1Hz = One Cycle and One Cycle = 360degrees
When you want to find the difference caused by a time separation of two waves, you can find how much of a wave cycle will pass during the time delay.
At 0.25Hz, it takes 4 seconds to make 1 Cycle. With some algebra you find that 1 cycle every 4 seconds = a quarter cycle every 1 second.
So now that you know a one second delay causes a quarter of a cycle to pass, you can convert that into degrees by multiplying 0.25 and 360. 0.25x360=90
Lo and behold, a time difference of 1 second, creates a phase seperation of 90 degrees.
x1 = A sin ω t = A sin (2 pi f t) equation for SHM with no phase
x1 =A sin (2 * pi * .25 t) = A sin (.5 * pi * t)
x2 = A sin (.5 pi * (t - 1)) = A sin (.5 pi t - pi/2) second oscillator lags by 1 sec
So x2 is pi / 2 behind x1 in phase
Or x2 = A cos .5 pi t since sin (Ф - pi/2) = cos Ф
