Chapter 9, Problem 2CRT

(a)

To determine

To find: The graph of polar equation.

Answer to Problem 2CRT

The graph of polar equation is shown in figure-(1).

Explanation of Solution

Given information:

Polar equation is $r=2\sin 2\theta$ .

Table of values are

$\theta$	$r$
$0$	$0$
$\frac{\pi }{4}$	$2$
$\frac{3\pi }{4}$	$2$
$\frac{5\pi }{4}$	$2$
$\frac{3\pi }{2}$	$0$

Calculation:

Plot the points to obtain the following four-leafed rose

    figure−(1)

Therefore, the graph of polar equation is shown in figure-(1).

(b)

To determine

To find: The polar equation in to the rectangular coordinates equation.

Answer to Problem 2CRT

The polar equation in to the rectangular coordinates of equation is ${(x^2+y^2)}^{3/2}=4xy$ .

Explanation of Solution

Given information:

Polar equation is $r=2\sin 2\theta$ .

Calculation:

Convert the polar equation in to the rectangular coordinatesequation

  $r=2\sin 2\theta$

  $\begin{array}{l}r=2(2)\sin \theta \cos \theta \\ r=4\sin \theta \cos \theta \end{array}$

Multiply by $r^2$ both sides

  $r^3=r^{2}4\sin \theta \cos \theta$

Substitute $x=r\sin \theta ,y=r\cos \theta$

  $r^3={(x^2+y^2)}^{3/2}$

  ${(x^2+y^2)}^{3/2}=4xy$

Therefore, the polar equation in to the rectangular coordinates of equation is ${(x^2+y^2)}^{3/2}=4xy$ .