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I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example:

$$f(x)=\cos(x/3)+\cos(x/4).$$

JMP
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Jaydeep
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  • Welcome to Math SE! We like questions that tell us what you have already tried, so that we can give appropriate answers. Also, if you know that the definition of "period" has two parts, did you succeed in proving any of them? – user21820 Jul 21 '14 at 15:11
  • http://math.stackexchange.com/questions/164221/period-of-the-sum-product-of-two-functions – lab bhattacharjee Jul 21 '14 at 15:18
  • This also : http://math.stackexchange.com/questions/681750/sum-of-two-periodic-functions-is-periodic – pitchounet Jul 21 '14 at 15:19

2 Answers2

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The period of $\cos\dfrac xk$ is $2\pi k$

So, the period of $\cos\dfrac x3$ is $2\pi\cdot3$ and that of $\cos\dfrac x4$ is $2\pi\cdot4$

As $\dfrac{2\pi\cdot4}{2\pi\cdot3}=\dfrac43$ is rational

So, the period of $\cos\dfrac x3+\cos\dfrac x4$ will be a divisor of lcm$(6\pi,8\pi)=24\pi$

Now try with the divisors of $24$

lab bhattacharjee
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Pick $x=0$. Then $f(x)=2$ is the maximum of $f(x)$. The next time this occurs is when $x/3$ and $x/4$ are both multiples of $2\pi$. This will happen next at $24\pi$ ($12\pi$ makes $f(x)=0$)

recmath
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