# Time Period of a periodic function

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.

• Have you tried plotting these three functions, each on it's own? This would solve the problem. – Semoi Jan 21 at 18:23

Let's first fix to some number. I'll choose $$\omega=2\pi f$$ with $$f=1Hz$$, but you can choose any other number".
• 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.