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Expert-verified Found in: Page 600 ### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508 # Testing for Symmetry Test the polar equation for symmetry to the polar axis, the pole, and the line $\theta =\pi /2$${r}^{2}=4\mathrm{cos}2\theta$ .

The polar equation ${r}^{2}=4\mathrm{cos}2\theta$ is symmetric to the polar axis, the pole, and the line$\theta =\frac{\pi }{2}$ .

See the step by step solution

## Step 1. Given information.

The given polar equation ${r}^{2}=4\mathrm{cos}2\theta$.

## Step 2. Finding given equation Symmetric or not.

If the polar equation ${r}^{2}=4\mathrm{cos}2\theta ,\left(r,\theta \right)$ can be replaced

by $\left(r,-\theta \right)$ or $\left(-r,\pi -\theta \right)$ , then the graph is symmetric to the pole.

If the polar equation ${r}^{2}=4\mathrm{cos}2\theta ,\left(r,\theta \right)$ can be replaced by $\left(-r,\theta \right)$ or $\left(r,\pi +\theta \right)$ , then the graph is symmetric to the pole.

If the polar equation ${r}^{2}=4\mathrm{cos}2\theta ,\left(r,\theta \right)$ can be replaced by $\left(r,\pi -\theta \right)$ or $\left(-r,-\theta \right)$ , then the graph is symmetric to the line $\theta =\frac{\pi }{2}$.

The given polar equation is ${r}^{2}=4\mathrm{cos}2\theta$

Testing for the symmetry of the line $\theta =\frac{\pi }{2}$ .

Replacing $\left(r,\theta \right)=\left(-r,-\theta \right)$ and simplifying the equation

If the resulting equation ${r}^{2}=4\mathrm{cos}2\theta$ is equal to the original equation then it is symmetry about the line$\theta =\frac{\pi }{2}$ .

$\begin{array}{l}\text{Invalid element}=4\mathrm{cos}\left(-2\theta \right)\\ {r}^{2}=4\mathrm{cos}2\theta \end{array}$

The polar equation ${r}^{2}=4\mathrm{cos}2\theta$ is not symmetric to the line $\theta =\frac{\pi }{2}$ .

## Step 3. Testing for symmetry about page polar axis.

Replacing $\left(r,\theta \right)=\left(r,-\theta \right)$ and simplifying the equation ${r}^{2}=4\mathrm{cos}2\theta$ then it is symmetry about the polar axis.

$\begin{array}{c}{r}^{2}=4\mathrm{cos}\left(-2\theta \right)\\ {r}^{2}=4\mathrm{cos}2\theta \end{array}$

The polar equation ${r}^{2}=4\mathrm{cos}2\theta$ is symmetric to the polar axis.

## Step 4. Testing for symmetry about the pole.

Replacing $\left(r,\theta \right)=\left(-r,\theta \right)$ and simplifying the equation

If the resulting equation is equal to the original equation ${r}^{2}=4\mathrm{cos}2\theta$ is symmetric to the pole.

$\begin{array}{c}{\left(-r\right)}^{2}=4\mathrm{cos}2\theta \\ {r}^{2}=4\mathrm{cos}2\theta \end{array}$

The polar equation ${r}^{2}=4\mathrm{cos}2\theta$ is symmetric to the pole. ### Want to see more solutions like these? 