Illustrations of a unit circle. t is an angle measure. 
The equation defining the points (x, y) of the unit circle is
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 functions 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:
The equation of the circle above also immediately gives us the wellknown "trigonometric 1":
It is also an intuitive way of realizing that:
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 triangles with angles greater than π/2, or negative angles, but in the unit circle both of these have sensible, intuitive meanings.
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