Main period of \(\displaystyle{f{{\left({x}\right)}}}={\cos{{5}}}{x}+{\cos{{10}}}{x}\) \(\displaystyle{f{{\left({x}\right)}}}={\cos{{5}}}{x}+{\cos{{10}}}{x}\) \(\displaystyle{f{{\left({x}\right)}}}={\cos{{5}}}{x}+{2}{{\cos}^{{2}}{\left({5}{x}\right)}}-{1}\) \(\displaystyle{f{{\left({x}\right)}}}={2}{{\cos}^{{25}}{x}}+{\cos{{5}}}{x}-{1}\) have

Jaylin Clements

Jaylin Clements

Answered question

2022-03-21

Main period of f(x)=cos5x+cos10x
f(x)=cos5x+cos10x
f(x)=cos5x+2cos2(5x)1
f(x)=2cos25x+cos5x1
have tried to further simplify the function to a complete square or a function like cos2 (something) but I was not able to. Then I factorised:
f(x)=2(cos5x+12)(cos5x-1)
where each multiplier has a period of 2π5 which makes me think that the main period of the function is 2π5.

Answer & Explanation

exinnaemekswr1k

exinnaemekswr1k

Beginner2022-03-22Added 10 answers

You don't have to simply anything.
Hint:
Time period of cos(ax)=2πa
Also, the time period of sum of two functions f1(x) and f2(x) with period T1  and  T2 is LCM(T1,T2)
Also note that:
LCM(p1q1,p2q2)=LCM(p1,p2)HCF(q1,q2)
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 T1T2 is rational (periodic) or not (aperiodic).

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