Saurabh Koranglekar
Last Activity: 5 Years ago
To graph the function , 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 , gives the largest integer less than or equal to . Therefore, we need to evaluate and separately.
Sine and Cosine Values
- For :
- When is in the range , .
- When is in the range , .
- For :
- When is in the range , .
- When is in the range , .
Combining the Results
Now, let's look at different intervals of to determine the value of .
Interval Analysis
- **For :**
- Here, varies from 0 to 1 and varies from 1 to 0.
- Thus, and , leading to .
- **For :**
- During this interval, goes from 1 to 0, while ranges from 0 to -1.
- This results in and , so .
- **For :**
- In this section, ranges from 0 to -1, and varies from -1 to 0.
- Here, and , giving .
- **For :**
- For these values, goes from -1 to 0, while goes from 0 to 1.
- We find and , resulting in .
Graphing the Function
Now that we have established the values of for specific intervals, we can summarize the results:
- From to ,
- From to ,
- From to ,
- From to ,
When you plot these sections on a graph, you will see horizontal lines corresponding to the values of across each interval. The graph will consist of the following segments:
- From to , a horizontal line at .
- From to , a horizontal line at .
- From to , a horizontal line at .
- From to , a horizontal line at .
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 . Making sure to label your axes and points will help in presenting your final graph clearly!