Sinusoidals

Sinusoidals, in mathematics and engineering, refer to signals or functions that have the form A sin(ωt + φ) or A cos(ωt + φ). They are the basic building blocks of periodic signals. The parameters are amplitude A, angular frequency ω, and phase φ. A sine and a cosine of the same frequency differ only by a phase shift: cos(ωt) = sin(ωt + π/2).

As solutions to the simple harmonic oscillator, sinusoids satisfy x'' + ω^2 x = 0. They are orthogonal over a period, enabling Fourier series representations. They can be written as the real part of complex exponentials: A e^{i(ωt+φ)}; the complex form clarifies linear superposition.

In signal processing, sinusoids are eigenfunctions of linear time-invariant systems: a sinusoidal input at frequency ω yields a sinusoidal output at the same frequency, scaled by the system's gain and shifted in phase. Any periodic signal can be decomposed into a sum of sinusoids; non-periodic signals can be treated via Fourier transforms, representing them as weighted sums (or integrals) of sinusoids across frequencies.

Applications include communications, acoustics, vibration analysis, and electronic signal analysis. In practice, sinusoids provide a convenient basis for spectral analysis, filtering, and modulation techniques. The terms sine and cosine are interchangeable for practical purposes; the choice of reference phase is a matter of convention.

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