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October 7th, 2015, 11:39 AM   #1
szz
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Math Focus: Calculus
Fundamental period of a composite function

Hi all,

I am a little bit confused about determining the fundamental period of a function:

$\displaystyle x(t) = \cos\left({2 \pi t\over \sqrt{5}}\right) + 7\sin\left({3\pi t\over\sqrt{125}}\right )$

I solve it as follows:

$\displaystyle
\begin{aligned}
& {2\pi \over {{2 \pi t\over \sqrt{5}}}} = \sqrt{5}\\
& {2\pi \over {{3 \pi t\over \sqrt{5}}}} = {2\sqrt{125}\over 3}\\
& (\sqrt{5}\sqrt{125}){\mathrm {lcm}(1,2)\over \mathrm {gcd}(1,3)} = 2\sqrt{5}\sqrt{125} = 50
\end{aligned}$

But graphing the function seems that I am wrong, and the period seems to be $\displaystyle \approx 22.36$.

What am I missing ?
Thank you un advance.

Last edited by szz; October 7th, 2015 at 12:22 PM.
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October 7th, 2015, 12:24 PM   #2
szz
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Math Focus: Calculus
Solved:

$\displaystyle \begin{aligned}
& {2\pi \over {{2 \pi t\over \sqrt{5}}}} = \sqrt{5}\\
& {2\pi \over {{3 \pi t\over \sqrt{5}}}} = {2\sqrt{125}\over 3} = {10\sqrt{5}\over 3}\\
& (\sqrt{5}){\mathrm {lcm}(1,10)\over \mathrm {gcd}(1,3)} = 10\sqrt{5} \approx 22.36
\end{aligned}
$
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