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Chapter 6: Problem 53

Describe the test for symmetry with respect to the polar axis.

Short Answer

Expert verified
The test for symmetry with respect to the polar axis is conducted by replacing 胃 with -胃 in the equation. If the resulting equation is the same as the original, then it is symmetric with respect to the polar axis.

Step by step solution

01

Introduction to symmetry in polar coordinates

In polar coordinates, there are three types of symmetries: symmetry about the polar axis (i.e., x-axis in Cartesian coordinates), symmetry about the pole, and symmetry about the line \(\theta = \pi / 2\) (i.e., y-axis in Cartesian coordinates).
02

Define the polar axis symmetry

An equation in polar coordinates is symmetric with respect to the polar axis if, for a point P with coordinates (r, 胃), the point P' with coordinates (r, -胃) is also on the graph. Essentially, this means if a point is on the graph, its reflection over the x-axis, which is the polar axis, will also be on the graph.
03

Describe the test for symmetry with respect to the polar axis

To test if a graph in polar coordinates is symmetric with respect to the polar axis, we can replace 胃 with -胃 in the equation. If the resulting equation is identical to the original equation, then the graph is symmetric with respect to the polar axis.

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