How would you find the period of $$f(x)=\cos(ax) + \sin(bx)$$
*Edit such that $a,b \in \mathbb{Z}$
A step by step proof would be appreciated!
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Albus Dumbledore
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1Let $a = 1, b = \pi$ and you will find out that there is no period... – Henricus V. Mar 07 '16 at 06:25
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1For "most" real number pairs $(a,b)$, $f(x)$ will not be periodic. But for example when $a$ and $b$ are integers, then $f(x)$ is periodic. – André Nicolas Mar 07 '16 at 06:27
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http://math.stackexchange.com/questions/164221/period-of-the-sum-product-of-two-functions – Martin R Mar 07 '16 at 06:28
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It only works for integers. There exists no general case here. Try graphing the function on a calculator, you'll see. – Sat D Mar 07 '16 at 06:28
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@SatD: Even if $a,b$ are not rationally related, the function will be almost periodic, so noticing that $f$ is not periodic will be difficult. – copper.hat Mar 07 '16 at 06:33
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See also: Period of sum of sinusoids. – Martin Sleziak Aug 30 '19 at 15:25
2 Answers
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This has a period if and only if $\frac{a}{b}$ is rational.
marty cohen
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Why? I tried to justify, but I don't know how to justify that if $h=f+g$, being $f$ of periods $p_1$ and $g$ of periods $p_2$, then for $h$ to be periodic there must be $m, n$ such that $p_1=p_2$. – Pierre Aug 05 '22 at 12:40
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Period of $cos(ax)$ and $sin(bx)$ are $2π/|a|$ and $2π/|b|$, respectively. Now
The period of the sum of two periodic functions are the LCM of their periods
So here the period of $f(x)= cos(ax) + sin(bx)$ will be LCM($2π/|a|$,$2π/|b|$)
PS: As other answers(or comments) pointed out this only works if $a,b \in$ $\mathbb{Z}$
hxri
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1You shall note that this works only for $a,b\in\mathbb{Z}$. Please look on this graph – Galc127 Mar 07 '16 at 06:39
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yes but i think the person needs an answer for integers. (By seeing the lack of detail in the question) – hxri Mar 07 '16 at 06:45
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