The wave-shaped oscillation that models sound, light, tides, and all periodic phenomena.
The sine function, written sin(x), is a fundamental trigonometric function defined using the unit circle: for any angle x, sin(x) is the y-coordinate of the point on the unit circle at that angle. Its graph is a smooth, continuously repeating wave called a sinusoid.
28), meaning the pattern repeats exactly every 2π units. It is one of the most important functions in mathematics, physics, and engineering due to its ability to model any periodic (repeating) behavior.
f(x) = A·sin(Bx + C) + D
Amplitude — the peak height above center (|A| = max deviation)
Angular frequency — controls the period: Period = 2π/|B|
Phase shift — horizontal offset: shift = -C/B
Vertical shift — moves the midline up or down
Domain
All real numbers: (-∞, +∞)
Range
[-A, A] (from -amplitude to +amplitude)
Period
2π / |B| (default 2π ≈ 6.28)
Amplitude
|A|
Zeros
x = nπ for any integer n (when A=1, B=1, C=D=0)
Odd Function
sin(-x) = -sin(x) — symmetric about the origin
f(x) = sin(x)Basic sine wave: amplitude 1, period 2π.
f(x) = 2·sin(x)Double amplitude — the wave reaches ±2.
f(x) = sin(2x)Double frequency — the wave completes in period π.
Adjust amplitude, period, and phase shift to explore wave transformations
Both are sinusoidal waves with the same shape, amplitude, and period. The only difference is a phase shift: cos(x) = sin(x + π/2). Cosine starts at its maximum value (1) when x=0, while sine starts at 0.