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Chapter 6: Problem 79

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

Short Answer

Expert verified
To create a high-quality plot of the given butterfly curve, utilise a graphing utility equipped for polar coordinates. Adjust the \(\theta\) step for smoothness and experiment with different range settings until the best quality plot is achieved. Unfortunately, specifics cannot be provided without the use of the actual graphing utility.

Step by step solution

01

Understand Polar Equation

The polar equation provided is of the form \(r = \cos^2 5 \theta +\sin 3 \theta + 0.3\). Here, \(r\) represents the distance from the origin (pole), and \(\theta\) the angle measured from the positive x-axis.
02

Use a Graphing Utility

Input the given equation into a graphing utility designed to handle polar equations. Commonly used tools can include software like GeoGebra, Desmos, or a graphic calculator.
03

Adjust \(\theta\) step

A key factor in the quality of the plotted curve is the step size for \(\theta\). Start with a small value and gradually increase it if necessary until the butterfly curve is smooth and well-defined.
04

Experiment with the Range

Range settings can greatly affect the appearance of the plot. Experiment with different range values until the best quality butterfly plot is achieved.

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