physics SHM

The difference between the two sets of equation is on the "initial condition" of the oscillating system.

"Initial condition", in this aspect, means the displacement of the oscillating object at time t = 0.

If the object starts oscillation at the equilibrium position (i.e. x= 0 m) , then we would expect to have an equation that gives x = 0 m when t=0 s, hence, the equation x = A.sin(wt) is appropriate.

On the other hand, if the object starts to oscillate at the +ve max displacement ( +ve amplitude A), then we would expect that the equation describing its motion to have x = +A at t = 0s. The equation x = A.cos(wt) is thus the right equation.

Along this line, if the object starts to oscillate at the -ve max displacement (-ve amplitude), the equation desrcribing the motion should give x = -A when t = 0 s, thus the equation x = (-A).cos(wt) should be used.