Kosinus

Kosinus, commonly known as the cosine function, is a fundamental trigonometric function. In a right triangle, kosinus of an angle is the ratio of the length of the adjacent side to the hypotenuse. On the unit circle, kosinus θ is the x-coordinate of the point corresponding to angle θ, where the point on the circle has coordinates (cos θ, sin θ).

The kosinus function is even, meaning kosinus(−θ) = kosinus(θ). It is periodic with period 2π and takes values in the interval [−1, 1]. It reaches 1 at θ = 2kπ and −1 at θ = (2k+1)π for any integer k. The complementary sine function satisfies kosinus^2 θ + sine^2 θ = 1, linking the two through the Pythagorean identity.

In calculus and analysis, the derivative of kosinus is −sine, and the integral of kosinus is sine plus a constant. Kosinus can be expressed via Euler’s formula e^{iθ} = kosinus θ + i sine θ, yielding kosinus θ = (e^{iθ} + e^{−iθ})/2. Its Maclaurin series is kosinus θ = 1 − θ^2/2! + θ^4/4! − θ^6/6! + …

Applications of kosinus are widespread, including physics, engineering, signal processing, and computer graphics, where it models periodic phenomena and waveforms. The inverse function, arccos, is defined on [−1, 1] with range [0, π], allowing the recovery of angles from cosine values.