Short Answer

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$r = \cos 4 θ$

The required equation is ${(x^{2} + y^{2})}^{\frac{3}{2}} = x^{2} - 3 y^{2}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given equation in polar coordinates is:

$r = \cos 4 θ$

Step 2. Find the equation in rectangular coordinates.

$r = \cos 4 θ r = 2 \cos^{2} 2 θ - 1 [\cos 4 θ = (2 \cos^{2} 2 θ - 1)] r = 2 (2 \cos^{2} 2 θ - 1) - 1 [\cos 2 θ = 2 \cos^{2} θ - 1] r = 4 \cos^{2} θ - 2 - 1 r = 4 \cos^{2} θ - 3$

Substitute $\frac{x}{r} = \cos θ$ ,

role="math" localid="1649323840721" $r = 4 \cdot \frac{x^{2}}{r^{2}} - 3 r = \frac{4 x^{2}}{r^{2}} - 3 r = \frac{4 x^{2} - 3 r^{2}}{r^{2}}$

Cross multiply,

$r^{3} = 4 x^{2} - 3 r^{2} {(\sqrt{x^{2} + y^{2}})}^{3} = 4 x^{2} - 3 (x^{2} + y^{2}) [r^{2} = x^{2} + y^{2} and r = \sqrt{x^{2} + y^{2}}] {(x^{2} + y^{2})}^{\frac{3}{2}} = 4 x^{2} - 3 x^{2} - 3 y^{2} {(x^{2} + y^{2})}^{\frac{3}{2}} = x^{2} - 3 y^{2}$

Therefore, the equation in rectangular coordinates is ${(x^{2} + y^{2})}^{\frac{3}{2}} = x^{2} - 3 y^{2}$ .

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Short Answer

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$r = \cos 4 θ$

Step by Step Solution

Step 1. Given information.

Step 2. Find the equation in rectangular coordinates.

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Calculus

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Short Answer

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.r=cos 4θ

Step by Step Solution

Step 1. Given information.

Step 2. Find the equation in rectangular coordinates.

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In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$r = \cos 4 θ$