A petal in the third quadrant is generated when r is negative.
[tex]\begin{gathered} \text{ On the interval (}\frac{\pi}{6},\frac{\pi}{3})\text{ ,} \\ \cos (3\theta)\in\lbrack0,-1\rbrack \end{gathered}[/tex]
Therefore,
[tex]r=6\cos (3\theta)\text{ is in \lbrack{}0,-6\rbrack on the interval (}\frac{\pi}{6},\frac{\pi}{3})[/tex]
