cosx

Cos x, written cos x, is the cosine function of a real variable x. In a right triangle it equals the ratio of the length of the adjacent side to the hypotenuse for an angle x; more generally, it is the x-coordinate of the point on the unit circle corresponding to the angle x.

It is an even function with period 2π and range [-1, 1]. It remains unchanged under x → -x, and it repeats every 2π units.

Its derivative is -sin x, and its second derivative is -cos x. The indefinite integral is ∫ cos x dx = sin x + C.

The Maclaurin series for cos x is cos x = 1 - x^2/2! + x^4/4! - x^6/6! + …, valid for all real x. Euler's formula e^{ix} = cos x + i sin x implies cos x = (e^{ix} + e^{-ix})/2.

Cos x = 0 at x = π/2 + kπ for integers k. The inverse function arccos is defined on [-1, 1], giving values in the interval [0, π].

Cosine appears in many contexts, including Fourier analysis and wave phenomena, where it models periodic signals and rotations. It underlies numerous trigonometric identities and is related to sine via cos x = sin(π/2 − x) and as the real part of e^{ix}.

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