---
tag: 0P4NQD746K
kind: statement
role: theorem
name: Sine is 2π-periodic
msc: [33B10]
uses: [028XEQCE88, 01A8JGR4JM, 089KWVYD69]
sources:
  - url: https://en.wikipedia.org/wiki/Sine_and_cosine
    locator: Graph of a function and its elementary properties
---

## Statement

For every real $x$,
$$\sin(x + 2\pi) = \sin x, \qquad \cos(x + 2\pi) = \cos x.$$
The number $2\pi$ is the smallest positive period of $\sin$ and of $\cos$.

## Remarks

From [[$\cos \pi = -1$ and $\sin \pi = 0$|01A8JGR4JM]], the [[addition formulas|028XEQCE88]] give $\cos 2\pi = 1$ and $\sin 2\pi = 0$. Apply the addition formulas again with $y = 2\pi$.
So $\sin$ and $\cos$ are [[periodic|089KWVYD69]] functions.
Also $\sin(x + \pi) = -\sin x$ and $\sin(x + \pi/2) = \cos x$. Mathlib states the periodicity as `Real.sin_periodic`.

## Explanations

### L1 human

A full turn around the unit circle has length $2\pi$, and it brings the point back to where it started.
So the graphs of sine and cosine repeat after each interval of length $2\pi$.