Chapter 9.2, Problem 8AYU

True or False The tests for symmetry in polar coordinates are always conclusive.

To determine

To check: Check whether the given statement is true or false.

Answer to Problem 8AYU

Solution:

It is a false statement.

Given:

Statement says that the tests for symmetry in polar coordinates are always conclusive

Explanation of Solution

Test for symmetry:

Symmetry with respect to the polar axis ($x$-Axis):

In a polar equation, replace $\theta$ by $\pi - \theta$. If an equivalent equation results, the graph is symmetric with respect to the polar axis.

Symmetry with respect to the line $\theta = \frac{\pi }{2}\left(y - Axis\right)$:

In a polar equation, replace $\theta$ by $\pi - \theta$. If an equivalent equation results, the graph is symmetric with respect to the line $\theta = \frac{\pi }{2}$.

Symmetry with respect to the Pole:

In a polar equation, replace $r$ by $- r$ or $\theta$ by $\theta + \pi$. If an equivalent equation results, the graph is symmetric with respect to the line $\theta = \frac{\pi }{2}$.

The three tests for symmetry given here are sufficient conditions for symmetry, but they are not necessary conditions. That is, an equation may fail these tests and still have a graph that is symmetric with respect to the polar axis, the line $\theta = \frac{\pi }{2}$, or the pole. For example, the graph of $r = \text{sin}\left(2\theta \right)$ turns out to be symmetric with respect to the polar axis, the line $\theta = \frac{\pi }{2}$, and the pole, but only the test for symmetry with respect to the pole (replace $\theta$ by $\theta + \pi$ works.

Therefore, the tests for symmetry in polar coordinates need not to be conclusive always.