Chapter 8.2, Problem 47E

To determine

To sketch the graph of the family of polar equation where

  $r=1+\sin n\theta$

  $n=1,2,3,4,5$

And find the number of loops and how are they related

Answer to Problem 47E

The graph of polar equation when $n=1$

  $r=1+\sin \theta ,0\le \theta \le 2\pi$

  

The graph of polar equation when $n=2$

  $r=1+\sin 2\theta ,0\le \theta \le 2\pi$

  

The graph of polar equation when $n=3$

  $r=1+\sin 3\theta ,0\le \theta \le 2\pi$

  

The graph of polar equation when $n=4$

  $r=1+\sin 4\theta ,0\le \theta \le 2\pi$

  

The graph of polar equation when $n=5$

  $r=1+\sin 5\theta ,0\le \theta \le 2\pi$

  

As the value of n increases the number of loops in the graphs also increases

Hence the number of loops in the graph is directly proportionate to n

  $n\propto$ No. of loops

Explanation of Solution

Given: Polar equation: $r=1+\sin n\theta$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

As the value of n increases the number of loops in the graphs also increases

Hence the number of loops in the graph is directly proportionate to n

  $n\propto$ No. of loops

Calculation:

Conclusion:

Hence the number of loops in the graph is directly proportionate to n

  $n\propto$ No. of loops