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Chapter 6: Problem 22

Test for symmetry and then graph each polar equation. $$r=1-2 \cos \theta$$

Short Answer

Expert verified
The graph of the equation \(r=1-2 \cos \theta\) is not symmetric about the x-axis, y-axis, or the origin. The graph should be sketched based on values for \(\theta\) at regular intervals.

Step by step solution

01

Test for Symmetry

To test for symmetry, the given equation is checked for the three cases: - Symmetry about the x-axis: Replace \(\theta\) with \(-\theta\) and see if the identical equation is obtained. - Symmetry about the y-axis: Replace \(\theta\) with \(\pi-\theta\) and check if we obtain the same equation.- Symmetry with respect to the origin: Replace \(\theta\) with \(\pi + \theta\) and check if we obtain the same equation. In our case, none of these replacements will yield the original equation, meaning the graph is not symmetric about the x-axis, y-axis, or the origin.
02

Plot the Polar Equation

At this point, one would use a polar graph paper to plot the points. It is advised to first find the length and direction of the radius vector r for sample values of \(\theta\) at regular intervals, and plot these points on the polar graph. Then, the graph can be completed by drawing a smooth curve through these points.

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