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Chapter 6: Problem 59

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. $$r=3-8 \cos \theta$$

Short Answer

Expert verified
The graph of the polar equation \(r=3-8 \cos \theta\) is only traced once in the interval \(-蟺 鈮� \theta < 蟺\).

Step by step solution

01

Plot the Graph

Firstly, the graph should be plotted. Input the polar equation \(r=3-8 \cos \theta\) into a graphing utility. Observe the resultant plot. The graph displays a lima莽on, which is a kind of spiraled figure, in this case with an internal loop.
02

Identify the Shape

Look at the shape to identify an interval during which the graph completes a single revolution without retracing itself. This shape has got a loop, implying there must be a certain interval where the graph passes through each point only once.
03

Finding the Interval

Limit the range of \(\theta\) to find the interval where the graph is only traced once. By observation, when \(\theta\) is restricted to the interval \(0 鈮� \theta < 2蟺\), the graph retraces itself twice. However, when \(\theta\) is limited to \(-蟺 鈮� \theta < 蟺\), the graph is traced only once.

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