The wave that starts at its peak — the twin of sine used in circular motion and signal analysis.
The cosine function, written cos(x), is a fundamental trigonometric function defined using the unit circle: for any angle x, cos(x) is the x-coordinate of the point on the unit circle at that angle. Its graph is a smooth, continuously repeating wave identical in shape to the sine wave, but shifted π/2 units to the left — cosine starts at its maximum value of 1 when x = 0, while sine starts at 0.
The cosine function is periodic with a period of 2π and is one of the most widely used functions in physics, engineering, and signal processing.
f(x) = A·cos(Bx + C) + D
Amplitude — peak height from 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|]
Period
2π / |B| (default 2π ≈ 6.28)
Amplitude
|A|
Zeros
x = π/2 + nπ for any integer n (basic form)
Even Function
cos(-x) = cos(x) — symmetric about the y-axis
f(x) = cos(x)Basic cosine: amplitude 1, period 2π, starts at maximum.
f(x) = 3·cos(x)Triple amplitude — the wave reaches ±3.
f(x) = cos(x) - 1Shifted down by 1, midline moves to y = -1.
Adjust amplitude, period, and phase shift to explore wave transformations
Both are periodic waves with the same shape, amplitude, and period. The key difference is their starting point: cos(x) equals 1 at x = 0 (starts at its peak), while sin(x) equals 0 at x = 0 (starts at zero). Mathematically, cos(x) = sin(x + π/2).