Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Question

Chapter 9.6, Problem 36E

To determine

To calculate: To identify and graph the polar equation. Also to identify any symmetry and zeros of $r$

Expert Solution & Answer

Answer to Problem 36E

Type of polar equation is limacon with inner loop

Polar equation is symmetric with respect to polar axis

Zeros of polar equations are $θ=π6,11π6$

Explanation of Solution

Given information: Polar equation is $r=3−2cosθ$

Formula Used:

Cardioid: Heart shape curve and general form is $r=a+acosθ$

Limacons: General form of cardioid and general form is $r=a+bcosθ$

When $ab<1$ , polar equation represents limacon with inner loop

When $ab=1$ , polar equation represents Cardiod

When $1<ab<2$ , polar equation represents Dimpled limacon

When $ab=2$ , polar equation represents convex limacon

Rose Curves: Sinusoidal curve that have a flower shape and general equation is $r=acos(kθ)$

Archimedian Spirals: Curve that extends indefinitely outward from a pole and general form is $r=a+bθ$

Lemniscate: Eight shaped curve and general form is $r2=a2cos(2θ)$

Test for Symmetry:

1. To test symmetry with respect to the line $θ=π2$ , replace $(r,θ)$ by $(r,π−θ)$

2. To test symmetry with respect to the polar axis, replace $(r,θ)$ by $(r,−θ)$

3. To test symmetry with respect to the pole, replace $(r,θ)$ by $(r,π+θ)$

Calculation:

Polar equation is given as follows:

$r=3−2cosθ$

Type of Polar Equation:

Polar equation is of form $r=a+bcosθ$ such that $ab<1$

Thus, graph of polar equation is a limacon with inner loop

Test Symmetric of Polar Equation:

To test Symmetry respect with respect to the line $θ=π2$ :

Replacing $(r,θ)$ by $(r,π−θ)$ , polar equation is

$r=3−2cos(π−θ)r=3+2cosθ$

Since above equation is NOT same as the given polar equation

Thus, polar equation is NOT symmetric with respect to the line $θ=π2$

To test Symmetry with respect to the polar axis

Replacing $(r,θ)$ by $(r,−θ)$ , polar equation is

$r=3−2cos(−θ)r=3−2cosθ$

Since above equation is same as the given polar equation

Thus, polar equation is symmetric with respect to the polar axis

To test Symmetry with respect to the pole

Replacing $(r,θ)$ by $(r,π+θ)$ , polar equation is

$r=3−2cos(π+θ)r=3+2cosθ$

Since above equation is NOT same as the given polar equation

Thus, polar equation is NOT symmetric with respect to the pole

Calculating Zeros of Polar Equation:

In order to calculate zeros of polar equation, substitute $r=0$

Thus,

$3−2cosθ=0cosθ=32θ=cos−132θ=π6,11π6$

Graph of Polar Equation:

Graph of polar equation is as follows:

Precalculus with Limits: A Graphing Approach, Chapter 9.6, Problem 36E

Conclusion:

