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measure. In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a unit circle is a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a pl ...
of unit
radius In classical geometry, a radius of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ''radius'', meaning ray but al ...
—that is, a radius of 1. Frequently, especially in
trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, ...
, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distances ...
in the Euclidean plane. In
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...
, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a
right triangle A right triangle (American English) or right-angled triangle (British English, British ) is a triangle in which one angle is a right angle (that is, a 90-Degree (angle), degree angle). The relation between the sides and angles of a right triangle ...
whose hypotenuse has length 1. Thus, by the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
, and satisfy the equation :$x^2 + y^2 = 1.$ Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points 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 number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
s of the form :$z = e^ = \cos t + i \sin t = \operatorname\left(t\right)$ for all (see also: cis). This relation represents . In
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quant ...
, this is referred to as phase factor.

# Trigonometric functions on the unit circle

The trigonometric functions cosine and sine of angle may be defined on the unit circle as follows: If is a point on the unit circle, and if the ray from the origin (0, 0) to makes an angle from the positive -axis, (where counterclockwise turning is positive), then :$\cos \theta = x \quad\text\quad \sin \theta = y.$ The equation gives the relation :$\cos^2\theta + \sin^2\theta = 1.$ The unit circle also demonstrates that sine and cosine are periodic functions, with the identities :$\cos \theta = \cos\left(2\pi k+\theta\right)$ :$\sin \theta = \sin\left(2\pi k+\theta\right)$ for any integer . Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point on the unit circle such that an angle with is formed with the positive arm of the -axis. Now consider a point and line segments . The result is a right triangle with . Because has length , length , and length 1, and . Having established these equivalences, take another radius OR from the origin to a point on the circle such that the same angle is formed with the negative arm of the -axis. Now consider a point and line segments . The result is a right triangle with . It can hence be seen that, because , is at in the same way that P is at . The conclusion is that, since is the same as and is the same as , it is true that and . It may be inferred in a similar manner that , since and . A simple demonstration of the above can be seen in the equality . When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than . However, when defined with the unit circle, these functions produce meaningful values for any real number, real-valued angle measure – even those greater than 2. In fact, all 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. Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the Trigonometric identity#Angle sum and difference identities, angle sum and difference formulas.

# Circle group

Complex numbers can be identified with points in the Euclidean plane, namely the number is identified with the point . Under this identification, the unit circle is a group (mathematics), group under multiplication, called the ''circle group''; it is usually denoted $\mathbb.$ On the plane, multiplication by gives a counterclockwise rotation by . This group has important applications in mathematics and science.

# Complex dynamics

Image:Erays.png, Unit circle in complex dynamics The Julia set of Dynamical system (definition), discrete nonlinear dynamical system with evolution function: :$f_0\left(x\right) = x^2$ is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems.

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# See also

*Angle measure *Pythagorean trigonometric identity * Riemannian circle *Unit angle *Unit disk *Unit sphere *Unit hyperbola *Unit square *Turn (unit) *z-transform Circles 1 (number) Trigonometry Fourier analysis Analytic geometry