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


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


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?

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Try graphing $r=cos\theta$ by hand. It creates a circle between 0 and $\pi$. It re-traces the same circle between $\pi$ and $2\pi$. The same thing occurs in your graph. – Paul Sundheim Aug 23 '14 at 14:54
@PaulSundheim how is that same with this graph, this graph does not trace itself before $5\pi$, and i am getting $10\pi$. – SHOBHIT GAUTAM Aug 23 '14 at 14:56
As you said, "the graph is completely traced in 5π" but now you say "this graph does not trace itself before 5π"? Which is the correct statement? – Paul Sundheim Aug 23 '14 at 14:59
I just traced it on a graphing calculator. Seemed to be complete in $5\pi$ and then repeated. – paw88789 Aug 23 '14 at 15:01
@PaulSundheim The graph is completely traced in 5pi means after that you are just tracing it again.I know 5pi is correct, do you mind telling where is the computation wrong? – SHOBHIT GAUTAM Aug 23 '14 at 15:04
up vote 3 down vote accepted

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

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$r(5\pi+t)=$?? what do you mean. – SHOBHIT GAUTAM Aug 23 '14 at 15:13
He means $r$ evaluated at $5\pi+t$ – Paul Sundheim Aug 23 '14 at 15:15
got it thanks for the help. – SHOBHIT GAUTAM Aug 23 '14 at 15:18
@Paul S. Yes. Thanks. – paw88789 Aug 23 '14 at 15:30

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

enter image description here


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very helpful and cool, which utility is this. – SHOBHIT GAUTAM Aug 23 '14 at 15:26
Hi @Shobhit. This was entirely done with Matlab. – Dmoreno Aug 23 '14 at 15:26
thank you again its cool – SHOBHIT GAUTAM Aug 23 '14 at 15:28
You're welcome mate. – Dmoreno Aug 23 '14 at 15:31

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