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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 36. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{3 c s c (胃)}{c s c (胃) - 1}$

Short Answer

Expert verified

The rectangular equation is

$x^{2} - 6 y = 9$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{3 c s c (胃)}{c s c (胃) - 1}$

02

Step 2. Convert into the rectangular equation

The polar equation is

$r = \frac{3 c s c (胃)}{c s c (胃) - 1}$

Multiply top and bottom by $\sin (胃)$

$r = \frac{3 c s c (胃) 脳 \sin (胃)}{c s c (胃) 脳 \sin (胃) - 1 脳 \sin (胃)} r = \frac{3 \frac{1}{\sin (胃)} 脳 \sin (胃)}{\frac{1}{\sin (胃)} 脳 \sin (胃) - 1 脳 \sin (胃)} r = \frac{3}{1 - \sin (胃)}$

Multiply denominator on both sides

$r 脳 (1 - \sin (胃)) = \frac{3}{1 - \sin (胃)} 脳 1 - \sin (胃) r 脳 (1 - \sin (胃)) = 3 r - r \sin (胃) = 3$

Apply the conversion formula

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

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

Add both sides by $y$

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

Take square on both sides and simplify it

${(\sqrt{x^{2} + y^{2}})}^{2} = {(3 + y)}^{2} x^{2} + y^{2} = 9 + 6 y + y^{2} x^{2} + y^{2} - 6 y - y^{2} = 9 x^{2} - 6 y = 9$

So, the rectangular equation is

$x^{2} - 6 y = 9$

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Most popular questions from this chapter

To graph $y = {(x - 3)}^{2} + 1$ , shift the graph of $y = x^{2}$ to the right________ units and then________ 1 unit.

Transform the equation $r = 6 \cos (胃)$ from polar coordinate to rectangular coordinate.

In Problems 15鈥�18, the graph of a hyperbola is given. Match each graph to its equation.

Equations:

$(A) \frac{x^{2}}{4} - y^{2} = 1 (B) x^{2} - \frac{y^{2}}{4} = 1 (C) \frac{y^{2}}{4} - x^{2} = 1 (D) y^{2} - \frac{x^{2}}{4} = 1$

Graph:

True or False

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