Introduction

The geometric construction of the cosine from the unit circle is developed in sine and cosine. Here the cosine is treated as a real function of a real variable.

The cosine function $f(x) = \cos(x)$ assigns to each angle $x,$ measured in radians, its corresponding cosine value. Its graph is a periodic wave with period $2\pi$ and amplitude $1,$ oscillating between $-1$ and $1.$ The function has all real numbers in its domain, and its range is the interval $[-1, 1].$ On $[0, \pi],$ the cosine is strictly decreasing from $1$ to $-1,$ and the inverse of this restriction is the arccosine function.

Cosine is a periodic function with period 2π. It completes one full cycle for every complete revolution on the unit circle. 1 -1 π 2π f(x) = cos(x)

Together with the sine function, the cosine models periodic phenomena. In simple harmonic motion the displacement of a mass on a spring or of a pendulum is a cosine of time, and the acceleration, as its second derivative, is again a cosine with opposite sign.

Properties

The following properties of the cosine function follow from its definition on the unit circle.

  • Domain: $x \in \mathbb{R}$
  • Range: $-1 \leq y \leq 1$
  • Periodicity: periodic in $x$ with period $2\pi$
  • Parity: even, with $\cos(-x) = \cos(x)$
  • Monotonicity: decreasing where $\sin(x) > 0$ and increasing where $\sin(x) < 0,$ on alternating intervals of length $\pi.$
  • Roots: $x = \dfrac{\pi}{2} + n\pi$ with $n \in \mathbb{Z}$
  • None of the roots is an integer, since $\dfrac{\pi}{2} + n\pi$ is irrational for every $n \in \mathbb{Z}.$
  • Maximum and minimum points: the maximum value $1$ is reached at $x = 2k\pi$ and the minimum value $-1$ at $x = \pi + 2k\pi,$ with $k \in \mathbb{Z}.$

Limits, derivatives, and integrals of the cosine function

A remarkable limit of the cosine function is:

$$\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$$

Near the origin the difference $1 - \cos(x)$ vanishes faster than $x,$ so the ratio tends to zero.

The function $\cos(x)$ is continuous and differentiable for every real value of $x.$ Its derivative is:

$$\frac{d}{dx}\cos(x) = -\sin(x)$$

Differentiating repeatedly, the function returns to itself after four steps:

$$ \begin{align} \frac{d^2}{dx^2}\cos(x) &= -\cos(x) \\[6pt] \frac{d^3}{dx^3}\cos(x) &= \sin(x) \\[6pt] \frac{d^4}{dx^4}\cos(x) &= \cos(x) \end{align} $$

The derivatives repeat with period four, so the $n$-th derivative has the closed form:

$$\frac{d^n}{dx^n}\cos(x) = \cos\left(x + \frac{n\pi}{2}\right)$$

Since the derivative of $\sin(x)$ is $\cos(x),$ the indefinite integral of the cosine function is:

$$\int \cos(x) \ dx = \sin(x) + c$$

A broader treatment of trigonometric integrals, with the transformation and substitution techniques for the more complex cases, is given in trigonometric function integrals.

The cosine function can also be written using imaginary numbers. With $e^{ix}$ the exponential function of base $e$ and $i$ the imaginary unit, Euler's formula gives:

$$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$

Maclaurin series

The Maclaurin series of a function is its Taylor series centred at the origin, a power series whose partial sums approximate the function near $x = 0.$ For the cosine function the series converges for every real number:

$$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$

Only even powers appear, in agreement with the cosine being an even function. Keeping the first two terms gives the approximation $\cos(x) \approx 1 - \dfrac{x^2}{2}$ for small $x.$ The difference $1 - \cos(x)$ is then close to $\dfrac{x^2}{2},$ which recovers the limit $\dfrac{1 - \cos(x)}{x} \to 0$ as $x \to 0.$