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Individual periods are $1$, $6$ and $10$ . Their LCM is $30$ . My question is: Is the LCM of the individual period always the shortest period of the net function ? and if not, is there a way to identify ?

In this question it is indeed the shortest period but as given in the third comment of this post

It is mentioned that it need not always be the case, how do i know if it is that case or not?

soutrik das
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1 Answers1

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The period of $\sin ax$ is $2\pi/a$ The periods of individual functiona are $T_1=1, T_2=6, T_3=10$ As their LCM is 30. So the period is 30. You may check that $f(x+T)=f(x),$for real values of $x$, where $T=30$. No number lesser than 30 will do so.

Yes LCM method is the shortes one but only some time it fails. For example the period of $f(x)=\sin^4 x+\cos^4 x$. By LCM rile is $\pi$ but by trigonometric reduction $f(x)=A+B \cos 4x$ so the correct period id $\pi/2$

Trigonometric reduction alway gives the period correctly. Here you may use $\sin C + \sin D.$ forula,

Z Ahmed
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  • yes , but will doing LCM give me the shortest functions of all such combinations of simple functions into a big functions ? and how would you know if it is not the smalles period? – soutrik das Jun 20 '20 at 03:38
  • @soutrik das you may see my edit. – Z Ahmed Jun 20 '20 at 03:47
  • your example $\sin^2x-\cos^2x$ can be written as $-\cos{2x}$ which still has $\pi$ as the period but i get what you wanted to say that sometimes it cannot give the shortest period. – soutrik das Jun 20 '20 at 04:57
  • @sourtik das sorry it was a typo it should have been $$\sin^4 x+\cos^4 x$, you may see my edit. – Z Ahmed Jun 20 '20 at 05:05