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Chapter 9: Q. 87 (page 585)

Explain why the following test for symmetry is valid: Replace r by -r and 胃by-胃in a polar equation. If an equivalent

equation results, the graph is symmetric with respect to the

line 胃=蟺2(y-axis).

(a) Show that the test on page 574 fails for r2=cos胃, yet this new test works.

(b) Show that the test on page 574 works for r2=sin胃, yet

this new test fails.

Short Answer

Expert verified

(a) Assuring all the conditions the new test works.

(b) Assuring all the conditions the new test does not works.

Step by step solution

01

Part (a) Step 1. Given 

Given polar coordinate

r2=cos胃

02

Step 2. Explanation

Consider

r2=cos胃

Now, by symmetry test with respect to polar coordinate

replace localid="1646736860762" 胃飞颈迟丑蟺-胃

therefore,

r2=cos蟺-胃=-肠辞蝉胃

the two are not equivalent therefore test fails.

Now replace r with -r and

胃with-胃therefore,(-r)2=cos(-胃)r2=肠辞蝉胃

hence the new test works.

03

Part (b) Step 1. Given

Given polar coordinate

r2=sin胃

04

Part (b) Step 2. Explanation

Consider,

r2=sin胃replace胃with蟺-胃thus,r2=sin蟺-胃=蝉颈苍胃thustwoareequalhencetestworksNowreplacerwith-rand胃with-胃(-r)2=sin(-胃)r2=-蝉颈苍胃Thetwoarenotequalsotestfails.

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