# Time period of a periodic function

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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.

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PeroK
Homework Helper
Gold Member
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
Homework Helper
Gold Member
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.

Kaushik