# Main period of $f(x)=\cos 5x+\cos 10x$

Here is what I have done so far: \begin{align*} f(x)&=\cos5x+\cos10x\\ f(x)&=\cos5x+2\cos^2(5x)-1\\ f(x)&=2\cos^25x+\cos5x-1\\ \end{align*} I have tried to further simplify the function to a complete square or a function like $$\cos^2(\text{something})$$ but I was not able to. Then I factorised: $$f(x)=2\left(\cos5x+\frac12\right)(\cos5x-1),$$ where each multiplier has a period of $$\dfrac{2π}{5}$$, which makes me think that the main period of the function is $$\dfrac{2π}{5}$$.

Please excuse me for the bad terminology. English is not my native language.

• Sep 25, 2018 at 6:34
• The problem is similar but I could not understand what divisor of icm(6π, 8π) means. Sep 25, 2018 at 6:43

You don't have to simply anything.

Hint:

Time period of $\displaystyle \cos (ax)=\frac{2\pi}{a}$

Also, the time period of sum of two functions $f_1(x)$ and $f_2(x)$ with period $T_1$ and $T_2$ is $\mathrm{LCM}(T_1$, $T_2)$

Also note that: $$\mathrm{LCM}\left(\frac{p_1}{q_1},\frac{p_2}{q_2}\right)=\frac{\mathrm{LCM}(p_1,p_2)}{\mathrm{HCF}(q_1,q_2)}$$

As a side note, before finding the time period of the sum always check if the sum is periodic or not. You can do that by checking if $\frac{T_1}{T_2}$ is rational (periodic) or not (aperiodic).

• Thank you. It's very helpful. Sep 25, 2018 at 6:50
• @DanielHalachev You are most welcome :) Sep 25, 2018 at 6:50

To calculate the period for any function of the form: $$f(x) = a(x) + b(x) + c(x)+ \cdots$$

At first, find the periods of individual functions $$a(x), b(x), c(x), \ldots$$

The period of $$f(x)$$ would then be the least common multiples of the periods of $$a(x), b(x), c(x), \ldots$$

The logic makes sense, since if $$a(x)$$ repeats in $$t_1$$ period, $$b(x)$$ repeats in t2 period, $$c(x)$$ repeats in $$t_3$$ period and so on, the whole function will repeat in multiples of $$t_1, t_2, t_3$$ and so on. Since period is the least interval where the function repeats, we calculate the least common multiple of all the multiples of $$t_1, t_2, t_3$$ and so on, to find the period of $$f(x)$$.

In this case, period of $$\sin(5x)$$ is $$\frac{2\pi}{5}$$, and for $$\sin(10x)$$ is $$\frac{2\pi}{10}$$. The lest common multiples of these two intervals is $$\frac{2\pi}{5}$$, which is the period of $$f(x)$$. Thanks.