# Unit circle

In mathematics, a **unit circle** is a circle with unit radius, i.e., a circle whose radius is 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. The unit circle is often denoted *S*^{1}; the generalization to higher dimensions is the unit ball.

If (*x*, *y*) is a point on the unit circle in the first quadrant, 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

Since *x*^{2} = (−*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 just those in the first quadrant.

One may also use other notions of "distance" to define other "unit circles"; see the article on normed vector space for examples.

## Trigonometric functions on the unit circle

The trigonometric functions cosine and sine may be defined on the unit circle as follows. If (*x*, *y*) is a point of the unit circle, and if the ray from the origin (0, 0) to (*x*, *y*) makes an angle *t* from the positive *x*-axis, (where the angle is measured in the counter-clockwise direction), then

The equation *x*^{2} + *y*^{2} = 1 gives the relation

The unit circle also gives an intuitive way of realizing that sine and cosine are periodic functions, with the identities

- for any integer
*k*.

These identities come from the fact that the *x*- and *y*-coordinates of a point on the unit circle remain the same after the angle *t* is increased or decreased by any number of revolutions (1 revolution = 2π radians).

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, using the unit circle, these functions have sensible, intuitive meanings for any real-valued angle measure.

In fact, not only sine and cosine, but all of the six standard trigonometric functions — sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant — can be defined geometrically in terms of a unit circle, as shown at right.

## Circle group

Complex numbers can be identified with points in the Euclidean plane, namely the number *a* + *bi* is identified with the point (*a*, *b*). Under this identification, the unit circle is a group under multiplication, called the circle group. This group has important applications in math and science; see circle group for more details.

## See also

da:Enhedscirklen de:Einheitskreis fr:Cercle trigonométrique he:מעגל היחידה ja:単位円 ko:단위원 (기하) nl:Eenheidscirkel sr:Јединични круг vi:Vòng tròn đơn vị