For a standard sinusoidal wave referenced at a zero-up crossing, what are the typical phase values for a crest, the next zero crossing, and the following trough along the wave?

Introduction / Context:Phase labeling of a sinusoid is fundamental in signal processing, optics, and communications. Establishing a consistent reference (such as the previous negative-to-positive zero crossing) allows us to assign standard phase angles to notable points on the waveform.

Given Data / Assumptions:

• A unit-amplitude sinusoid is considered.
• The phase reference is taken at the zero crossing where the wave passes upward through zero (zero-up crossing).
• Phase increases in the direction of propagation.

Concept / Approach:For a cosine-like reference shifted to start at zero-up crossing, the crest occurs a quarter cycle later (π/2), the next zero crossing (downward) occurs at half a cycle (π), and the trough appears at three-quarters of a cycle (3π/2). These canonical phase points are frequently used in phasor diagrams and time-series analysis.

Step-by-Step Solution:Set the phase origin at zero-up crossing (phase 0).Advance by a quarter cycle → crest at π/2.Advance by another quarter cycle → downward zero crossing at π.Advance by another quarter cycle → trough at 3π/2.Thus, all listed phase assignments are correct.

Verification / Alternative check:Plotting a sine wave with phase origin at zero-up crossing reproduces these angles exactly.

Why Other Options Are Wrong:

• Assigning crest at π or trough at 2π misplaces features by half a cycle or a full cycle.

Common Pitfalls:Mixing sine and cosine phase references; forgetting to specify the reference crossing (upward vs. downward).

Final Answer:All of these