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Q 44.

Found in: Page 731


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

r=sin 4θ

The required equation is x2+y252=2xyx2-y2.

See the step by step solution

Step by Step Solution

Step 1. Given information.

The given equation in polar coordinates is:

r=sin 4θ

Step 2. Find the equation in rectangular coordinates.

r=sin 4θr=2 sin 2θ cos 2θ Since sin 4θ=2 sin 2θ cos 2θ

Now substitute sin2θ=2sinθ cosθ and cos 2θ=2cos2θ-1,

r=2 sinθ cosθ·2cos2θ-1

Substitute xr=cosθ and yr=sinθ,

localid="1649319886678" r=2·yr·xr2·x2r2-1r=2·xyr22·x2r2-1r=2xyr22x2-r2r2

Now cross multiply,

r5=2xy2x2-r2x2+y25=2xy2x2-x2+y2 r2=x2+y2 and r=x2+y2x2+y252=2xy2x2-x2+y2x2+y252=2xyx2-y2

Therefore, the equation in rectangular coordinates is x2+y252=2xyx2-y2.

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