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

Question

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.

See https://math.stackexchange.com/questions/873723/how-to-find-the-period-of-the-sum-of-two-trigonometric-functions — lab bhattacharjee, Sep 25 '18 at 06:34
The problem is similar but I could not understand what divisor of icm(6π, 8π) means. — Kotaka Danski, Sep 25 '18 at 06:43

paulplusx · Accepted Answer · 2018-09-25T06:50:15.667

1

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).

edited Sep 25 '18 at 06:50

answered Sep 25 '18 at 06:43

paulplusx

1,636

Thank you. It's very helpful. – Kotaka Danski Sep 25 '18 at 06:50
@DanielHalachev You are most welcome :) – paulplusx Sep 25 '18 at 06:50

score 1 · Answer 2 · edited Sep 25 '18 at 08:39

1

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.

edited Sep 25 '18 at 08:39

José Carlos Santos

427,504

answered Sep 25 '18 at 06:53

Kong

41
4

Please format your answer with MathJax – paulplusx Sep 25 '18 at 06:55
Another helpful answer! Thank you. If you have the knowledge and time, could you tell me how to proceed in the case, when we have f(x)*g(x)? I am asking out of curiousity. – Kotaka Danski Sep 25 '18 at 08:53
@DanielHalachev I suppose you can try simplifying it by reducing it to a sum. – paulplusx Sep 25 '18 at 10:07

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

2 Answers2