Illustration of a unit circle. The variable t is an angle measure. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

In mathematics, a unit circle is a circle with a unit radius, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers 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. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The unit circle is often denoted S¹; the generalization to higher dimensions is the unit sphere. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed

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. Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised Thus, by the Pythagorean theorem, x and y satisfy the equation

x² + y² = 1. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

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 just those in the first quadrant.

One may also use other notions of "distance" to define other "unit circles;" see the article on mathematical norms for examples. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

Trigonometric functions on the unit circle

All of the trigonometric functions of the angle θ can be constructed geometrically in terms of a unit circle centered at O.

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 counterclockwise turning is positive), then

The equation x² + y² = 1 gives the relation

Note that cos²(t)=(cos(t))². In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called This is the standard shorthand for expressing powers of trigonometric functions.

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

for any integer k. In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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 Mathematics, the real numbers may be described informally in several different ways

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. The versed sine, also called the versine and in Latin, the sinus versus ("flipped sine" or the sagitta ("arrow" is a The exsecant, also abbreviated exsec, is a Trigonometric function defined in terms of the secant function sec(&theta \operatorname{exsec}(\theta

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). Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Under this identification, the unit circle is a group under multiplication, called the circle group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex This group has important applications in mathematics and science.

See also

• Angle measure
• Unit square
• Unit disc
• Circle group

External links

• Eric W. Weisstein, Unit circle at MathWorld. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The unit square is a square with all of the side lengths equalling 1 In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• An excellent Flash animation for learning the unit circle
• Printable, full page, unit circle handout
• GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions