We have this formula: $$ \displaystyle \arg (G(j \omega))=-\omega\left(t_{2}-t_{1}\right) $$
(We understand the amplitude of course, please help us to find a proof for the phase shift)
without a proof (control theory course). Can someone remind us how it is calculated?
It was not that hard in the end!
We consider the time points when the two curves passes 0:
$$ \begin{aligned} \sin \left(\omega t_{1}\right) &=0 \\ K \sin \left(\omega t_{2}+\phi\right) &=0 \end{aligned} $$
$$ \Longrightarrow \omega t_{1} = \omega t_{2}+\phi \\ \phi = \omega t_{1} - \omega t_{2} \\ \phi = -\omega(t_2-t_1) $$