cos

Cosine, denoted cos, is a trigonometric function that associates to each real number x a value cos(x). In trigonometry, it is commonly defined as the x-coordinate of the point on the unit circle corresponding to angle x measured in radians, and equivalently as the real part of e^{ix} by Euler's formula. It is an even function: cos(-x) = cos(x).

The function is defined for all real x, with range [-1, 1]. It has period 2π, and oscillates between -1 and 1. Its derivative is -sin x, and its second derivative is -cos x, reflecting its role as a harmonic oscillator. The fundamental Pythagorean identity with sine is cos^2 x + sin^2 x = 1. Addition and double-angle identities include cos(a+b) = cos a cos b − sin a sin b and cos 2x = 2 cos^2 x − 1 (also cos 2x = 1 − 2 sin^2 x). Zeros occur at x = π/2 + kπ for integers k.

Cosine can be expressed as a power series around 0: cos x = ∑_{n=0}^∞ (-1)^n x^{2n}/(2n)!. It also has a complex exponential form cos x = (e^{ix} + e^{-ix})/2. The graph is a smooth wave with amplitude 1 and period 2π.

Cosine plays a central role in trigonometry, Fourier analysis, signal processing, physics, engineering, and computer graphics. It is used to model periodic phenomena, to solve trigonometric equations, and in vector and complex number calculations.