Sine Wave
The graph y = A·sin(ωx) of pure oscillation — the atom every periodic signal is built from.
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Formulas and key values
- Cartesian equation
- PeriodIn the graph ≈ 6.283
- Zeros
A is the amplitude and ω the angular frequency.
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Start the video01 · Definition
What is the Sine Wave?
The sine wave y = A·sin(ωx) is the shadow of uniform circular motion: project a point moving around a circle onto a line, stretch time along an axis, and this is the graph you get. Amplitude A sets the height, angular frequency ω sets how many radians fit per unit — the period is 2π/ω.
Its special status comes from Fourier’s theorem: any reasonable periodic signal decomposes into a sum of sines and cosines. That single fact makes the sine wave the working currency of acoustics, AC power, radio, and signal processing — and the first curve every trigonometry student learns to transform.
02 · Properties
Key properties
- Period 2π/ω, amplitude A, zeros at x = kπ/ω.
- It is the projection of uniform circular motion onto a diameter.
- Derivative is a cosine — the same wave shifted left by a quarter period.
- Fourier’s theorem: periodic signals are sums of sines and cosines.
- Simple harmonic motion (mass on a spring, small pendulum) graphs as a sine wave in time.
03 · Questions
Sine Wave: common questions
- What is the equation of the Sine Wave?
- Cartesian equation: y = A sin(ω x). A is the amplitude and ω the angular frequency.
04 · Labs
Interactive labs
Sine and cosine on the unit circle
Why does a point going round a circle draw a sine wave?
Unit circle · Radian measure · Special angles
Open laby = A sin(ωx + φ): amplitude, period and phase shift
What do A, ω and φ each do to the graph of y = sin x?
Amplitude · Period T = 2π/ω · Phase shift
Open lab05 · Wonders
See it in a math wonder
06 · Keep exploring