi}{2} \): \( y = 0 \)
• At \( x = \frac{\pi}{2} \): \( y = 1 \)
• From \( \frac{\pi}{2} \) to \( \pi \): \( y = 0 \)
• From \( \pi \) to \( \frac{3\pi}{2} \): \( y = -1 \)
• At \( x = \frac{3\pi}{2} \): \( y = -1 \)
• From \( \frac{3\pi}{2} \) to \( 2\pi \): \( y = -1 \)
• At \( x = 2\pi \): \( y = 0 \)

In summary, the graph consists of horizontal segments at \( y = 0 \) and \( y = -1 \), with a single point at \( y = 1 \) when \( x = \frac{\pi}{2} \). This creates a step-like graph that reflects the behavior of the greatest integer function applied to the sine function.