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Chapter 6: Problem 59

Use a graphing utility to graph the polar equation. $$r=4 \sin 5 \theta$$

Short Answer

Expert verified
The polar equation \(r=4 \sin 5 \theta\) will be graphed as a 'rose curve' with 5 petals, symmetric around the horizontal axis. This can be plotted by using a suitable graphing utility like Desmos or GeoGebra.

Step by step solution

01

Understanding Polar Graphs

Firstly, it is important to acknowledge that the nature of the polar graphs is based on the radial distance, \(r\), and the angle, \( \theta \), values. As \( \theta \) takes values from 0 to \(2\pi\), your graphing utility will run through each corresponding value of \(r\), given by your equation \(r=4 \sin 5 \theta\), to vizualize these coordinates in a polar way.
02

Choose a Graphing Utility

To graph the polar equation, choose a graphing utility or software that supports polar graphing such as Desmos or GeoGebra.
03

Input the Polar Equation

Input the given polar equation \(r=4 \sin 5 \theta\) into the graphing utility. Make sure to clarify to the software which variables designate the radial distance and the angle.
04

Adjust the values for \( \theta \)

Generally, for such a type of graph, it would be helpful to set up \( \theta \) to vary from 0 to \(2\pi\), corresponding to a full round. Some graphing utilities automatically do this when creating a polar plot.
05

Interpret the Plot

After the equation is graphed, observe that the graph is a 'rose curve' with 5 petals. The sine function gives symmetry around the horizontal axis, and the factor 5 inside the function determines the number of petals in the graph.

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