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In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1.[1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.[2][note 1]

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

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

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 unit circle with angles

Trigonometric functions on the unit circle

trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then

${\displaystyle \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}$

The equation x2 + y2 = 1 gives the relation

${\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}$

The unit circle also demonstrates that sine and x2 + y2 = 1 gives the relation

${\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}$

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The unit circle also demonstrates that sine and cosine are periodic functions, with the identities

${\displaystyle \cos \theta =\cos(2\pi k+\theta )}$