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Chapter 6: Problem 56

If an equation fails the test for symmetry with respect to the polar axis, what can you conclude?

Short Answer

Expert verified
If an equation fails the test for symmetry with respect to the polar axis, it means that the graph of the function is not symmetric with respect to the polar axis. However, it may still have symmetry with respect to the pole or the line theta = pi/2. This will depend on the nature of the equation itself.

Step by step solution

01

Understanding symmetry in polar coordinates

In polar coordinates, symmetry falls into three categories: Symmetry with respect to the polar axis, symmetry with respect to the pole, and symmetry with respect to the line theta = pi/2. The polar coordinates (r, 胃) and (r, -胃) refer to the same point for the equation to be symmetric with respect to the polar axis. Therefore, if replacing 胃 with -胃 in the equation does not give the same equation back, it means it fails the test for symmetry with respect to the polar axis.
02

Determine the conclusion when failing the polar axis symmetry test

If any equation fails the test of symmetry with respect to the polar axis, it means the function is not symmetric around the polar axis. That does not imply that the graph will not have symmetry; the equation may be still symmetric with respect to the pole, or the line theta = pi/2. This depends on the equation itself, hence such cases must be analyzed individually.

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