Given the equation
$ A_1\sin(\omega t + \theta_1) + A_2\sin(\omega t + \theta_2) + A_3\sin(\omega t + \theta_3) = 0 $
which holds for all $t$.
It seem apparent that this implies that the phase shifts should be equal $\theta_1 = \theta_2 = \theta_3$ (possibly $\pi$ out of phase which could be represented by a negative amplitude).
However I can't seem to formulate a solid mathematical proof for this. Any suggestions?