Time period of a periodic function

[Kaushik]

Homework Statement

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Relevant Equations

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Consider the following periodic function:
$f(t) = \sin(ωt) + \cos(2ωt) + \sin(4ωt)$
What is the time period of the above periodic function?
The following is given in my book:

Period is the least interval of time after which the function repeats. Here, $\sin(ωt)$ has a period $T_o = \frac{2π}{ω}$, $\cos(2ωt)$ has a period $\frac{T_o}{2}$ and $\sin(4ωt)$ has a period of $\frac{T_o}{4}$. The period of the first term is a multiple of the periods of the last two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is $T_o$, and thus, Time period is $T_o$.

I don't understand what the above lines mean.

 

[PeroK]

What don't you understand?

 

[Kaushik]

What don't you understand?

Oh wait! So by the time the $\sin(ωt)$ function finishes one oscillation the other two finishes more than one i.e, 2 and 4 respectively. Is that what they mean when they say multiple? So we need the shortest time at which all three terms of the function $f(t)$ are in the initial position and that time is $T_o$. Is it?

 

[haruspex]

Oh wait! So by the time the $\sin(ωt)$ function finishes one oscillation the other two finishes more than one i.e, 2 and 4 respectively. Is that what they mean when they say multiple? So we need the shortest time at which all three terms of the function $f(t)$ are in the initial position and that time is $T_o$. Is it?

Yes.