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Chapter 9: Q. 88 (page 585)

Write down two different tests for symmetry with respect to

the polar axis. Find examples in which one test works and

the other fails. Which test do you prefer to use? Justify your

answer.

Short Answer

Expert verified

The first test works for r=cos胃while fails for r=sin胃but the second test works for r=sin胃and fails forr=cos胃{"x":[[5,4,5,4,4,19,30,33],[52,79],[53,79],[122,115,95,94,107,120,124],[137,129,129,133,144,154,159,158,152,136],[195,193,181,169,166,173,183,194,196,192,182,170,165],[224.5,223.5,222.5,219.5,218.5,216.5,216.5,216.5,218.5,223.5,231.5,241.5,253.5,267.5,276.5,282.5,286.5,289.5,291.5,289.5,284.5,276.5,270.5,264.5,257.5,248.5,236.5,226.5],[228.5,228.5,229.5,230.5,238.5,251.5,275.5,297.5,314.5,322.5,323.5]],"y":[[9,73,73,18,18,11,9,16],[31,31],[53,52],[21,5,17,59,75,71,60],[73,64,33,15,9,14,33,60,75,73],[21,11,7,12,28,36,39,46,58,70,75,70,57],[20.25390625,20.25390625,20.25390625,21.25390625,24.25390625,27.25390625,33.25390625,42.25390625,51.25390625,60.25390625,67.25390625,71.25390625,72.25390625,70.25390625,66.25390625,60.25390625,54.25390625,45.25390625,37.25390625,28.25390625,21.25390625,14.25390625,10.25390625,9.25390625,9.25390625,11.25390625,15.25390625,18.25390625],[39.25390625,39.25390625,39.25390625,39.25390625,40.25390625,40.25390625,39.25390625,36.25390625,34.25390625,33.25390625,32.25390625]],"t":[[0,0,0,0,0,0,0,0],[0,0],[0,0],[0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0],[1646737436105,1646737436116,1646737436132,1646737436150,1646737436166,1646737436183,1646737436200,1646737436216,1646737436233,1646737436250,1646737436267,1646737436283,1646737436299,1646737436317,1646737436332,1646737436350,1646737436365,1646737436383,1646737436399,1646737436416,1646737436432,1646737436450,1646737436467,1646737436482,1646737436499,1646737436516,1646737436532,1646737436544],[1646737436857,1646737436877,1646737436925,1646737436933,1646737436949,1646737436965,1646737436982,1646737436998,1646737437015,1646737437032,1646737437049]],"version":"2.0.0"}.

Step by step solution

01

Step 1. Given

A polar axis.

02

Step 2. Explanation

Symmetrytestwithrespecttopolaraxisi.e.replace胃with-胃andsecondalyreplace胃with蟺-胃thereforer=cos(-胃)r=肠辞蝉胃thusfirsttestworksforr=肠辞蝉胃Nowreplace胃with蟺-胃inr=肠辞蝉胃thus,r=cos(蟺-胃)=-肠辞蝉胃thistestfailsforr=肠辞蝉胃

03

Step 3. Explanation

Nowconsiderr=sin胃replace胃with-胃r=sin(-胃)r=-sin胃thustestispassed.Nowreplace胃with蟺-胃r=sin(蟺-胃)=蝉颈苍胃testfailsforr=蝉颈苍胃

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