Cosines

Cosine, usually denoted cos, is a trigonometric function that describes the ratio of the length of the adjacent side to the hypotenuse in a right triangle with a given angle. On the unit circle, cos θ equals the x-coordinate of the point corresponding to the angle θ, so it determines the horizontal projection of a unit vector. The function is defined for all real θ, with range between −1 and 1, is even (cos(−θ) = cos θ), and has period 2π.

Key identities and properties include the Pythagorean identity sin^2 θ + cos^2 θ = 1; the addition formulas cos(α ± β) = cos α cos β ∓ sin α sin β; and the double-angle identity cos 2θ = 2 cos^2 θ − 1, which is equivalent to cos 2θ = 1 − 2 sin^2 θ. The cosine function is continuous, differentiable, and its graph forms a smooth wave with maximum 1 and minimum −1.

Applications of cosines span geometry, trigonometry, and analysis. In triangle geometry, the Law of Cosines states c^2 = a^2 + b^2 − 2ab cos C, linking side lengths to an included angle. In vector mathematics, cos θ = (u · v)/(|u||v|) gives the angle between vectors. In Fourier analysis and signal processing, cosine functions serve as fundamental waveforms and basis functions. The inverse function arccos provides principal values in the interval [0, π].

Cosine can be extended beyond real numbers to complex arguments, with cos z = (e^{iz} + e^{−iz})/2, making it an entire, even function.