polar coordinates ..question about the answer from the solution manual

Question

Im trying to figure out but for some reason I dont know how to...could someone please tell me how did they get this answer from the solution manual....they skipped steps so I have no idea

Answer 1

I assume you can see how to draw the graph on the left. The point of doing so is to find the interesting points for the polar plot. This is similar to finding the interesting points of a polynomial, like roots and relative maxima/minima to plot it.

What interesting points do we find? Any place $r=0$ the polar graph goes through the origin, so $(0,\frac \pi 6)$ and $(0,\frac {5\pi}6)$ qualify. Also maxima and minima in $r$, so $(-1,\frac \pi2)$ and $(3, \frac {3\pi}2)$ These are the points identified on the polar plot. It doesn't get you the whole graph, but it gets you many of the major features.