Sinusoidal

Sinusoidal refers to anything related to the sine function or the sine wave. In mathematics, a sinusoid is a function of the form f(x) = A sin(ωx + φ) or f(x) = A cos(ωx + φ), with amplitude A ≥ 0, angular frequency ω > 0, and phase shift φ. The graph is a smooth, periodic wave with period T = 2π/ω and range [-A, A]. Sine and cosine are phase-shifted versions of each other and are fundamental solutions to many differential equations; they are linked by Euler's formula e^{iθ} = cos θ + i sin θ.

In physics and engineering, sinusoidal waves model simple harmonic motion, where a quantity oscillates with constant amplitude and a restoring force proportional to displacement. The general motion is x(t) = A sin(ωt + φ), with angular frequency ω and ordinary frequency f = ω/(2π). In signal processing, periodic signals are often decomposed into sums of sinusoids (Fourier analysis); pure sine waves serve as idealized basis signals. In alternating current circuits, sinusoidal voltages and currents are described using phasors and impedance in the complex plane.

Sinusoidal waves also appear in acoustics and optics as idealized pure tones and light waves, respectively. The term can describe any waveform with the characteristic sine shape or properties resembling a sine wave, or any function that exhibits sinusoidal behavior in time or space.