sinfamily

The sinfamily is a set of sine-based functions used in mathematics and signal processing. It is commonly defined as all functions of the form f(x) = a sin(bx + c), where a, b, and c are real parameters with a ≠ 0 and b ≠ 0. In this form, a determines the amplitude, b the angular frequency, and c the phase shift. The sinfamily thus encompasses the class of sine waves that can be scaled, stretched, and shifted horizontally and, to a lesser extent, vertically.

The period of a member is T = 2π/|b|, and its horizontal shift is -c/b. The sine wave attains a maximum of a and a minimum of -a, giving an overall amplitude of |a|. The family is not closed under addition or multiplication: summing two sine functions generally yields a different waveform that may not be representable as a single a sin(bx + c).

In practice, the sinfamily underpins Fourier analysis and sinusoidal regression, where complex periodic signals are decomposed into sums of sine waves. It also models simple oscillators in physics and engineering, and is used in digital signal processing for modulation, filtering, and synthesis. Examples include f1(x) = 3 sin(2x + π/4) and f2(x) = sin(x). See also cosine family and Fourier series.