Plotting a polar curve

Question

The question is, to generate a polar graph using a graphing utility, and to choose parameter interval so that the complete graph is generated.

$$r=\cos\frac{\theta}{5}$$

To find such an interval, we are looking for smallest number of complete revolutions until value of $r$ begins to repeat. Algebraically,this amounts to

$$\cos\frac{\theta}{5}=\cos\frac{\theta+2n\pi}{5}$$

For this equality to hold,$\frac{2n\pi}{5}$ must be an even multiple of $\pi$,the smallest n for which it occurs is $n=5$.Therefore, the graph will be traced completely in $5$ revolutions ($10\pi$).

But when I draw it the graph is completely traced in $5\pi$, where have I gone wrong?

Answer 1

$r(5\pi+t)=\cos(\pi+\frac{t}5)=-\cos(\frac{t}{5})=-r(t)$. Also the angle at $5\pi+t$ points in the opposite direction of the angle at $t$. Hence you get repetition.

Answer 2

Since the question has been already answered, I post here a visual answer that I hope may help you to understand the problem better:

Cheers!