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Chapter 6: Problem 42

Test for symmetry and then graph each polar equation. $$r=\frac{3 \sin 2 \theta}{\sin ^{3} \theta+\cos ^{3} \theta}$$

Short Answer

Expert verified
The polar equation was tested for symmetry about the x-axis, y-axis, and the origin resulting in no symmetry. Using a graphing tool, the equational pattern should be plotted to visually present the solution.

Step by step solution

01

Test for Symmetry about the x-axis

Replace \(\theta\) with \(-\theta\) in the equation \(r=\frac{3 \sin 2\theta}{\sin^3\theta+\cos^3\theta}\), if the equation remains unchanged, then it is symmetrical about the x-axis.
02

Test for Symmetry about the y-axis

Replace \(\theta\) with \(\pi - \theta\) in the original equation. If the equation remains the same, then it is symmetrical about the y-axis.
03

Test for Symmetry about the Origin

Replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\) in the original equation. If the equation remains the same, then it is symmetrical about the origin.
04

Sketch the Graph

This step requires converting the polar equation into Cartesian coordinates (x and y) and plotting the points on a graph. The range for \(\theta\) is usually from 0 to \(\pi\) or -\(\pi/2\) to \(\pi/2\). The graph is then drawn by connecting points with smooth curves, keeping in mind the determined symmetries.

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