y = \cos x

Graph of the Cosine Function $y = \cos x$

$y = \cos x$ is the $x$ -coordinate of the point at angle $x$ , measured in radians, on the unit circle¹. Where the sine gives the height of that point, the cosine gives its horizontal position. The graph is the same wave as the sine, only shifted.

Domain and range

The domain is all real numbers and the range is the interval $[-1, 1]$ .

The domain is all real numbers
The range is $-1 \leq y \leq 1$
The amplitude is $1$ and the period is $2\pi$
The maximum $1$ falls at $x = 2n\pi$ and the minimum $-1$ at $x = (2n+1)\pi$

Periodicity

Since $\cos(x + 2\pi) = \cos x$ , the period is $2\pi$ .

Symmetry

Since $\cos(-x) = \cos x$ , it is an even function, symmetric about the $y$ -axis. This is the sharpest difference from the sine, which is odd.

Monotonicity and extrema

The derivative is $y' = -\sin x$ . The function attains its maximum $1$ at $x = 2n\pi$ and its minimum $-1$ at $x = (2n + 1)\pi$ . In particular it starts at a crest, since $\cos 0 = 1$ .

The second derivative $y'' = -\cos x$ changes sign at each zero, so the inflection points sit at $x = \dfrac{\pi}{2} + n\pi$ and coincide with the zeros.

Notable values

$x$	$\cos x$
$0$	$1$
$\dfrac{\pi}{6}$	$\dfrac{\sqrt{3}}{2}$
$\dfrac{\pi}{4}$	$\dfrac{\sqrt{2}}{2}$
$\dfrac{\pi}{3}$	$\dfrac{1}{2}$
$\dfrac{\pi}{2}$	$0$
$\pi$	$-1$

The zeros are the solutions of $\cos x = 0$ , namely $x = \dfrac{\pi}{2} + n\pi$ .

Approximation near the origin

The tangent at $(0, 1)$ is horizontal, and near that crest a downward parabola approximates the curve closely.

\cos x \approx 1 - \frac{x^2}{2}

The Taylor series has only even powers and converges for every real number.

\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots

Comparison with the sine

Item	$y = \sin x$	$y = \cos x$
Value at $x = 0$	$0$	$1$
Symmetry	odd	even
Where the maximum falls	$\dfrac{\pi}{2} + 2n\pi$	$2n\pi$
Zeros	$n\pi$	$\dfrac{\pi}{2} + n\pi$
Derivative	$\cos x$	$-\sin x$

Since $\cos x = \sin\left( x + \dfrac{\pi}{2} \right)$ , the cosine is the sine wave shifted left by $\dfrac{\pi}{2}$ . The points where $\cos x = 0$ fix the positions of the vertical asymptotes of $\tan x$ and $\sec x$ . In Euler's formula $e^{ix} = \cos x + i\sin x$ , the cosine supplies the real part.

Tying angles to lengths

The cosine links angles with lengths, as in the dot product and the law of cosines².

\begin{align*} \vec{a} \cdot \vec{b} &= |\vec{a}||\vec{b}|\cos\theta \\ c^2 &= a^2 + b^2 - 2ab\cos C \end{align*}

Applications

Computing the angle between two vectors
The law of cosines in solving triangles
The power factor of alternating current
The cosine components of a Fourier expansion

As a wave it is identical to the sine, but because it peaks at $x = 0$ it is the natural reference for phase.

Sine and cosine, Wikipedia
Law of cosines, Wikipedia

Graph of the Cosine Function y=cos⁡xy = \cos xy=cosx