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Q. 45

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Precalculus Enhanced with Graphing Utilities
Found in: Page 583
Precalculus Enhanced with Graphing Utilities

Precalculus Enhanced with Graphing Utilities

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

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Short Answer

Identify and graph each polar equation. Verify your graph using a graphing utility.
r=1+2 sin θ

The equationr=1+2 sin θ is limacon with an inner loop and the graph is

See the step by step solution

Step by Step Solution

Step 1. Given Information 

The given equation is

r=1+2 sin θ

Step 2. Symmetry with the polar axis 

Substitute -θ forθ in the equation

r=1+2 sin- θr=1-2 sin θ

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 π-θfor θin the equation

role="math" localid="1646681410227" r=1+2 sin π-θr=1+2 sin θ

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

the graph is symmetric with respect to the line θ=π2

Step 4. Symmetry with the pole 

Substitute-r for r in the equation

-r=1+2 sin θr=-1-2 sin θ

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 θ in interval-π2,π2 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 sin θusing a graphing utility

Graph matches so our graph is correct.

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