How to draw the graph of the function y=[sinx]+[cosx] where enclosed brackets represent greatest integer function?

To graph the function \( y = [\sin x] + [\cos x] \), where the brackets denote the greatest integer function (also known as the floor function), we need to analyze how the sine and cosine functions behave and how their integer parts contribute to the overall function. Let's break it down step-by-step.

Understanding the Components

The sine and cosine functions oscillate between -1 and 1. The greatest integer function, denoted as \( [x] \), gives the largest integer less than or equal to \( x \). Therefore, we need to evaluate \( [\sin x] \) and \( [\cos x] \) separately.

Sine and Cosine Values

• For \( \sin x \):
  • When \( \sin x \) is in the range \([-1, 0)\), \( [\sin x] = -1\).
  • When \( \sin x \) is in the range \([0, 1)\), \( [\sin x] = 0\).
• For \( \cos x \):
  • When \( \cos x \) is in the range \([-1, 0)\), \( [\cos x] = -1\).
  • When \( \cos x \) is in the range \([0, 1)\), \( [\cos x] = 0\).

Combining the Results

Now, let's look at different intervals of \( x \) to determine the value of \( y = [\sin x] + [\cos x] \).

Interval Analysis

• **For \( x \in [0, \frac{\pi}{2}) \):**
  • Here, \( \sin x \) varies from 0 to 1 and \( \cos x \) varies from 1 to 0.
  • Thus, \( [\sin x] = 0 \) and \( [\cos x] = 0 \), leading to \( y = 0 + 0 = 0\).
• **For \( x \in [\frac{\pi}{2}, \pi) \):**
  • During this interval, \( \sin x \) goes from 1 to 0, while \( \cos x \) ranges from 0 to -1.
  • This results in \( [\sin x] = 0 \) and \( [\cos x] = -1\), so \( y = 0 - 1 = -1\).
• **For \( x \in [\pi, \frac{3\pi}{2}) \):**
  • In this section, \( \sin x \) ranges from 0 to -1, and \( \cos x \) varies from -1 to 0.
  • Here, \( [\sin x] = -1 \) and \( [\cos x] = -1\), giving \( y = -1 - 1 = -2\).
• **For \( x \in [\frac{3\pi}{2}, 2\pi) \):**
  • For these values, \( \sin x \) goes from -1 to 0, while \( \cos x \) goes from 0 to 1.
  • We find \( [\sin x] = -1 \) and \( [\cos x] = 0\), resulting in \( y = -1 + 0 = -1\).

Graphing the Function

Now that we have established the values of \( y \) for specific intervals, we can summarize the results:

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

When you plot these sections on a graph, you will see horizontal lines corresponding to the values of \( y \) across each interval. The graph will consist of the following segments:

• From \( x = 0 \) to \( x = \frac{\pi}{2} \), a horizontal line at \( y = 0 \).
• From \( x = \frac{\pi}{2} \) to \( x = \pi \), a horizontal line at \( y = -1 \).
• From \( x = \pi \) to \( x = \frac{3\pi}{2} \), a horizontal line at \( y = -2 \).
• From \( x = \frac{3\pi}{2} \) to \( x = 2\pi \), a horizontal line at \( y = -1 \).

By examining these intervals and understanding how the sine and cosine functions interact with the greatest integer function, you can successfully sketch the graph of \( y = [\sin x] + [\cos x] \). Making sure to label your axes and points will help in presenting your final graph clearly!