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Q. 52

Expert-verified
Found in: Page 583

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# Identify and graph each polar equation. Verify your graph using a graphing utility.$r=3\mathrm{cos}\left(4\theta \right)$

The equation $r=3\mathrm{cos}\left(4\theta \right)$is a rose with eight petals and the graph is

See the step by step solution

## Step 1. Given Information

The given equation is

$r=3\mathrm{cos}\left(4\theta \right)$

## Step 2. Symmetry with the polar axis

Substitute $-\theta$ for $\theta$ in the equation

$r=3\mathrm{cos}\left(4\left(-\theta \right)\right)\phantom{\rule{0ex}{0ex}}r=3\mathrm{cos}\left(4\theta \right)$

The equation matches with the original equation so the test is satisfied.

the graph is symmetric with respect to the polar axis

## Step 3. Symmetry with line θ=π2

Substitute $\pi -\theta$for $\theta$in the equation

role="math" localid="1646686393219" $r=3\mathrm{cos}\left(4\left(\mathrm{\pi }-\theta \right)\right)\phantom{\rule{0ex}{0ex}}r=3\mathrm{cos}\left(4\mathrm{\pi }-4\theta \right)\phantom{\rule{0ex}{0ex}}r=3\mathrm{cos}\left(4\theta \right)$

The equation matches the original equation so the test is satisfied.

the graph is symmetric with respect to the line $\theta =\frac{\mathrm{\pi }}{2}$

## Step 4. Symmetry with the pole

The equation is symmetric with respect to the polar axis and line $\theta =\frac{\mathrm{\pi }}{2}$

so the equation is symmetric with respect to the pole

## Step 5. Points for the graph

Consider different values for $\theta$ in interval$\left(0,\frac{\mathrm{\pi }}{2}\right)$ and determine coordinates of several points for graph

## Step 6. Graph of the equation

Locates points and use them to plot the graph of the equation

The graph state that the equation is a rose with eight petals

## Step 7. Verification

Plot the graph of the equation $r=3\mathrm{cos}\left(4\theta \right)$using a graphing utility

Graph matches so our graph is correct.