Chapter H.1, Problem 65E

a.

To determine

To graph: The polar curve $r=\sin n\theta$

Explanation of Solution

Given: $r=\sin n\theta$

Using graphing utility to graph $r=\sin n\theta$

For n=1

  

For n=2

  

For n=3

  

Above graphs shape depends on value of n. If the value of n is odd number of loop is equivalent to n. If the value of n is even then number of loop double.

b.

To determine

To graph: The polar curve $r=\left|\sin n\theta \right|$

Explanation of Solution

Given: $r=\left|\sin n\theta \right|$

Using graphing utility to graph $r=\left|\sin n\theta \right|$

For n=1

  

For n=2

  

For n=3

  

Above graphs shape depends on value of n. Number of loop is double the value of n.