Checking if the function is periodic or not

Question

The questions asks you to show if $\sin(\mid t \mid)$ is periodic.

I know that a sine wave is periodic if $\sin (x+T)=\sin(x)$ where $T$ is the period.

But how do I find the period of $\sin(\mid t \mid)$ in order to check whether it is periodic?

Answer 1

Assume $f(t)=\sin|t|$ has some periodicity $p$. Since $f(t)=0$ exactly when $t\in\pi\mathbb Z$, $p$ must be a multiple of $\pi$, that is, $p=k\pi$ for some $k\in\mathbb N_{>0}$.

If $k$ was odd, we'd have $f(-\pi/2)=f(-k\pi-\pi/2)$, which gives $1=-1$, a contradiction.

If $k$ was even, we'd have $f(-\pi/2)=f(+k\pi-\pi/2)$, which again gives $1=-1$,which is absurd.

Thus $f$ cannot be periodic.