Unit circle

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

[[Image:Unit circle.svg|thumb|alt=Unit circle|Illustration of a unit circle. The variable ''t'' is an angle measure.]] 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}}.

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 {{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 technical distinction between a circle and a disk.<ref name = "Unit sphere" />}}

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 <math display="inline">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.

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== thumb|Animation of the unit circle with angles In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers {{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.)

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

==Trigonometric functions on the unit circle== 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''. thumb|Sine function on unit circle (top) and its graph (bottom) The trigonometric functions cosine and sine of angle {{math|''θ''}} are defined using the unit circle.

In this geometric construction, the angle {{math|''θ''}} is formed by two rays: the '''initial arm''', which remains fixed along the positive {{math|''x''}}-axis, and the '''terminal arm''', a ray extending from the origin to a point {{math|(''x'', ''y'')}} on the circumference of the unit circle. The value of {{math|''θ''}} represents the measure of rotation from the initial arm to the terminal arm, where counterclockwise rotation is designated as positive and clockwise rotation as negative.<ref>{{cite web| title = Unit Circle Cheat Sheet: Everything You Need to Succeed | website = Easy Sevens Education | url = https://www.easysevens.com/unit-circle-cheat-sheet-everything-you-need-to-succeed/| date = Jan 28, 2025| access-date = Jan 31, 2026}}</ref> Consequently, the trigonometric functions are defined by the coordinates of the point where the terminal arm intersects the circle: <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 display="block"> \cos^2\theta + \sin^2\theta = 1.</math>

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities <math display="block">\cos \theta = \cos(2\pi k+\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-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 angle sum and difference formulas.

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

==Complex dynamics== {{Main|Complex dynamics}} right|thumb|Unit circle in complex dynamics The Julia set of 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== * Angle measure * Pythagorean trigonometric identity * Riemannian circle * Radian * Unit disk * Unit sphere * Unit hyperbola * Unit square * Turn (angle) * z-transform * Smith chart

==Notes== {{Reflist|group=note}}

==References== {{commons}} {{Reflist}}

Category:Circles Category:1 (number) Category:Trigonometry Category:Fourier analysis Category:Analytic geometry