cos1

Cos1 commonly denotes the value of the cosine function at the input 1, with the argument measured in radians. The cosine function, written cos, is a fundamental trigonometric function that maps real numbers to the interval [−1, 1]. It can be defined by its power series, by the unit circle, or as the solution to the differential equation y'' + y = 0 with appropriate initial conditions.

Numerically, cos(1) ≈ 0.540302305868139717... This value is irrational and does not have a simple closed-form expression in terms of elementary constants. It arises from evaluating the cosine function at a real number that is not a standard special angle.

One common way to compute cos(1) is via its Maclaurin series: cos x = sum_{n=0}^∞ (−1)^n x^{2n}/(2n)!. For x = 1, cos(1) = 1 − 1/2! + 1/4! − 1/6! + 1/8! − ... The series converges rapidly, and truncated sums provide accurate approximations.

An alternative expression uses complex exponentials: cos(1) = (e^{i} + e^{−i})/2. This highlights its relation to the exponential function and the broader complex exponential framework used in Fourier analysis and signal processing.

Note that the value cos(1) assumes radians as the unit for the input. If the input is specified in degrees, cos(1°) ≈ 0.999847695, which is substantially different. See also the cosine function, radian measure, and numerical methods for evaluating trigonometric functions.