How can i find the period T of a complex continuous signal

Question

I'm new to this kind of mathematics and i came across a really complex signal in a course of my university

I know that this signal x(t) = sinωt has a period of T = 2π because of ω = 2π/T but how can i find the period of this one?

Answer 1

$T$ is said to be period of $x(t)$ if it is the minimum value satisfying $x(t) = x(t+T)$. In your case, this becomes $$3\cos(2\pi t) + 2\cos(3\pi t + \frac{\pi}{6}) + 4\sin(4\pi t ) = 3\cos(2\pi t + 2\pi T) + 2\cos(3\pi t +3\pi T + \frac{\pi}{6}) + 4\sin(4\pi t + 4\pi T ) = $$ for all $t$. So $T$ should be chosen as the minimum number such that $2\pi T$,$3\pi T$ and $4\pi T$ are multiples of $2\pi$, which in this case would be $T=2$.