Chapter H.1, Problem 66E

+6

To determine

To explain:

The changes in the shape of curve $r=1+c\sin n\theta$ if:

Case 1: $n$ increases

Case 2: $c$ changes

Answer to Problem 66E

On increasing $\left(n\right)$ in the curve equation number of lobes increases for the curve. The number of lobes in the curve comes in forward to origin if the value $c$ increases to $1$ .

Explanation of Solution

Given information:

  $r=1+c\sin n\theta$

The given equation of the curve is $r=1+c\sin n\theta$ . Now need to find the changes occur in the curve equation when $n$ increases and $c$ changes:

Consider:

Put $n=1$ : $r=1+c\sin n\theta =1+c\sin \left(1\theta \right)=1+c\sin \theta$

The curve has one lobe.

Put $n=2$ : $r=1+c\sin n\theta =1+c\sin \left(2\theta \right)=1+c\sin 2\theta$

The curve has two lobes.

Consider,

  $-1>c>1$ , then the curves are congruent.

The lobes analysis depends on number of lobes $\left(n\right)$ :

If $n$ is even then the curve has $n$ big and $n$ small lobes.

If $n$ is odd then the curve has $n$ inner and $n$ outer lobes.

  

If $c$ is between $0$ and $1$ , then the number of lobes $n$ do not pinch the origin point. The number of lobes in the curve comes in forward to origin if the value $c$ increases to $1$ .

So, when $c>1$ , the $n$ big and $n$ small lobes are present. Here, if

If $n$ is odd then in the big lobes the small lobes are contained.

If $n$ is even then big lobes the small lobes are used alternatives.