Sinusoidal Function
A sinusoidal function is any function that can be written in the form y = A*sin(Bx + C) + D or y = A*cos(Bx + C) + D, representing a smooth periodic wave.
Formula
y = A\sin(Bx + C) + D \text{ or } y = A\cos(Bx + C) + D
Definition
A sinusoidal function is a smooth, repeating wave described by sine or cosine, taking the general form $y = A\sin(Bx + C) + D$ (or the cosine equivalent), where $A$ scales vertically (amplitude), $B$ compresses horizontally (period), $C$ shifts horizontally (phase shift), and $D$ shifts vertically (midline); the sine and cosine versions are equivalent, differing only by a $\pi/2$ phase shift. Using the phasor representation, $A\,\text{Im}(e^{i(\omega x + \phi)}) + D$, a pure sinusoidal function has a Fourier spectrum consisting of exactly two Dirac deltas at frequencies $\pm\omega/(2\pi)$ with conjugate complex amplitudes.
Example
The height of a Ferris wheel seat over time is sinusoidal: it smoothly rises, reaches the top, smoothly falls, reaches the bottom, and repeats. Modeling lake water temperature with a high of $75^\circ$F in August and a low of $45^\circ$F in February gives amplitude $15$, midline $60$, and period $12$ months: $T(t) = 15\sin(2\pi t/12 - \pi/2) + 60$. Superposing two close sinusoids, $\sin(\omega t) + \sin((\omega + \Delta\omega)t) = 2\cos(\Delta\omega t/2)\sin((\omega + \Delta\omega/2)t)$, produces a beat phenomenon, the audible "wah-wah" heard when two instruments are slightly out of tune.
Key Insight
Sinusoidal functions model anything that oscillates smoothly, tides, breathing, AC electricity, pendulums, and music, the mathematical DNA of repeating motion; any sinusoidal function can be written as either a sine or cosine with an appropriate phase shift, a matter of convenience rather than mathematics. The fundamental theorem of Fourier analysis states that every "nice" function on an interval can be approximated by sums of sinusoidal functions, making the sinusoidal function the universal building block of periodic phenomena across electrical engineering, acoustics, climate modeling, and quantum field theory.