Chapter 9.6, Problem 4E

To determine

To calculate: To fill in the blanks

Answer to Problem 4E

The equation $r=1+\sin \theta$ represents a Cardiod.

Explanation of Solution

Given information: Statement is “The equation $r=1+\sin \theta$ represents a ___________.”

Formula Used:

Lima cons are the graph of below polar equations:

  $\begin{array}{l}r=a\pm b\cos \theta \\ r=a\pm b\sin \theta \end{array}$

When $\frac{a}{b}<1$ , polar equation represents limacon with inner loop

When $\frac{a}{b}=1$ , polar equation represents Cardiod

When $1<\frac{a}{b}<2$ , polar equation represents Dimpled limacon

When $\frac{a}{b}=2$ , polar equation represents convex limacon

Calculation:

Polar equation is given as

  $r=1+\sin \theta$

Comparing the above equation with $r=a\pm b\sin \theta$ ,

  $\begin{array}{l}a=1\\ b=1\end{array}$

Thus,

  $\begin{array}{l}\frac{a}{b}=\frac{1}{1}\\ \frac{a}{b}=1\end{array}$

Therefore, polar equation represents Cardiod

Conclusion:

Hence, equation $r=1+\sin \theta$ represents a Cardiod.