$x=A \sin \omega t$

When displacement

$x=\frac{A}{2}$

$\frac{A}{2}=A \sin (\omega t+\phi)$

$\sin ^{-1} \frac{1}{2}=\omega t+\phi$

$\omega t+\phi=30^{\circ} \text { or } 150^{\circ}$

When particles are in opposite direction at one lime phase is $30^{\circ}$ and at the other $150^{\circ}$. So phase difference is $120^{\circ}$.