IMO Method A is total nonsense. It gets the correct answer
primarily because, on dimensional grounds, the correct answer is
essentially the only answer. Method B is fine.
With regard to your concern, I think you are simply mistaking the
mathematical representation of a sine function with the sinusoidal
shape of the physical waveform. Often, (usually?) the small angle
approximation is completely reasonable. Consider waves on a
guitar string. Long before you would ever see a 45 degree angle,
you'd see a very broken string (or guitar!)