What is a good way to pick points for polar equations?

If I want to plot $r = 4\sin3\theta$ what is a good way to pick points? The period is $2\pi/3$ so it will repeat after that, but how do I pick points so that I get a good range of values so that I will be able to tell in which direction or how the curve is moving?

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First, the period is $\frac{2\pi}{3}$.
As to picking points, what I'd recommend is not to—that is, instead of picking some specific values for $\theta$ and calculating $r$, sketch a graph of $y=4\sin 3x$ for $0\le x\le 2\pi$. The maximums, minimums, and zeros on that graph are most likely the points you're looking for, you can probably draw that graph as quickly (or more quickly) than computing specific values for $r$, and the graph also tells you what's happening between those values.
If you're determined to pick points, note that simple sinusoidal functions (things like $f(x)=a\sin bx+c$) have maximums/minimums/zeros spaced by quarter-periods.
It depends upon how you plot it. If I wanted to plot it in Excel, for instance, I would just figure the period $2\pi /3 \approx 2.1$ and plot 100 points at an interval of 0.021. If you are plotting by hand but calculating by spreadsheet, maybe 20 points evenly spaced is reasonable. You could start there and see if it meets your needs. More aggressive would be to figure out where the maximum and minimum r come from ($\pi /6, \pi /2$ and $0, \pi /3, 2\pi /3)$ and plot those, plus halfway in between. But then we are getting close to 20.