Derivation formula for Lissajous method

[u04f061]

in cro to measure the phase difference b/w two waveforms we take inverse sine of the ratio of two vertical displacements.can any one give me the derivation of this formula

 

[icdesignusa]

lissajous method formula

It is actually the arcsin of the ratio of the vertical and horizontal displacements. The derivation is straightforward. The Lissajous curve is ploting the first waveform (x=V1sin wt) versus the second waveform (y=V2sin wt + phi). So the steps would be to get an expression for the curve, and then the two zero crossings would be the displacement. From that expression, you can isolate phi = arcsin (d1/d2).