# Cosine Function
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/cosine-function/
Fetched from algebrica.org post 6430; source modified 2026-03-06T22:27:24.
## Cosine function
The cosine [function](../functions.md) \\( f(x) = \\cos(x) \\) assigns to each angle \\( x \\), expressed in radians, its corresponding [cosine](../sine-and-cosine.md) value. Its graph is a periodic wave with a period of \\( 2 \\pi \\) and an amplitude of 1, oscillating between \\(-1\\) and \\(1\\). The function \\( f(x) = \\cos x \\) has all real numbers in its [domain](../determining-the-domain-of-a-function.md), but its range is \\( -1 \\leq \\cos(x) \\leq 1 \\).
##### Together with the [sine function](../sine-function.md), it represents one of the fundamental models of periodic waves, and is widely used to describe cyclic phenomena in physics, engineering, and mathematics. For example, in simple [harmonic motion](../simple-harmonic-motion.md) in physics, the cosine function often appears in the equations for displacement and [acceleration](../acceleration.md), describing the oscillatory behavior of systems like springs and pendulums.
## Properties
- [Domain](../determining-the-domain-of-a-function.md): \\( x \\in \\mathbb{R} \\)
- Range: \\( y \\in \\mathbb{R} : -1 \\leq y \\leq 1 \\)
- Periodicity: periodic in \\( x \\) with period \\( 2\\pi \\)
- Parity: [even](../even-and-odd-functions.md), \\( \\cos(-x) = \\cos(x) \\)
- Roots: \\( x = \\dfrac{\\pi}{2} + n \\pi, \\quad n \\in \\mathbb{Z} \\)
- [Integer](../integers.md) root: \\( x = \\dfrac{\\pi}{2} \\)
- [Maximum and minimum points](../maximum-minimum-and-inflection-points.md): \\( \\cos(x) \\) reaches its \\(1\\) at \\( x = 2k \\pi \\) with \\( k \\in \\mathbb{Z} \\) and its minimum \\(-1\\) at \\( x = \\pi + 2k \\pi \\) with \\( k \\in \\mathbb{Z} \\).
## Limits, derivatives, and integrals of the cosine function
A fundamental [limit](../limits.md) involving the cosine function is:
\\[
\\lim\\limits\_{x \\to 0} \\frac{1 - \\cos(x)}{x} = 0
\\]
* * *
The function is [continuous](../continuous-functions.md) and differentiable at all real numbers. The [derivative](../derivatives.md) is:
\\[
\\frac{d}{dx} \\cos(x) = -\\sin(x)
\\]
* * *
[Indefinite integral](../indefinite-integrals.md):
\\[
\\int \\cos(x) dx = \\sin(x) + c
\\]
##### A comprehensive overview of trigonometric integrals, together with the most useful transformation and substitution techniques for handling more complex cases, is available in the page on [trigonometric function integrals](../integral-of-trigonometric-functions.md).
* * *
An alternative form of the function \\( \\cos(x) \\) using imaginary numbers is given by Euler’s formula, where \\( e^{ix} \\) is the [exponential function](../exponential-function.md) with base \\( e \\) and \\( i \\) is the [imaginary](../complex-numbers.md) unit:
\\[
\\cos(x) = \\frac{e^{ix} + e^{-ix}}{2}
\\]
