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Chapter 6: Problem 18

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

Short Answer

Expert verified
The polar equation \(r=2-2 \cos \theta\) is not symmetrical with respect to the x-axis, y-axis, or the origin, and its graph is a circle with a radius of 1 and center at the pole.

Step by step solution

01

Test for Symmetry

Start testing for the three types of symmetry. For symmetry about the x-axis, replace \(\theta\) with \(-\theta\). If we obtain the original equation, then it's symmetrical with respect to x-axis. Similarly, for symmetry about y-axis, replace \(\theta\) with \(\pi - \theta\) or \(\theta\) with \(-\pi - \theta\). For symmetry about origin, replace \(\theta\) with \(\theta + \pi\). In this case, none of these substitutions will yield the original equation. So, the polar equation \(r=2-2 \cos \theta\) is not symmetrical with respect to the x-axis, y-axis, or the origin.
02

Create a table of values

Next, create a table of values for \(\theta\) and \(r\). Often, angles like 0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2 \pi\) are good starting points because these are easy to work with.
03

Calculate corresponding r-values

Calculate the corresponding r-values by plugging the \(\theta\) values into the polar equation. This will give a set of \((r, \theta)\) coordinates.
04

Sketch the graph

Finally, plot the \((r, \theta)\) coordinates on the polar coordinate system. Connect the plotted points to make the graph of the polar equation \(r=2-2 \cos \theta\).

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