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Chapter 6: Problem 66

Use a graphing utility to graph the polar equation. $$r=2+4 \cos \theta$$

Short Answer

Expert verified
The graph of the polar equation \(r=2+4cos(\theta)\) results in a circle with a radius of 3 and a smaller enclosed circle with a radius of 1. The large circle is centered at (2,0) in polar coordinates, and the smaller one is centered at the origin.

Step by step solution

01

Understanding Polar Coordinates

Polar coordinates are represented as \((r,\theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. In this equation, \(r = 2+4cos(\theta)\), \(r\) varies as \(\cos(\theta)\) changes.
02

Converting the Equation

The given equation is already in polar form. Plot appropriate values for \(\theta\) which vary in the range of \([0, 2\pi]\), and the corresponding \(r\) values can be calculated using the equation \(r = 2+4cos(\theta)\).
03

Plotting the Graph

Each point in the graph is represented by (r, \theta). It's recommended to start plotting with small angle increments, for instance 10掳 intervals. For each angle value, calculate \(r = 2+4cos(\theta)\) and plot the point. As you go around the circle from \(\theta = 0\) to \(\theta = 2\pi\), \(r\) will change according to the equation.
04

Interpretation of the Graph

After plotting the points, we should see the graph forming a shape. For the equation \(r=2+4cos(\theta)\), we will get a circle with a radius of 3 and a smaller enclosed circle with a radius of 1. The circle will have a center (2,0) and the smaller circle will have a center at the origin.

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