Chapter 14.3, Problem 32WE

To determine

To find: The polar graph of given polar equation.

Answer to Problem 32WE

  

Explanation of Solution

Given information:

An equation in polar coordinate is given as $r=4\sin 3\theta$ .

Concepts used: Conversion formula for coordinate systems are given as-

1. From polar to rectangular:

  $x=r\cos \theta$

  $y=r\sin \theta$

2. From rectangular to polar:

  $r^2=x^2+y^2\Rightarrow r=\pm \sqrt{x^2+y^2}$

  $\cos \theta =\frac{x}{r}$ ; $\sin \theta =\frac{y}{r}$ ; $\tan \theta =\frac{x}{y}$ .

Calculation:

Given equation in polar-coordinate is $r=4\sin 3\theta$ .

Constructing a table for $\theta$ .

$\theta$	$r=4\sin 3\theta$
$0^0$	$0$
${15}^0$	$2\sqrt{2}$
${30}^0$	$4$
${45}^0$	$2\sqrt{2}$
${60}^0$	$0$
${75}^0$	$-2\sqrt{2}$
${90}^0$	$-4$
${105}^0$	$-2\sqrt{2}$
${120}^0$	$0$
${135}^0$	$2\sqrt{2}$
${150}^0$	$4$
${165}^0$	$2\sqrt{2}$
${180}^0$	$0$

Plotting graph as below.