Cosine Function
Cosine function
The cosine function $f ( x ) = cos ( x )$ assigns to each angle $x$, expressed in radians, its corresponding cosine 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, but its range is $- 1 \leq cos ( x ) \leq 1$.
Together with the sine function, 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 in physics, the cosine function often appears in the equations for displacement and acceleration, describing the oscillatory behavior of systems like springs and pendulums.
Properties
- Domain: $x \in \mathbb{R}$
- Range: $y \in \mathbb{R} : - 1 \leq y \leq 1$
- Periodicity: periodic in $x$ with period $2 \pi$
- Parity: even, $cos ( - x ) = cos ( x )$
- Roots: $x = \frac{\pi}{2} + n \pi , n \in \mathbb{Z}$
- Integer root: $x = \frac{\pi}{2}$
- Maximum and minimum points: $cos ( x )$ reaches its $1$ at $x = 2 k \pi$ with $k \in \mathbb{Z}$ and its minimum $- 1$ at $x = \pi + 2 k \pi$ with $k \in \mathbb{Z}$.
Limits, derivatives, and integrals of the cosine function
A fundamental limit involving the cosine function is: \(\underset{x \rightarrow 0}{lim} \frac{1 - cos ( x )}{x} = 0\)
The function is continuous and differentiable at all real numbers. The derivative is: \(\frac{d}{d x} cos ( x ) = - sin ( x )\)
Indefinite integral: \(\int cos ( x ) d x = sin ( x ) + c\)
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.
An alternative form of the function $cos ( x )$ using imaginary numbers is given by Euler’s formula, where $e^{i x}$ is the exponential function with base $e$ and $i$ is the imaginary unit: \(cos ( x ) = \frac{e^{i x} + e^{- i x}}{2}\)