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.

Illustrations of a unit circle.
t is an angle measure.

The equation defining the points (x, y) of the unit circle is

${\displaystyle 1=x^{2}+y^{2}}$

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 ${\displaystyle t}$  from the ${\displaystyle x}$ -axis has the coordinates:

${\displaystyle x=\cos(t)}$ 

${\displaystyle y=\sin(t)}$ 

The equation of the circle above also immediately gives us the well-known "trigonometric 1":

${\displaystyle 1=\cos ^{2}(t)+\sin ^{2}(t)}$ 

It is also an intuitive way of realizing that:

${\displaystyle \cos(t)=\cos(2\pi n+t)}$ 

since ${\displaystyle (x,y)}$  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.

See also

• Trigonometric function
• Angle measure