Cosineshaped

Cosineshaped is an adjective used to describe something that follows or resembles the profile of a cosine wave. In mathematics and related fields, a cosineshaped curve is smooth, periodic, and symmetric, characterized by a single dominant curvature oscillating between a maximum and a minimum in a cosine pattern.

A typical cosineshaped curve is modeled by y = A cos(Bx + φ) + D, where A is the amplitude, B controls frequency (the period is 2π/B), φ is the phase shift, and D is a vertical offset. The curve completes a full cycle over an interval of length 2π/B and, when φ = 0, shows even symmetry about the y-axis. The parameterization allows the shape to be shifted horizontally, vertically, or scaled in height and period.

Properties of cosineshaped curves include smoothness and infinite differentiability. The first derivative is -AB sin(Bx + φ) and the second derivative is -AB^2 cos(Bx + φ). These features reflect the standard behavior of cosine functions, including periodic repetition and a single extremum per half-period.

Applications of cosineshaped curves appear across science and engineering. They are used to model smooth periodic phenomena such as optical intensity variations, acoustic or electrical signals, and envelopes in signal processing. They also underpin Fourier analysis as the cosine component in decomposing complex periodic signals, and they inform the design of easing and transition profiles in computer graphics and animation. See also cosine, sine wave, Fourier series, and harmonic analysis.