How to find the period of the sum of two trigonometric functions

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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:

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

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2 Answers

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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 $24$

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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 $f(x)=0$)

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