Chapter 9.6, Problem 13E

To determine

To calculate: To determine the equation of polar curve whose graph is shown

Answer to Problem 13E

Polar equation is $r=1-2\sin \theta$

Explanation of Solution

Given information: Polar graph is

  

Formula Used:

Cardioid: Heart shape curve and general form is $r=a+a\cos \theta$

Limacons: General form of cardioid and general form is $r=a+b\cos \theta$

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

Rose Curves: Sinusoidal curve that have a flower shape and general equation is $r=a\cos \left(k\theta \right)$

Archimedian Spirals: Curve that extends indefinitely outward from a pole and general form is $r=a+b\theta$

Lemniscate: Eight shaped curve and general form is $r^2=a^2\cos \left(2\theta \right)$

Calculation:

Polar graph is given as follows:

  

Above is a graph of limacon with inner loop

Also, graph is below the polar axis and symmetry with respected to the line $\theta =\frac{\pi }{2}$

Thus, general equation is given as follows

  $r=a-b\sin \theta$ , $\frac{a}{b}<1$

Here,

  $\begin{array}{l}a=1\\ b=2\end{array}$

Substituting the values in general equation,

  $r=1-2\sin \theta$

Conclusion:

Hence, polar equation is $r=1-2\sin \theta$