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Chapter 6: Problem 62

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. $$r=3 \sin \left(\frac{5 \theta}{2}\right)$$

Short Answer

Expert verified
The interval for \(\theta\) for which the polar graph, \(r = 3 \sin \left(\frac{5\theta}{2}\right)\), is traced only once is \(\theta \in [0, \frac{2蟺}{5}]\).

Step by step solution

01

Graph the polar equation

Use a graphing utility to plot the equation \(r = 3 \sin \left(\frac{5\theta}{2}\right)\). The graph will look like a rose with 5 petals.
02

Determine the interval for \(\theta\)

In a polar equation \(r=a \sin n\theta\) or \(r=a \cos n\theta\), the graph makes a complete rotation when \(\theta\) runs from 0 to \(\frac{2蟺}{n}\) for \(n\) odd; and when \(\theta\) runs from 0 to \(\frac{4蟺}{n}\) for \(n\) even. The rose will have \(n\) petals if \(n\) is odd and \(2n\) petals if \(n\) is even. Given \(r = 3 \sin \left(\frac{5\theta}{2}\right)\), here, n=5, which is an odd number and thus the interval for \(\theta\) that makes the graph complete is from 0 to \(\frac{2蟺}{5}\). It implies that for our polar equation, it will trace each petal once for the interval 0 to \(\frac{2蟺}{5}\).
03

Validate the interval

After determining the theoretical interval, you can validate it by plugging the values of \(\theta\) in this interval into the polar equation and observing that it indeed traces the full graph exactly once.

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