Period of $\sin(ax)+\cos(bx)$

Asked Dec 08 '14 at 03:09

Active Dec 08 '14 at 05:24

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Take the function $f(x)=\sin(ax)\cos(bx)$, with $a,b>0$. Suppose there are positive integer $p$ and $q$ such that $ap=bq=r$.

Then $r$ is a period of $f$ but non always the shortest : for $f(x)= \sin x\cdot \cos(3x)$ the shortest period isn't $2\pi$, but $pi$.

If $f(x)= \sin(ax)+\cos(bx)$, $r$ should always be the fundamental period of $f$ ? And if so, how to prove it?

edited Dec 08 '14 at 03:21

thanasissdr

asked Dec 08 '14 at 03:09

user198469

1

You're title says $\sin(ax)+\cos(bx)$, but your question says $\sin(ax)\cos(bx)$. – Thomas Andrews Dec 08 '14 at 03:22
Don't you mean something like $\frac{2\pi}{a} p + \frac{2\pi}{b} q = r$? – epimorphic Dec 18 '14 at 03:50
@ThomasAndrews - The question begins with assertions about $\sin(ax)\cos(bx)$, but concludes with a question about $\sin(ax)+\cos(bx)$. – Ted Hopp Mar 07 '19 at 16:04

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