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Chapter 6: Problem 61

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. $$r=2 \cos \left(\frac{3 \theta}{2}\right)$$

Short Answer

Expert verified
The graph of the polar equation \( r = 2 \cos(\frac{3}{2} \theta ) \) is traced only once for \(\theta\) in the range \([0, \frac{4}{3} \pi]\).

Step by step solution

01

Plotting the Polar Function

To start off, use a graphing utility to plot the given polar equation \( r = 2 \cos(\frac{3}{2} \theta ) \). Notice how the graph behaves, particularly when \(\theta\) varies.
02

Understanding the plotted graph

Once the graph of the polar function is plotted, observe that the function is periodic with respect to \(\theta\). For \(\theta\) in the range \([0, 4\pi]\), the graph completes one period. This can be attributed to the nature of the cosine function when \( \frac{3}{2} \theta \) equals its period of \(2\pi\), thus giving \(\theta = \frac{4}{3} \pi\). The graph is traced only once for \(\theta\) in the range \([0, \frac{4}{3} \pi]\).
03

Final Answer

In conclusion, the graph of the polar function \( r = 2 \cos(\frac{3}{2} \theta ) \) is traced only once for \(\theta\) in the range \([0, \frac{4}{3} \pi]\).

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