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Q. 45

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.$r=1+2\mathrm{sin}\theta$

The equation$r=1+2\mathrm{sin}\theta$ is limacon with an inner loop and the graph is

See the step by step solution

## Step 1. Given Information

The given equation is

$r=1+2\mathrm{sin}\theta$

## Step 2. Symmetry with the polar axis

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

$r=1+2\mathrm{sin}\left(-\theta \right)\phantom{\rule{0ex}{0ex}}r=1-2\mathrm{sin}\theta$

The equation does not match with the original equation so the test fails

the graph may or may not be 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="1646681410227" $r=1+2\mathrm{sin}\left(\mathrm{\pi }-\theta \right)\phantom{\rule{0ex}{0ex}}r=1+2\mathrm{sin}\theta$

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

Substitute$-r$ for r in the equation

$-r=1+2\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}r=-1-2\mathrm{sin}\theta$

The equation does not match with the original equation so the test fails

the graph may or may not be symmetric with respect to the pole

## Step 5. Points for the graph

Consider different values for $\theta$ in interval$\left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ 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 limacon with an inner loop

## Step 7. Verification

Plot the graph of the equation $r=1+2\mathrm{sin}\theta$using a graphing utility

Graph matches so our graph is correct.