Q.The displacement of a particle in S.H.M. is $x = 4\sin\!\left(5t + \dfrac{\pi}{6}\right)\ \text{cm}$, with $t$ in seconds. Find the amplitude, angular frequency, period, frequency, and phase constant of the motion.

The phase $(\omega t+\phi)$ of an S.H.M. at time $t$ is the single quantity that fixes both the particle's displacement and the direction it is moving at that instant. The phase constant $\phi$ is the value of the phase at $t=0$, set by the initial displacement and velocity. Two S.H.M.s of the same frequency with phase difference $0$ are in phase (move together); with phase difference $\pi$ they are exactly out of phase (always opposite); with phase difference $\pi/2$ they are in quadrature -- one is at an extreme exactly when the other passes through its mean position at maximum speed, which is precisely the relationship between an S.H.M. particle's own displacement and velocity.