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Chapter 6: Problem 44

Test for symmetry and then graph each polar equation. $$r=2-4 \cos 2 \theta$$

Short Answer

Expert verified
After testing for symmetry, it turns out that the equation is symmetric with respect to the origin. After filling out the table of values and plotting the points on a polar graph, we obtain a rose curve with 4 petals.

Step by step solution

01

Check for symmetry

To perform the symmetry test for the given polar equation \(r=2-4 \cos 2 \theta\), plug in \(-\theta\) for \(\theta\). If we get the same equation by doing so, the function is symmetric with respect to the polar axis. If the results differ, check for symmetry with respect to the origin by plugging in \(\theta + \pi\) for \(\theta\). The equation will be symmetric with respect to the origin if under these conditions we get the same equation.
02

Fill out a table of values

Use the formula for r and plug in a few different values for \(\theta\). Start with \(\theta = 0\) and increment by \(\frac{\pi}{4}\) up to \(2\pi\). Calculate the corresponding r values for each \(\theta\). Remember that if r comes out as negative, it means the actual point is located opposite to the position indicated by \(\theta\) (you can reflect over the polar axis).
03

Plot the graph

Using the table of values created in step 2, plot the corresponding points for each set of r and \(\theta\) on a polar graph. Connect the dots to get a continuous curve graph. Remember to put arrows at the ends of the curve to indicate that the graph continues beyond what is drawn.

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