Unit circle

Illustration of a unit circle. The variable t is an angle measure.
Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since C = 2πr, the circumference of a unit circle is .

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.

If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation

${\displaystyle x^{2}+y^{2}=1.}$

Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.

The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.

One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.

In the complex plane

The unit circle can be considered as the unit complex numbers, i.e., the set of complex numbers z of the form

${\displaystyle z=\,\mathrm {e} ^{it}\,=\cos(t)+i\sin(t)=\operatorname {cis} (t)}$

for all t (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as phase factor.

Animation of the unitycircle with angles
Other Languages
العربية: دائرة وحدة
Bân-lâm-gú: Tan-ūi-îⁿ
bosanski: Jedinični krug
Deutsch: Einheitskreis
Esperanto: Unuocirklo
français: Cercle unité
한국어: 단위원
Bahasa Indonesia: Lingkaran satuan
Кыргызча: Бирдик айлана
македонски: Единична кружница
монгол: Нэгж тойрог
Nederlands: Eenheidscirkel

norsk nynorsk: Einingssirkel
português: Círculo unitário
Simple English: Unit circle
slovenščina: Enotska krožnica
српски / srpski: Јединични круг
srpskohrvatski / српскохрватски: Jedinični krug
svenska: Enhetscirkel
Türkçe: Birim çember
українська: Одиничне коло
ייִדיש: איינס קרייז