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Chapter 7: Problem 31

Test for symmetry and then graph each polar equation. $$r=1-3 \sin \theta$$

Short Answer

Expert verified
The given polar equation \(r = 1 - 3 \sin \theta\) shows no symmetry about the x-axis, y-axis or origin after testing. To graph this equation, plot several points using different \(\theta\) values and then connect these points to form the polar curve.

Step by step solution

01

Test for Symmetry

First, test the polar equation for three types of symmetries. 1. Symmetry about x-axis: Replace \( \theta\) by \(-\theta\) . If the original equation remains unchanged, then it has x-axis symmetry. 2. Symmetry about y-axis: Replace \( r\) by \(-r\) . If the original equation remains unchanged, then it has y-axis symmetry. 3. Symmetry about the origin: Replace \( \theta\) by \(\theta + \pi\) and \( r\) by \(-r\) . If the original equation remains unchanged, then it has symmetry about the origin.
02

Plot the Polar Equation

To plot the given equation in a polar graph, it's necessary to select several values of \(\theta\) and calculate corresponding \(r\) values using our given equation. Make a theta-r table and plot these points on a polar grid.
03

Draw the Graph

After determining the points from the table in step 2, plot those points on a polar coordinate plane. Connect these points smoothly to form the complete polar curve representing the given equation.

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