Chapter 8.2, Problem 52E

To determine

To match the below equation with the graph labeled I-IV where

  $r=1+3\cos \left(3\theta \right)$

  

Answer to Problem 52E

The graph that matches the equation $r=1+3\cos \left(3\theta \right)$ is II

  

  $\begin{array}{l}\theta =0,r=4\\ \theta =30,r=1\\ \theta =60,r=-2\\ \theta =90,r=1\\ \theta =120,r=4\\ \theta =-30,r=1\\ \theta =-60,r=4\end{array}$

Explanation of Solution

Given: Polar equation: $r=1+3\cos \left(3\theta \right)$

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.

After plotting the points in the graph it can be seen that $r=1+3\cos \left(3\theta \right)$ gives the below graph and matches with the label II graph

Calculation:

When

  $\begin{array}{l}\theta =0,r=4\\ \theta =30,r=1\\ \theta =60,r=-2\\ \theta =90,r=1\\ \theta =120,r=4\\ \theta =-30,r=1\\ \theta =-60,r=4\end{array}$

Hence it matches to label II from the above graph.

Conclusion:

Hence the graph labelled II matches the given equation $r=1+3\cos \left(3\theta \right)$