Problem 80 - Algebra and Trigonometry

Question

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\sin ^{4} 4 \theta+\cos 3 \theta$$

Step-by-Step Solution

Answer

This function produces a butterfly curve. By graphing it using a tool that can produce polar graphs, and finely adjusting the step size for \(\theta\), the quality of the butterfly can be increased.

Step 1: Understand the function

The function given to be graphed is \(r=\sin ^{4} 4 \theta+\cos 3 \theta\), which is in polar form. This means that \(r\) is the distance from the origin to a point on the graph, and \(\theta\) is the angle made by this point and the positive x-axis. The value of \(r\) changes as \(\theta\) changes, creating the butterfly shape.

Step 2: Graphing the function using a graphing tool

To graph this, you can use any mathematical software that supports polar graphing. The \(\theta\) values are usually taken from 0 to 2𝜋 to cover the full range of the trigonometric functions. In the graphing tool, input the equation \(r=\sin ^{4} 4 \theta+\cos 3 \theta\). Set the variable \(\theta\) to vary from 0 to 2𝜋, and select the option to produce a polar graph.

Step 3: Refining the graph

Initially, you may not see a graph that looks like a butterfly, this will depend on the graphing tool settings. You can refine your graph by adjusting the range settings, the step size for \(\theta\) as well as the window display. Decreasing the step size increases the number of points calculated, which can improve the quality of the graph. In typical graphing software, the step size can be adjusted by changing the precision or detail level of the graphing tool.