Effective (rms) value of a sinusoid: The effective value of a sine wave, expressed relative to its peak value, is equal to which of the following?

Introduction:
Root-mean-square (rms) value is the DC-equivalent heating value of an AC waveform. For a pure sine wave, rms has a fixed ratio to peak (maximum) and to peak-to-peak, which is essential when converting between datasheet ratings, scope measurements, and power calculations.

Given Data / Assumptions:

• Ideal sinusoidal waveform.
• We want rms in terms of peak.

Concept / Approach:

For a pure sine: Vrms = Vp / √2. Numerically, 1 / √2 ≈ 0.707106. Also, sin 45° = √2 / 2 ≈ 0.707106. Thus 0.707 of peak and sin 45° of peak are equal statements. Note that 0.636 of peak is the average of a full-wave rectified sine, not the rms of a pure sine.

Step-by-Step Solution:

Verification / Alternative check:

Check with Vp = 10 V: Vrms = 10 / √2 ≈ 7.071 V; sin 45° * 10 V gives the same, confirming equivalence.

Why Other Options Are Wrong:

• 0.636 of peak voltage: This is the average of a full-wave rectified sine (2/π), not rms.
• 0.5 of peak voltage: No standard sinusoidal relation yields this as rms.

Common Pitfalls:

• Mixing up average and rms for rectified signals.
• Confusing peak with peak-to-peak (Vp-p = 2 * Vp).

Final Answer:

both 0.707 of peak voltage and sin 45° of peak voltage