Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 799: 56

Answer

The equation may be symmetric with respect to the polar axis, the line $\theta=\frac{\pi}{2}$, and the pole. See graph.
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Work Step by Step

Step 1. To test the symmetry with respect to the polar axis, replace $(r,\theta)$ with $(r,-\theta)$; we have $r^2 = cos(-2\theta)$ or $r^2 = cos(2\theta)$ . Thus the equation is symmetric with respect to the polar axis. Step 2. To test the symmetry with respect to the line $\theta=\frac{\pi}{2}$, replace $(r,\theta)$ with $(-r,-\theta)$; we have $(-r)^2 = cos(-2\theta)$ or$r^2 = cos(2\theta)$ . Thus the equation is symmetric with respect to the line $\theta=\frac{\pi}{2}$. Step 3. To test the symmetry with respect to the pole, replace $(r,\theta)$ with $(-r,\theta)$; we have $(-r)^2 = cos(2\theta)$ or $r^2 = cos(2\theta)$. Thus the equation is symmetric with respect to the pole. Step 4. Using test points with $0\leq\theta\leq2\pi$, we can graph the equation as shown in the figure.
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