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Chapter 6: 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 \(r = 4 \sin 6 \theta\) presents a rose shape with 12 petals.

Step by step solution

01

Converting Polar to Cartesian Coordinates

Polar equations are more conveniently represented in a graph by converting them into Cartesian coordinates. Here, to convert the polar coordinates to Cartesian coordinates, we use the formulas: \[x = r \cdot \cos(\theta)\]\[y = r \cdot \sin(\theta)\]Substituting \(r = 4 \sin 6\theta\), we get:\[x = 4 \sin 6\theta \cdot \cos(\theta)\]\[y = 4 \sin 6\theta \cdot \sin(\theta)\]This will simplify the process of plotting the equation.
02

Setting up the Graphing Utility

Input the Cartesian coordinates into a graphing utility. Note to use the parametric mode in your graphing utility, as two equations are provided and they are functions of a variable \(\theta\). Ensure to set the value of \(\theta\) to a scope that shows a complete graph. Typically, \(\theta\) from 0 to \(2\pi\) is sufficient.
03

Analyzing the Graph

After plotting, you will see a graph. This graph exhibits a 'flower' or 'rose' pattern which is characteristic of these types of polar equations where the function of \(r\) involves multiple angles of \(\theta\). The coefficient \(6\) of the \(\theta\) inside the sinusoidal function determines the number of 'leaves' or 'petals' on the graph.

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Most popular questions from this chapter

Group members should research and present a report on unusual and interesting applications of vectors.

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.

Find the work done when a crane lifts a 6000-pound boulder through a vertical distance of 12 feet. Round to the nearest foot-pound.

Help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect?
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