# Unit circle

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Unit circle

In mathematics, a unit circle is a circle with a 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 sphere.

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

:$x^2 + y^2 = 1.$

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", such as the Riemannian circle; see the article on mathematical norms for additional 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 counterclockwise turning is positive), then

:$cos\left(t\right) = x ,!$:$sin\left(t\right) = y. ,!$

The equation "x"2 + "y"2 = 1 gives the relation

:$cos^2\left(t\right) + sin^2\left(t\right) = 1. ,!$Note that cos2(t)=(cos(t))2. 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

:$cos t = cos\left(2pi k+t\right) ,!$ :$sin t = sin\left(2pi k+t\right) ,!$ 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 = 2&pi; radians = 360º).

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than &pi;/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 &mdash; sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant &mdash; 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 mathematics and science.

ee also

*Angle measure
*Unit square
*Unit disc
*Circle group
*Riemannian circle

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* [http://www.dudefree.com/unitcircle/ An excellent Flash animation for learning the unit circle]
* [http://www.nbritton.org/uploads/unit_circle.pdf Printable, full page, unit circle handout]
* [http://glab.trixon.se/ GonioLab] : Visualization of the unit circle, trigonometric and hyperbolic functions

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### Look at other dictionaries:

• unit circle — noun : a circle whose radius is one unit of length long * * * Math. a circle whose radius has a length of one unit. [1950 55] * * * unit circle or unit sphere, Mathematics. a circle or sphere whose radius is one unit of distance …   Useful english dictionary

• unit circle — Math. a circle whose radius has a length of one unit. [1950 55] * * * …   Universalium

• unit circle — noun a) A circle of radius 1. b) The circle of radius 1 with centre at the origin, used in defining trigonometric functions …   Wiktionary

• unit circle — noun Date: 1955 a circle having a radius of 1 …   New Collegiate Dictionary

• Orthogonal polynomials on the unit circle — In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced …   Wikipedia

• Unit — may refer to:In mathematics: * Unit vector, a vector with length equal to 1 * Unit circle, the circle with radius equal to 1, centered at the origin * Unit interval, the interval of all real numbers between 0 and 1 * Imaginary unit, i , whose… …   Wikipedia

• unit sphere — unit circle or unit sphere, Mathematics. a circle or sphere whose radius is one unit of distance …   Useful english dictionary

• Circle group — For the jazz group, see Circle (jazz band). Lie groups …   Wikipedia