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Chapter 7: Problem 61

Use a graphing utility to graph the polar equation. $$r=4 \sin 6 \theta$$

Short Answer

Expert verified
The graph of the polar equation \(r=4 \sin 6 \theta\) is a rose curve with six petals because the coefficient of \(\theta\) in the sine function determines the number of petals.

Step by step solution

01

Understanding the Polar Equation

The first step is to understand the polar equation \(r=4 \sin 6 \theta\). Here \(r\) is the distance from the origin and \(\theta\) is the angle measured counter-clockwise from the positive x-axis. The polar equation represents a graph that consists of plotting all points (r, \(\theta\)) such that the equation is satisfied.
02

Converting into Cartesian Coordinates

The polar equation can be converted into Cartesian coordinates for easier visualization which is given by \(x = r\cos(\theta)\), \(y = r\sin(\theta)\). Hence the Cartesian equation becomes \(x= 4\sin(6\theta)\cos(\theta)\) and \(y= 4\sin(6\theta)\sin(\theta)\). This conversion can help to visualize and recognize the graph in a familiar format.
03

Using Graphing Utility

Now, open a graphing utility and input the polar equation \(r=4 \sin 6 \theta\). Adjust the values of \(\theta\) to cover all possible values from 0 to 2\(\pi\). The utility will graph the polar equation, and you can see how the graph changes as \(\theta\) changes.
04

Examining the Graph

The last step is to examine the characteristics of the graph. Since it's a sine graph with a given period, the graph will show multiple 'loops' or 'petals' formulated by the changes in \(\theta\) and \(r\). Notice also how the coefficient of \(\theta\) in the sine function, which is 6 in this case, will determine the number of 'loops' or 'petals' in the polar graph.

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Most popular questions from this chapter

Exercises \(99-101\) will help you prepare for the material covered in the next section. Refer to Section 1.4 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i .\) $$(-1+i \sqrt{3})(-1+i \sqrt{3})(-1+i \sqrt{3})$$

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=2 \sin 3 \theta, r_{2}=2 \sin 3\left(\theta+\frac{\pi}{6}\right)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A complex number \(a+b i\) can be interpreted geometrically as the point \((a, b)\) in the \(x y\) -plane.

Use a graphing utility to graph the polar equation. $$r=\cos \frac{5}{2} \theta$$

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fifth roots of \(-1+i\)
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