RMS comparison of sin and cos waves: Given two sinusoidal waveforms A = 100 sin(ωt) and B = 100 cos(ωt), compare their root-mean-square (RMS) values.

Introduction / Context:RMS (root-mean-square) value quantifies the effective magnitude of a time-varying signal in terms of its ability to deliver power to a resistive load. For sinusoids, the RMS depends on amplitude, not on phase shift. This question checks whether phase-shifted sinusoids of the same amplitude share the same RMS value.

Given Data / Assumptions:

• Two signals: A(t) = 100 sin(ωt) and B(t) = 100 cos(ωt).
• Both have the same amplitude 100 V (or A in general units).
• Steady-state sinusoidal waveforms, same frequency ω.

Concept / Approach:

For a sinusoid x(t) = X_m sin(ωt + φ), the RMS value is X_rms = X_m / √2. The phase φ does not affect the RMS because the squaring operation removes the phase dependence when averaging over a full period. Therefore, any two sinusoids at the same amplitude have identical RMS irrespective of being sin or cos or any phase shift between them.

Step-by-Step Solution:

Verification / Alternative check:

Time-average over one period T: (1/T) ∫_0^T 100^2 sin^2(ωt) dt = 100^2/2, same as for cos^2(ωt). Square root yields 100/√2 for both. Simulation or measurement with a true-RMS meter yields identical readings for equal-amplitude sin and cos waveforms at the same frequency.

Why Other Options Are Wrong:

• Claims that one RMS exceeds the other rely on phase, which cancels in RMS computation.
• “May or may not be equal” is incorrect for sinusoids with the same amplitude; they are always equal.
• Equality does not require identical phase—only identical amplitude and waveform type.

Common Pitfalls:

• Confusing instantaneous values (which depend on phase) with RMS, a time-averaged quantity.
• Assuming different starting phases change energy content over a full cycle; they do not.

Final Answer:

rms values of the two waves are equal