Problem 81 - 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. $$r=\sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta$$

Step-by-Step Solution

Answer

The butterfly curve can be generated using graphing software by inputting the given formula in the polar coordinates form, adjusting the \(\theta\) step to a small value (like 0.01), and experimenting with the range settings. The final graph representation is obtained by iteratively adjusting these parameters until the graph resembles a butterfly.

Step 1: Open the Graphing Utility

Open your graphing software or utility of choice. This could include options such as GeoGebra, Desmos, or others.

Step 2: Input the Given Function

Input the given function \(r = \sin ^{5} \theta + 8 \sin \theta \cos ^{3} \theta\) into the software. Make sure to input it in the polar coordinates form, since the given formula is in the polar coordinates form.

Step 3: Adjusting the \(\theta\) Step

Next, try to adjust the step for \(\theta\). This might depend on the capabilities of your specific software, but a smaller step size will generally yield a more precise graph. Set a small step size for \(\theta\), for example, \(\theta\) = 0.01.

Step 4: Experiment with the Range Setting

Now, experiment with the range setting values to produce a butterfly of the best possible quality. Try different ranges until you get a result that resembles a butterfly.

Step 5: Viewing the Graph

Examine the graph generated by your software. If adjustments are needed, return to steps 3 or 4.