Hello,

We know that sinus is 2pi-periodic.

sin(X+2pi)=sin(X)

sin(2x) -> find t such as sin(2(x+t))=sin(2x)

sin(2(x+t))=sin(2x+2t)=sin(2x)

This means that 2t=2pi -> t=pi

Do the same for sin(x/2) and finding t' such as sin((x+t')/2)=sin(x/2)

The period of the function will be gcd(t,t')