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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q 27. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{8}{4 + 3 \sin (胃)}$

Short Answer

Expert verified

The rectangular equation is

$16 x^{2} + 7 y^{2} + 48 y = 64$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{8}{4 + 3 \sin (胃)}$

02

Step 2. Convert into the polar equation

The polar equation is

$r = \frac{8}{4 + 3 \sin (胃)}$

Multiply both sides by the denominator

$r 脳 (4 + 3 \sin (胃)) = \frac{8}{4 + 3 \sin (胃)} 脳 4 + 3 \sin (胃) r 脳 (4 + 3 \sin (胃)) = 8 4 r + 3 r \sin (胃) = 8$

Apply conversion equation formula

$r = \sqrt{x^{2} + y^{2}}, y = r \sin (胃)$

So, the rectangular equation is

$4 \sqrt{x^{2} + y^{2}} + 3 y = 8$

03

Step 3. Remove square root

The rectangular equation is

$4 \sqrt{x^{2} + y^{2}} + 3 y = 8$

Subtract both sides by $3 y$

$4 \sqrt{x^{2} + y^{2}} + 3 y - 3 y = 8 - 3 y 4 \sqrt{x^{2} + y^{2}} = 8 - 3 y$

Take square on both sides

${(4 \sqrt{x^{2} + y^{2}})}^{2} = {(8 - 3 y)}^{2} 16 (x^{2} + y^{2}) = 64 - 48 y + 9 y^{2} 16 x^{2} + 16 y^{2} + 48 y - 9 y^{2} = 64 16 x^{2} + 7 y^{2} + 48 y = 64$

So, the rectangular equation is

$16 x^{2} + 7 y^{2} + 48 y = 64$

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