Unit circle
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
S1.
Illustration of a unit circle.
t is an
angle measure.
Image:UnitCircle.png
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 from the -axis has the coordinates:
-
The equation of the circle above also immediately gives us the well-known "trigonometric 1":
The unit circle also gives an intuitive way of realizing that sine and
cosine are periodic functions, with the identity
- and for any integer n.
This identity comes from the fact that (
x,
y) coordinates remain the same after the angle
t is increased or decreased by one revolution in the circle (2π). The notion of sine, cosine, and other trigonometric functions only makes sense with
angles more than zero or less than π/2 when working with right triangles, but in the unit circle, angles outside this range have sensible, intuitive meanings.
See also