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{{Short description|Circle with radius of one}}

The '''unit circle''' is a concept of [[mathematics]] (used in several contexts, especially in [[trigonometry]]). In essence, this is a [[circle]] constituted by all points that have [[Euclidean distance]] 1 from the [[origin]] (0,0) in a two-dimensional [[coordinate system]]. It is denoted by ''S''¹.

<div style="float:right;margin:0 0 1em 1em;text-align:center;">[[Image:Unit_circleUnit circle.pngsvg|thumb|alt=Unit circle]]<br>''Illustrations|Illustration of a unit circle.<br>' The variable ''t''' is an [[angle]] measure.''</div>]]

[[File:2pi-unrolled.gif|thumb|Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since {{math|1=''C'' = 2''πr''}}, the circumference of a unit circle is {{math|2π}}.]]

The equation defining the points (''x'', ''y'') of the unit circle is

In [[mathematics]], a '''unit circle''' is a [[circle]] of unit [[radius]]—that is, a radius of 1.<ref>{{Cite web|title=Unit Circle |url=https://mathworld.wolfram.com/UnitCircle.html|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com |language=en|access-date=2020-05-05}}</ref> 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 {{math|''S''¹}} because it is a one-dimensional unit [[n-sphere|{{math|''n''}}-sphere]].<ref name="Unit sphere">{{Cite web|title=Hypersphere|url=https://mathworld.wolfram.com/Hypersphere.html|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2020-05-06}}</ref>{{refn|group="note"|For further discussion, see the [[Circle|technical distinction between a circle and a disk]].<ref name = "Unit sphere" />}}

:<math>1 = x^2 + y^2</math>

If {{math|(''x'', ''y'')}} is a point on the unit circle's [[circumference]], then {{math|{{abs|''x''}}}} and {{math|{{abs|''y''}}}} are the lengths of the legs of a [[right triangle]] whose hypotenuse has length 1. Thus, by the [[Pythagorean theorem]], {{math|''x''}} and {{math|''y''}} satisfy the equation

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

:<math>1 display= "block">x^2 + y^2 = 1.</math>

Since {{math|1=''x''² = (−''x'')²}} for all {{math|''x''}}, and since the reflection of any point on the unit circle about the {{math|''x''}}- or {{math|''y''}}-axis is also on the unit circle, the above equation holds for all points {{math|(''x'', ''y'')}} on the unit circle, not only those in the first quadrant.

==Trigonometric functions in the unit circle==

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.

In a unit circle, several interesting things relating to [[trigonometric function]]s may be defined, with the given notation:

One may also use other notions of "distance" to define other "unit circles", such as the [[Riemannian circle]]; see the article on [[normednorm vector(mathematics)|mathematical spacenorms]] for additional examples.

A point on the unit circle, pointed to by a certain vector from the origin with the [[angle]] <math>t</math> from the <math>x</math>-axis has the coordinates:

==In the complex plane==

:<math>x = \cos(t)</math>

[[File:Unitycircle-complex.gif|thumb|Animation of the unit circle with angles]]

:<math>y = \sin(t)</math>

In the [[complex plane]], numbers of unit magnitude are called the [[unit complex numbers]]. This is the set of [[complex number]]s {{mvar|z}} such that <math>|z| = 1.</math> When broken into real and imaginary components <math>z = x + iy,</math> this condition is <math>|z|^2 = z\bar{z} = x^2 + y^2 = 1.</math>

The complex unit circle can be parametrized by angle measure <math>\theta</math> from the positive real axis using the complex [[exponential function]], <math>z = e^{i\theta} = \cos \theta + i \sin \theta.</math> (See [[Euler's formula]].)

The equation of the circle above also immediately gives us the well-known "trigonometric 1":

Under the complex multiplication operation, the unit complex numbers form a [[group (mathematics)|group]] called the ''[[circle group]]'', usually denoted <math>\mathbb{T}.</math> In [[quantum mechanics]], a unit complex number is called a [[phase factor]].

:<math>1 = \cos^2(t) + \sin^2(t)</math>

==Trigonometric functions inon the unit circle==

It is also an intuitive way of realizing that:

[[Image:Unit-circle_sin_cos_tan_cot_exsec_excsc_versin_vercos_coversin_covercos.svg|thumb|All of the trigonometric functions of the angle {{math|''θ''}} (theta) can be constructed geometrically in terms of a unit circle centered at ''O''.]]

