Sine & Cosine Waves

Explore amplitude, frequency, and phase shift of sinusoidal waves

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1.0
1.0
0°
sin y = 1.0 sin(1.0x + 0°) cos y = 1.0 cos(1.0x + 0°)

Values at Tracking Point

sin(x): 0.0000
cos(x): 1.0000
Wavelength: 6.283
Period: 6.283

Sine & Cosine Functions

The general form of a sinusoidal wave is:

y = A · sin(ωx + φ)
  • A = amplitude (peak height)
  • ω = angular frequency (cycles per unit)
  • φ = phase shift (horizontal offset)

Key Relationships

  • sin(x) starts at 0, rises to 1
  • cos(x) starts at 1, falls to 0
  • cos(x) = sin(x + π/2) — cosine leads sine by 90°
  • sin²(x) + cos²(x) = 1

Amplitude

The amplitude controls the height of the wave. It is the distance from the center line to the peak (or trough). A wave with amplitude A oscillates between ±A.

Frequency

Frequency controls how many cycles occur in a given interval. Higher frequency = more oscillations. The period T = 2π/ω is the length of one full cycle.

Phase Shift

Phase shift moves the wave horizontally. A positive phase shifts the wave to the left (advancing), while a negative phase shifts it to the right (delaying).

Did You Know?

Sine and cosine waves describe everything from sound and light to alternating current electricity. The Fourier Transform — one of the most important mathematical tools — decomposes any signal into a sum of sine and cosine waves!