Chapter 9.6, Problem 12E

To determine

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

Answer to Problem 12E

Polar equation is $r=2\sin 3\theta$

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 rose curve with three petals

Thus, general equation is given as follows

  $r=a\sin \left(k\theta \right)$

Here, number of petals is given as

  $\begin{array}{l}Petals=k\\ 3=k\\ k=3\end{array}$

Also, from graph, $a=2$

Substituting the values in general equation,

  $r=2\sin 3\theta$

Conclusion:

Hence, polar equation is $r=2\sin 3\theta$