Hence, type of polar equation is limacon with inner loop

Polar equation is symmetric with respect to polar axis

Zeros of polar equations are $θ=π6,11π6$

Chapter 9.6, Problem 35E

Chapter 9.6, Problem 37E

Chapter 9 Solutions

Precalculus with Limits: A Graphing Approach

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Prob. 66RE Ch. 9 - Prob. 67RE Ch. 9 - Prob. 68RE Ch. 9 - Prob. 69RE Ch. 9 - Prob. 70RE Ch. 9 - Prob. 71RE Ch. 9 - Prob. 72RE Ch. 9 - Prob. 73RE Ch. 9 - Prob. 74RE Ch. 9 - Prob. 75RE Ch. 9 - Prob. 76RE Ch. 9 - Prob. 77RE Ch. 9 - Prob. 78RE Ch. 9 - Prob. 79RE Ch. 9 - Prob. 80RE Ch. 9 - Prob. 81RE Ch. 9 - Prob. 82RE Ch. 9 - Prob. 83RE Ch. 9 - Prob. 84RE Ch. 9 - Prob. 85RE Ch. 9 - Prob. 86RE Ch. 9 - Prob. 87RE Ch. 9 - Prob. 88RE Ch. 9 - Prob. 89RE Ch. 9 - Prob. 90RE Ch. 9 - Prob. 91RE Ch. 9 - Prob. 92RE Ch. 9 - Prob. 93RE Ch. 9 - Prob. 94RE Ch. 9 - Prob. 95RE Ch. 9 - Prob. 96RE Ch. 9 - Prob. 97RE Ch. 9 - Prob. 98RE Ch. 9 - Prob. 99RE Ch. 9 - Prob. 100RE Ch. 9 - Prob. 101RE Ch. 9 - Prob. 102RE Ch. 9 - Prob. 103RE Ch. 9 - Prob. 104RE Ch. 9 - Prob. 105RE Ch. 9 - Prob. 106RE Ch. 9 - Prob. 107RE Ch. 9 - Prob. 108RE Ch. 9 - Prob. 109RE Ch. 9 - Prob. 110RE Ch. 9 - Prob. 111RE Ch. 9 - Prob. 112RE Ch. 9 - Prob. 113RE Ch. 9 - Prob. 114RE Ch. 9 - Prob. 115RE Ch. 9 - Prob. 116RE Ch. 9 - Prob. 117RE Ch. 9 - Prob. 118RE Ch. 9 - Prob. 119RE Ch. 9 - Prob. 120RE Ch. 9 - Prob. 121RE Ch. 9 - Prob. 122RE Ch. 9 - Prob. 123RE Ch. 9 - Prob. 124RE Ch. 9 - Prob. 125RE Ch. 9 - Prob. 126RE Ch. 9 - Prob. 127RE Ch. 9 - Prob. 128RE Ch. 9 - Prob. 129RE Ch. 9 - Prob. 130RE Ch. 9 - Prob. 131RE Ch. 9 - Prob. 132RE Ch. 9 - Prob. 133RE Ch. 9 - Prob. 134RE Ch. 9 - Prob. 135RE Ch. 9 - Prob. 136RE Ch. 9 - Prob. 137RE Ch. 9 - Prob. 138RE Ch. 9 - Prob. 139RE Ch. 9 - Prob. 140RE Ch. 9 - Prob. 141RE Ch. 9 - Prob. 142RE Ch. 9 - Prob. 143RE Ch. 9 - Prob. 144RE Ch. 9 - Prob. 145RE Ch. 9 - Prob. 146RE Ch. 9 - Prob. 147RE Ch. 9 - Prob. 148RE Ch. 9 - Prob. 149RE Ch. 9 - Prob. 150RE Ch. 9 - Prob. 151RE Ch. 9 - Prob. 152RE Ch. 9 - Prob. 153RE Ch. 9 - Prob. 154RE Ch. 9 - Prob. 155RE Ch. 9 - Prob. 156RE Ch. 9 - Prob. 157RE Ch. 9 - Prob. 158RE Ch. 9 - Prob. 1CT Ch. 9 - Prob. 2CT Ch. 9 - Prob. 3CT Ch. 9 - Prob. 4CT Ch. 9 - Prob. 5CT Ch. 9 - Prob. 6CT Ch. 9 - Prob. 7CT Ch. 9 - Prob. 8CT Ch. 9 - Prob. 9CT Ch. 9 - Prob. 10CT Ch. 9 - Prob. 11CT Ch. 9 - Prob. 12CT Ch. 9 - Prob. 13CT Ch. 9 - Prob. 14CT Ch. 9 - Prob. 15CT Ch. 9 - Prob. 16CT Ch. 9 - Prob. 17CT Ch. 9 - Prob. 18CT Ch. 9 - Prob. 19CT Ch. 9 - Prob. 20CT Ch. 9 - Prob. 21CT Ch. 9 - Prob. 22CT Ch. 9 - Prob. 23CT Ch. 9 - Prob. 24CT Ch. 9 - Prob. 25CT Ch. 9 - Prob. 1CLT Ch. 9 - Prob. 2CLT Ch. 9 - Prob. 3CLT Ch. 9 - Prob. 4CLT Ch. 9 - Prob. 5CLT Ch. 9 - Prob. 6CLT Ch. 9 - Prob. 7CLT Ch. 9 - Prob. 8CLT Ch. 9 - Prob. 9CLT Ch. 9 - Prob. 10CLT Ch. 9 - Prob. 11CLT Ch. 9 - Prob. 12CLT Ch. 9 - Prob. 13CLT Ch. 9 - Prob. 14CLT Ch. 9 - Prob. 15CLT Ch. 9 - Prob. 16CLT Ch. 9 - Prob. 17CLT Ch. 9 - Prob. 18CLT Ch. 9 - Prob. 19CLT Ch. 9 - Prob. 20CLT Ch. 9 - Prob. 21CLT Ch. 9 - Prob. 22CLT Ch. 9 - Prob. 23CLT Ch. 9 - Prob. 24CLT Ch. 9 - Prob. 25CLT Ch. 9 - Prob. 26CLT Ch. 9 - Prob. 27CLT Ch. 9 - Prob. 28CLT Ch. 9 - Prob. 29CLT Ch. 9 - Prob. 30CLT Ch. 9 - Prob. 31CLT Ch. 9 - Prob. 32CLT Ch. 9 - Prob. 33CLT Ch. 9 - Prob. 34CLT Ch. 9 - Prob. 35CLT Ch. 9 - Prob. 36CLT Ch. 9 - Prob. 37CLT Ch. 9 - Prob. 38CLT Ch. 9 - Prob. 39CLT Ch. 9 - Prob. 40CLT Ch. 9 - Prob. 41CLT Ch. 9 - Prob. 42CLT Ch. 9 - Prob. 43CLT Ch. 9 - Prob. 44CLT Ch. 9 - Prob. 45CLT Ch. 9 - Prob. 46CLT Ch. 9 - Prob. 47CLT Ch. 9 - Prob. 48CLT Ch. 9 - Prob. 49CLT Ch. 9 - Prob. 50CLT Ch. 9 - Prob. 51CLT Ch. 9 - Prob. 52CLT Ch. 9 - Prob. 53CLT Ch. 9 - Prob. 54CLT Ch. 9 - Prob. 55CLT

Knowledge Booster

To identify and graph the polar equation. Also to identify any symmetry and zeros of r

To calculate: To identify and graph the polar equation. Also to identify any symmetry and zeros of r<m:math><m:mi>r</m:mi></m:math>

Answer to Problem 36E

Explanation of Solution

Chapter 9 Solutions

To calculate: To identify and graph the polar equation. Also to identify any symmetry and zeros of $r$