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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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    $\begingroup$ Have you tried plotting these three functions, each on it's own? This would solve the problem. $\endgroup$ – Semoi Jan 21 at 18:23
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Let's first fix to some number. I'll choose $\omega=2\pi f$ with $f=1Hz$, but you can choose any other number".

Now let's plot the three functions waves

  • After the time $t = T_0 = \frac{2\pi}{\omega} = \frac{1}{f} = 1s$ the blue function $\sin{(\omega t)}$ has done a full oscillation. Thus, it has also done full oscillations after $t=\{2, 3, 4, \ldots\}$.
  • After the time $t = T_0/2 = \frac{1}{2}s$ the red function $\cos{(2\omega t)}$ has done a full oscillation. Thus, it has also done full oscillations after $t=\{1, 1.5, 2, 2.5, \ldots\}$.
  • After the time $t = T_0/4 = \frac{1}{4}s$ the green function $\cos{(2\omega t)}$ has done a full oscillation. Thus, it has also done full oscillations after $t=\{0.5, 0.75, 1, 1.25, \ldots\}$.

Thus, after $1s$ the blue curve has done one full oscillation, the red wave has done two and the green curve has done four full oscillations.

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