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Derivation formula for Lissajous method

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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
 

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).
 

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