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Chapter 7: Problem 59

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

Short Answer

Expert verified
The graph of \(r=4 \sin 5 \theta\) is a rose curve with 5 petals, each extending 4 units from the origin.

Step by step solution

01

Understand the Polar Equation

The equation given is \(r=4 \sin 5 \theta\), which is a rose curve in polar coordinates. The number 5 in sin function indicates that there will be 5 petals in the rose curve as it is an odd number. The coefficient 4 determines the length of the petals.
02

Set Up the Graphing Utility

You'll need to set up the graphing utility to sketch polar graphs. This step often involves switching the graphing mode from rectangular to polar. In the polar graphing mode, enter the equation \(r=4 \sin 5 \theta\).
03

Plot the Polar Equation

Plot the graph using your graphing utility. For \(\theta\) values ranging from 0 to 2\(\pi\), the graph should plot the rose curve. The result should be a rose curve with 5 petals, each with a length of 4 units.
04

Analyze the Graph

Examine the graph of \(r=4 \sin 5 \theta\), and make sure it aligns with the properties of a rose curve. That is, there should be 5 petals, and each should stretch out 4 units from the origin.

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Most popular questions from this chapter

In calculus, it can be shown that $$e^{i \theta}=\cos \theta+i \sin \theta$$ In Exercises \(87-90,\) use this result to plot each complex number. $$ e^{\frac{\pi i}{4}} $$

Explaining the Concepts What is the graph of a polar equation?

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=4 \cos 2 \theta, r_{2}=4 \cos 2\left(\theta-\frac{\pi}{4}\right)$$

Explain how to plot a complex number in the complex plane. Provide an example with your explanation.

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. $$\begin{aligned}&r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15} \quad \text { (Use } \quad \theta \min =0 \quad \text { and }\\\&\theta \max =20 \pi .)\end{aligned}$$
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