Sine-wave phase reference: At which phase angle does a standard sine wave reach its maximum positive value relative to its zero-phase reference?

Introduction:Phase angle descriptions help locate key points on a sinusoid. Knowing where maxima and minima occur is essential for phasor analysis, AC power, and signal timing in communication systems.

Given Data / Assumptions:

• Reference sine wave: v(θ) = Vm * sin(θ).
• Phase angle θ measured in degrees from 0° to 360°.
• Conventional positive maximum sought.

Concept / Approach:

The sine function attains its maximum value +1 at θ = 90°. Therefore the waveform’s maximum positive amplitude occurs at 90°. At 270° (or −90°), the sine reaches −1 (negative peak). At 0° and 180°, the waveform crosses zero and reaches zero, respectively, but not the maximum.

Step-by-Step Solution:

Verification / Alternative check:

Graph of a sine wave or unit-circle evaluation confirms that the peak positive amplitude is at 90°. A derivative check shows dv/dθ = Vm * cos(θ) = 0 at 90°, indicating an extremum; second derivative test confirms it is a maximum.

Why Other Options Are Wrong:

• 0° and 360°: Zero value (start/end of cycle), not maximum.
• −90°: Negative peak, not positive.
• 180°: Another zero crossing with negative slope.

Common Pitfalls:

• Confusing sine with cosine; cosine has its maximum at 0°.
• Mixing degree and radian measures; 90° corresponds to π/2 radians.

Final Answer:

90°