Problem 83 - Algebra and Trigonometry

Question

Use a graphing utility to graph \(r=\sin n \theta\) for \(n=1,2,3,4,5\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of loops that occur corresponding to each value of \(n ?\) What is happening to the shape of the graphs as \(n\) increases? For each graph, what is the smallest interval for \(\theta\) so that the graph is traced only once?

Step-by-Step Solution

Answer

The number of loops in the graph equals to \(n\) when the polar function \(r=\sin n \theta\) is graphed. As \(n\) increases, the loops become tighter and more intricate. The smallest interval for \(θ\) where the graph is traced only once is \([0, 2\pi/n]\) for each \(n\).

Step 1: Graph the function

Start by graphing the polar function \(r=\sin n \theta\) using a graphing utility for \(n=1,2,3,4,5,6\). You should use a separate viewing window for each graph to make sure that you can see all of the details of the graph.

Step 2: Count the loops

Once you have the graphs, count the number of loops in each of them. A loop is a section of the graph where it starts from the pole (or the origin), goes out and circles around to return back to the pole.

Step 3: Identify the pattern

With the number of loops for each value of \(n\) noted down, you should see that the number of loops equals to \(n\) for each of the graphs.

Step 4: Note the shape changes

Now, look at how the shape of the graphs changes as \(n\) increases. You should see that as \(n\) increases, the loops become tighter and more intricate.

Step 5: Determine the Smallest Interval for \(θ\)

Lastly, for each graph determine the smallest interval for \(θ\) so that the graph is traced only once. This should be \([0, 2\pi/n]\) for each \(n\), where \(n\) is the number of loops.