On a CRO (oscilloscope), a Lissajous figure appears as a straight line at 45° to the x-axis. If the X-plate is driven by x(t) = 2 sin(ωt), what Y-plate input y(t) produces this pattern?

Introduction / Context:
Lissajous figures on a CRO result from applying sinusoidal voltages to the horizontal (X) and vertical (Y) plates. The observed shape encodes the relative amplitude, frequency, and phase between the two signals, which is a classic laboratory method to compare frequencies and phases.

Given Data / Assumptions:

• X input: x(t) = 2 sin(ωt).
• Observed figure: straight line at 45° to the x-axis.
• Same frequency applied to both axes (standard Lissajous measurement for phase).

Concept / Approach:
A straight line indicates zero phase difference or 180° phase difference with amplitude ratios defining slope. A 45° line with positive slope requires equal amplitudes and zero relative phase (or both inverted together). Therefore, the Y-input must match the X-input in both amplitude and phase: y(t) = 2 sin(ωt). Any phase shift would rotate or thicken the line into an ellipse, and an amplitude difference would change the slope away from 45°.

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