Problem 79 - 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. $$r=\cos ^{2} 5 \theta+\sin 3 \theta+0.3$$

Step-by-Step Solution

Answer

To generate a high-quality butterfly curve, plot the given polar equation \(r=\cos^{2} 5\theta+\sin 3\theta+0.3\), adjust the \(\theta\) step size for a smoother curve, and experiment with the range and scales on the graph for better visualization. Note that this has been done with a graphical tool, so there isn't a definitive graph or exact numeric endpoints to provide here.

Step 1: Set up the graph

Enter the given equation, \(r=\cos^{2} 5\theta+\sin 3\theta+0.3\), into the graphing utility. Ensure that the mode is set to polar and adjust the window size to a suitable range to capture the intricacy of the butterfly curve. A general starting point might be \(\theta\) ranging from -10 to 10 with incremental steps of 0.1.

Step 2: Plot the graph

Plot the graph of the function within the specified range. The graphing tool creates the plot by substituting various values of \(\theta\) into the equation and calculating the corresponding \(r\) value for each one. It then maps these pairs of \(r\) and \(\theta\) onto the Cartesian plane.

Step 3: Adjust the \(\theta\) step size

To improve the quality of the butterfly curve, adjust the step size of \(\theta\). A smaller step size will yield a smoother curve as more data points are plotted. Test different step sizes until the graph appears fine-tuned and smooth.

Step 4: Examine the graph

Review the produced graph to ensure a clear, visible representation of the butterfly curve has been achieved. Keep in mind that the scales on the graph may need to be adjusted for better visualization. This could include expanding or reducing the range of \(\theta\) or modifying the increment size. Experiment with these parameters until satisfied with the graphic output.