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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''<sup>1</sup>.
<div style="float:right;margin:0 0 1em 1em;text-align:center;">[[Image:Unit_circle.png|Unit circle]]<br>''Illustrations of a unit circle.<br>'''t''' is an [[angle]] measure.''</div>
The equation defining the points (''x'', ''y'') of the unit circle is
:<math>1 = x^2 + y^2</math>
One may also use other notions of "distance" to define other "unit circles"; see the article on [[normed vector space]] for examples.
==Trigonometric functions in the unit circle==
In a unit circle, several interesting things relating to [[trigonometric function]]s may be defined, with the given notation:
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:
:<math>x = \cos(t)</math>
:<math>y = \sin(t)</math>
The equation of the circle above also immediately gives us the well-known "trigonometric 1":
:<math>1 = \cos^2(t) + \sin^2(t)</math>
It is also an intuitive way of realizing that:
:<math>\cos(t) = \cos(2\pi n+t)</math>
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 π/2, or negative angles, but in the unit circle both of these have sensible, intuitive meanings.
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