Unit circle

[1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.

In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.

The complex unit circle can be parametrized by angle measure

from the positive real axis using the complex exponential function,

Under the complex multiplication operation, the unit complex numbers form a group called the circle group, usually denoted

In quantum mechanics, a unit complex number is called a phase factor.

The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities

for any integer k. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions.

First, construct a radius OP from the origin O to a point P(x1,y1) on the unit circle such that an angle t with 0 < t < ⁠π/2⁠ is formed with the positive arm of the x-axis.

Now consider a point Q(x1,0) and line segments PQ ⊥ OQ.

Having established these equivalences, take another radius OR from the origin to a point R(−x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis.

Now consider a point S(−x1,0) and line segments RS ⊥ OS.

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than ⁠π/2⁠.

However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π.

In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right.

Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be easily calculated by hand using the angle sum and difference formulas.

The Julia set of discrete nonlinear dynamical system with evolution function:

It is a simplest case so it is widely used in the study of dynamical systems.