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Chapter 9: Q 44. (page 731)

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.
$r = \sin 4 胃$

Short Answer

Expert verified

The required equation is ${(x^{2} + y^{2})}^{\frac{5}{2}} = 2 x y (x^{2} - y^{2})$ .

Step by step solution

01

Step 1. Given information.

The given equation in polar coordinates is:

$r = \sin 4 胃$

02

Step 2. Find the equation in rectangular coordinates.

$r = \sin 4 胃 r = 2 \sin 2 胃 \cos 2 胃 [Since \sin 4 胃 = 2 \sin 2 胃 \cos 2 胃]$

Now substitute $\sin 2 胃 = 2 \sin 胃 \cos 胃 and \cos 2 胃 = 2 \cos^{2} 胃 - 1$ ,

$r = 2 \sin 胃 \cos 胃路 (2 \cos^{2} 胃 - 1)$

Substitute $\frac{x}{r} = \cos 胃 and \frac{y}{r} = \sin 胃$ ,

localid="1649319886678" $r = 2 路 \frac{y}{r} 路 \frac{x}{r} (2 路 \frac{x^{2}}{r^{2}} - 1) r = 2 路 \frac{x y}{r^{2}} (2 路 \frac{x^{2}}{r^{2}} - 1) r = \frac{2 x y}{r^{2}} (\frac{2 x^{2} - r^{2}}{r^{2}})$

Now cross multiply,

$r^{5} = 2 x y (2 x^{2} - r^{2}) {(\sqrt{x^{2} + y^{2}})}^{5} = 2 x y (2 x^{2} - (x^{2} + y^{2})) [r^{2} = x^{2} + y^{2} and r = \sqrt{x^{2} + y^{2}}] {(x^{2} + y^{2})}^{\frac{5}{2}} = 2 x y (2 x^{2} - (x^{2} + y^{2})) {(x^{2} + y^{2})}^{\frac{5}{2}} = 2 x y (x^{2} - y^{2})$

Therefore, the equation in rectangular coordinates is ${(x^{2} + y^{2})}^{\frac{5}{2}} = 2 x y (x^{2} - y^{2})$ .

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Most popular questions from this chapter

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

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