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.
For a sequence whose Z-transform converges on the unit circle, its restriction 
is the discrete-time Fourier
transform. For a finite sequence of length 
, evaluating at the 
th roots of unity gives the
discrete Fourier transform (Oppenheim
2011).
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.