A unit circle is a circle of unit radius, i.e., of radius 1.
The unit circle plays a significant role in a number of different areas of mathematics. For example, the functions of trigonometry are most
simply defined using the unit circle. As shown in the figure above, a point 
on the terminal
side of an angle 
in angle standard
position measured along an arc of the unit circle has
as its coordinates 
so that 
is the horizontal coordinate of 
and 
is its vertical component.
As a result of this definition, the trigonometric functions are periodic with period 
.
Another immediate result of this definition is the ability to explicitly write the coordinates of a number of points lying on the unit circle with very little computation.
In the figure above, for example, points 
, 
,
, and 
correspond to angles of 
, 
,
, and 
radians, respectively, whereby it follows that 
, 
, 
, and 
. Similarly, this method can be used to find
trigonometric values associated to integer multiples of 
, plus a number of other angles obtained by half-angle,
double-angle, and other multiple-angle
formulas.
The unit circle can also be considered to be the contour in the complex plane defined by 
, where 
denotes the complex modulus.
This role of the unit circle also has a number of significant results, not the least of which occurs in applied complex analysis as the subset of the complex plane where the Z-transform reduces to the discrete Fourier transform.
From yet another perspective, the unit circle is viewed as the so-called ideal boundary of the two-dimensional hyperbolic plane 
in both the Poincaré
hyperbolic disk and Klein-Beltrami models
of hyperbolic geometry. In both these models,
the hyperbolic plane is viewed as the open unit disk,
whereby the unit circle represents the collection of infinite limit
points of sequences in 
.
-Transform." 6.003--Signals and Systems. MIT OpenCourseWare,
2011.