Chapter 9.2, Problem 89AYU

The tests for symmetry given on page 591 are sufficient, but not necessary. Explain what this means.

To determine

Explain the tests for symmetry are sufficient, but not necessary.

Answer to Problem 89AYU

Explanation of Solution

Check for symmetry test:

Polar axis:

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

The line $\theta = \frac{\pi }{2}$:

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

The Pole:

Replace $r$ by $- r$ and replace $\theta$ by $\theta + \pi$. If an equivalent equation results, the graph is symmetric with respect to the pole.

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 line works.

The graph of the equation $r = \text{sin}\left(2\theta \right)$ is rose with four petals.

This is symmetric with respect to the polar axis, the line $\theta = \frac{\pi }{2}$ and the pole.

Check for symmetry test:

Polar axis:

Replace $\theta$ by $- \theta$. If an equivalent equation results, the graph is symmetric with respect to the polar axis. The result is $r = \text{sin}\left(2\left(- \theta \right)\right) = - \text{sin}\left(2\theta \right)$. The test fails.

The line $\theta = \frac{\pi }{2}$:

Replace $\theta$ by $\pi - \theta$. If an equivalent equation results, the graph is symmetric with respect to the line $\theta = \frac{\pi }{2}$. The result is $r = \text{sin}\left(\pi - 2\theta \right) = \text{sin}\pi \text{cos}2\theta - \text{cos}\pi \text{sin}2\theta = \text{sin}\left(2\theta \right)$. The test satisfied.

The Pole:

Replace $r$ by $- r$ and replace $\theta$ by $\theta + \pi$. If an equivalent equation results, the graph is symmetric with respect to the pole. The result is $- r = \text{sin}\left(2\theta \right), r = - \text{sin}\left(2\theta \right)$. The test fails.

The result is $r = \text{sin}\left(2\theta + \pi \right) = \text{sin}2\theta \text{cos}\pi + \text{cos}2\theta \text{sin}\pi = - \text{sin}\left(2\theta \right)$. The test fails.

Hence, the tests for symmetry are sufficient, but not necessary.