Enter function parameters to compute horizontal shift and graph starting point
Function
Sign Convention
a (amplitude)
b (frequency)
c (phase)
y = 1 sin(1x - 1.57)
Result
Phase Shift
-
Direction
-
Detailed Derivation
Phase Shift Formula
y = a sin(bx - c) + d: phase shift = c/b (right)
y = a sin(bx + c) + d: phase shift = -c/b (left)
Key point: (c/b, d) is where one cycle starts
Identity: sin(x) = cos(x - pi/2)
Phase shift describes the horizontal displacement of a trigonometric graph from its standard position. It is essential for understanding wave interference, signal timing, and graph transformations.
⚠Distinguish the convention: a sin(bx-c) has shift c/b to the right. a sin(bx+c) has shift -c/b to the left.
What Is Phase Shift?
Phase shift is the horizontal translation of a trigonometric graph. It determines where the wave pattern starts relative to the vertical axis. Understanding phase shift is crucial for graphing, wave analysis, and solving equations.
Phase Shift Formula
For a sin(bx-c): shift = c/b. A cycle starts at x = c/b. The graph moves right by this amount.
Sine to Cosine
sin(x) = cos(x - pi/2). A sine wave is a cosine wave shifted right by pi/2. They differ only in phase.
Wave Interference
In physics, waves with different phases interfere. In-phase waves amplify, out-of-phase cancel. Phase shift determines interference patterns.
Graphing Steps
1) Find period 2pi/b. 2) Find phase shift c/b. 3) Divide period by 4 for key points. 4) Plot starting from shift.
Teaching Example: y = sin(2x - pi/2). Convention: a sin(bx-c). a=1, b=2, c=pi/2=1.57. Phase shift = c/b = 1.57/2 = 0.785 = pi/4 right. Period = 2pi/2 = pi. One cycle starts at x = 0.785.
Applications
Wave PhysicsAC ElectricitySignal ProcessingSound EngineeringGraphingControl Systems
FAQs about Phase Shift
What is phase shift?▼
The horizontal shift of a trig graph. For y = a sin(bx-c): shift = c/b right. For y = a sin(bx+c): shift = -c/b left.
How to find phase shift of sin(x)?▼
sin(x) = 1 sin(1x-0) has phase shift = 0/1 = 0. Starts at origin. sin(x - pi/2) = cos(x) has shift = pi/2 right.
What is the phase shift of cos(x)?▼
cos(x) = sin(x + pi/2), so its phase shift is -pi/2 from sine. Cos starts at its maximum at x=0, while sine starts at 0.
How does phase shift affect wave interference?▼
Waves with same phase (shift 0) add constructively. Waves with opposite phase (shift pi) cancel destructively. This is the basis of noise cancellation.
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