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

Difficulty: Easy

Correct Answer: 90°

Explanation:


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:

Consider v(θ) = Vm * sin(θ).sin(0°) = 0 → zero crossing.sin(90°) = +1 → maximum positive value Vm.sin(270°) = −1 → maximum negative value −Vm.


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°

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