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 21E

To determine

To calculate: To test for symmetry with respect to the line $θ=π2$ , the polar axis and the pole

Expert Solution & Answer

Answer to Problem 21E

The polar equation is NOT symmetric with respect to the line $θ=π2$ and polar axis, while equation is symmetric with respect to pole

Explanation of Solution

Given information: Polar equation is $r2=16sin2θ$

Formula Used:

Test for Symmetry:

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

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

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

Calculation:

Polar equation is given as follows:

$r2=16sin2θ$

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

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

$r2=16sin2(π−θ)r2=16sin(2π−2θ)r2=16(−sin2θ)r2=−16sin2θ$

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

$r2=16sin2(−θ)r2=16sin(−2θ)r2=16(−sin2θ)r2=−16sin2θ$

Since above equation is not same as the given polar equation

Thus, polar equation is NOT 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

$r2=16sin2(π+θ)r2=16sin(2π+2θ)r2=16(sin2θ)r2=16sin2θ$

Since above equation is same as the given polar equation

Thus, polar equation is with respect to the pole

Conclusion:

Hence, polar equation is NOT symmetric with respect to the line $θ=π2$ and polar axis, while equation is symmetric with respect to pole

Chapter 9.6, Problem 20E

Chapter 9.6, Problem 22E

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 test for symmetry with respect to the line θ = π 2 , the polar axis and the pole

To calculate: To test for symmetry with respect to the line θ=π2<m:math><m:mrow><m:mi>θ</m:mi><m:mo>=</m:mo><m:mfrac><m:mi>π</m:mi><m:mn>2</m:mn></m:mfrac></m:mrow></m:math> , the polar axis and the pole

Answer to Problem 21E

Explanation of Solution

Chapter 9 Solutions

To calculate: To test for symmetry with respect to the line $θ=π2$ , the polar axis and the pole