[[File:Periodic sine.svg|thumb|Sine function on unit circle (top) and its graph (bottom)]]

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

<math display="block">\cos \theta = x \quad\text{and}\quad \sin \theta = y.</math>

The equation {{math|1=''x''² + ''y''² = 1}} gives the relation

:<math>\cos(t) = \cos(2\pi n+t)</math>

:<math>1 display="block"> \cos^2(t)\theta + \sin^2(t)\theta = 1.</math>

The unit circle also demonstrates that [[sine]] and [[cosine]] are [[periodic function]]s, with the identities

since <math>(x,y)</math> coordinates are obviously the same after one revolution in the circle. The notion of [[sine]] and [[cosine]], as well as several other trigonometric functions make little sense for [[triangle]]s with [[angle]]s greater than &pi;/2, or negative angles, but in the unit circle both of these have sensible, intuitive meanings.

:<math display="block">\cos(t) \theta = \cos(2\pi nk+t\theta)</math>

<math display="block">\sin \theta = \sin(2\pi k+\theta)</math>

for any [[integer]] {{math|''k''}}.

Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius {{math|OP}} from the origin {{math|O}} to a point {{math|P(''x''₁,''y''₁)}} on the unit circle such that an angle {{math|''t''}} with {{math|0 < ''t'' < {{sfrac|π|2}}}} is formed with the positive arm of the {{math|''x''}}-axis. Now consider a point {{math|Q(''x''₁,0)}} and line segments {{math|PQ ⊥ OQ}}. The result is a right triangle {{math|△OPQ}} with {{math|1=∠QOP = ''t''}}. Because {{math|PQ}} has length {{math|''y''₁}}, {{math|OQ}} length {{math|''x''₁}}, and {{math|OP}} has length 1 as a radius on the unit circle, {{math|1=sin(''t'') = ''y''₁}} and {{math|1=cos(''t'') = ''x''₁}}. Having established these equivalences, take another radius {{math|OR}} from the origin to a point {{math|R(−''x''₁,''y''₁)}} on the circle such that the same angle {{math|''t''}} is formed with the negative arm of the {{math|''x''}}-axis. Now consider a point {{math|S(−''x''₁,0)}} and line segments {{math|RS ⊥ OS}}. The result is a right triangle {{math|△ORS}} with {{math|1=∠SOR = ''t''}}. It can hence be seen that, because {{math|1=∠ROQ = π − ''t''}}, {{math|R}} is at {{math|(cos(π − ''t''), sin(π − ''t''))}} in the same way that P is at {{math|(cos(''t''), sin(''t''))}}. The conclusion is that, since {{math|(−''x''₁, ''y''₁)}} is the same as {{math|(cos(π − ''t''), sin(π − ''t''))}} and {{math|(''x''₁,''y''₁)}} is the same as {{math|(cos(''t''),sin(''t''))}}, it is true that {{math|1=sin(''t'') = sin(π − ''t'')}} and {{math|1=−cos(''t'') = cos(π − ''t'')}}. It may be inferred in a similar manner that {{math|1=tan(π − ''t'') = −tan(''t'')}}, since {{math|1=tan(''t'') = {{sfrac|''y''₁|''x''₁}}}} and {{math|1=tan(π − ''t'') = {{sfrac|''y''₁|−''x''₁}}}}. A simple demonstration of the above can be seen in the equality {{math|1=sin({{sfrac|π|4}}) = sin({{sfrac|3π|4}}) = {{sfrac|1|{{sqrt|2}}}}}}.

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than {{sfrac|{{pi}}|2}}. However, when defined with the unit circle, these functions produce meaningful values for any [[real number|real]]-valued angle measure&nbsp;– even those greater than 2{{pi}}. In fact, all six standard trigonometric functions&nbsp;– sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like [[versine]] and [[exsecant]]&nbsp;– 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 easily calculated by hand using the [[Trigonometric identity#Angle sum and difference identities|angle sum and difference formulas]].

[[Image:Unit circle angles color.svg|thumb|The unit circle, showing [[Exact trigonometric constants|coordinates of certain points]]]] <!--Get a picture with the tangent values show-->

==Complex dynamics==

{{Main|Complex dynamics}}

[[Image:Erays.svg|right|thumb|Unit circle in complex dynamics]]

The [[Julia set]] of [[Dynamical system (definition)|discrete nonlinear dynamical system]] with [[evolution function]]:

<math display="block">f_0(x) = x^2</math>

is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems.

==See also==

* [[TrigonometricAngle functionmeasure]]

* [[Pythagorean trigonometric identity]]

*[[Angle measure]]

* [[Riemannian circle]]

* [[Radian]]

* [[Unit disk]]

* [[Unit sphere]]

* [[Unit hyperbola]]

* [[Unit square]]

* [[Turn (angle)]]

* [[z-transform]]

* [[Smith chart]]

==Notes==

{{Reflist|group=note}}

==References==

{{Reflist}}

[[Category:Circles]]

[[Category:1 (number)]]

[[Category:Trigonometry]]

[[Category:Fourier analysis]]

[[Category:Analytic geometry]]