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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
A graphing utility can be used to plot the polar equation \(r = 4 \sin 6\theta\). The equation represents a rose polar graph with six petals, each petal having a length of 4. By inputting the equation into the graphing utility, it will visualize the graph.

Step by step solution

01

Understand the polar equation

The given polar equation is \(r = 4 \sin 6\theta\). This equation depicts a rose polar graph. Rose polar graphs are represented by equations of the form \(r = a\sin(bc\theta)\) or \(r = a\cos(bc\theta)\), where 'a' is the length of the petals and 'b' is generally the number of petals. Here, 'a' is 4, 'b' is 6, and the sine tells the position of the petals.
02

Set values for \(\theta\)

To plot the graph, start by choosing values for \(\theta\). You can start at \(\theta = 0\) and end at \(\theta = 2\pi\) with increment of \(\pi/6\) or \(\pi/12\) to get a smoother graph.
03

Calculate corresponding 'r' values

Calculate 'r' for each value of \(\theta\) by substituting the values of \(\theta\) into the polar equation \(r = 4 \sin 6\theta\).
04

Plot the coordinates

Each '(r, \(\theta\))' pair is a set of polar coordinates. Plot these points on the polar grid and connect them to get the graph.
05

Utilize a graphing utility

To graph the polar equation using a graphing utility, input the polar equation \(r = 4 \sin 6\theta\) into the utility. The utility will do the computation and provide a visualization of the polar graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing utilities
Graphing utilities are incredible tools that simplify the complex task of plotting graphs, especially for polar equations like the one in the exercise, \(r = 4 \sin 6 \theta\). These utilities, available as software programs or calculators, automate the graph plotting process by handling the computations behind the scenes. Instead of manually calculating each point, you can:
  • Input the equation into the utility.
  • Adjust the view settings to see the entire graph.
  • Observe the graph's behavior as parameters change.
By leveraging these tools, you can focus more on understanding the graph's properties rather than being bogged down by numerous calculations. Use graphing utilities to visualize patterns, confirm manual plots, and gain deeper insights into equations.
rose polar graphs
Rose polar graphs depict beautiful flower-like shapes, and are commonly seen in polar coordinate systems. They are represented by equations of the form \(r = a \sin(b \theta)\) or \(r = a \cos(b \theta)\). Here:
  • '\(a\)' determines the petal length.
  • '\(b\)' influences the number of petals.
In our equation \(r = 4 \sin 6\theta\), 'a' is 4 and 'b' is 6, indicating that the graph will have six petals. The sin function positions the petals evenly around the origin on the polar grid. Additionally, if 'b' is even, the number of petals is exactly 'b', while if 'b' is odd, the number of petals is '2b'. Thus, for this equation, you can expect a symmetrical, six-petal flower shape.
polar coordinates
Polar coordinates provide a unique way to describe locations on a plane. Instead of using the traditional \(x\) and \(y\) axes like in Cartesian coordinates, polar coordinates involve:
  • \(r\): the radius or distance from the origin.
  • \(\theta\): the angle measured from the positive x-axis.
This system is particularly useful for problems dealing with cycles, rotations, and angles. To plot a point in polar coordinates, you measure the angle \(\theta\) from the x-axis and then move outward from the origin by the radius \(r\). For example, with our equation \(r = 4 \sin 6\theta\), you'll calculate and plot each \(r\) for different values of \(\theta\), revealing the dynamic and circular nature that polar coordinates excel in capturing.

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Most popular questions from this chapter

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=12 \cos \theta$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r^{2} \sin 2 \theta=4$$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$\left(2, \frac{\pi}{3}\right)$$

Convert each rectangular equation to a polar equation that expresses \(r\) in terms of \(\theta\) $$x^{2}+y^{2}=9$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=6 \sec \theta$$
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