Sine wave fundamentals: For a pure sine wave with a peak value of 12 V, what is the average (mean) value over one complete cycle?

Introduction / Context:
The average (mean) value of an alternating sinusoidal voltage over a full cycle is a frequent point of confusion. While RMS and average-rectified values are nonzero, the algebraic average across one complete cycle of a symmetric sine is zero. This question checks recognition of that property.

Given Data / Assumptions:

• Sine wave peak Vp = 12 V.
• Waveform is symmetrical about zero (no DC offset).
• Average is taken over one full period (0 to 2π in angle).

Concept / Approach:

For v(t) = Vp * sin(ωt), the positive half-cycle area equals the negative half-cycle area. Over 0 to 2π, the integral of sin is zero, hence the average value is zero volts.

Step-by-Step Solution:

Verification / Alternative check:

The RMS of this wave is Vp/√2 (≈ 8.49 V here) and the average of the absolute (half-wave average doubled) is 2Vp/π (≈ 7.64 V), but the algebraic average over a full cycle remains 0 V.

Why Other Options Are Wrong:

6.37 V is the RMS of a 9 V peak, not relevant. 7.64 V is the average-rectified value for 12 V peak, not the algebraic average. 1.27 V is unrelated.

Common Pitfalls:

Confusing average-rectified or RMS with algebraic average; overlooking DC offsets.

Final Answer:

0 V