Simple Harmonic Motion and Phase

In the description of harmonic motion, the choice between sine and cosine functions depends on how the oscillation was launched at the initial time. Since a harmonic oscillation repeats exactly after one period, different launch conditions only have two possible effects: they determine the amplitude (the maximal deformation during the oscillation) and also the times at which that maximum is reached. In a graph, changing the time of maximum deformation corresponds to shifting the plot left or right along the time axis without changing its overall shape.

A sine function can be mathematically thought of as a cosine function that is shifted by a specific amount.

This video discusses this idea mathematically, but allows the shift along the time axis to have an arbitrary value to account for arbitrary launch conditions at a chosen time (usually called t = 0). This introduces the concept of the phase of oscillation.