Unit circle

In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers such that |z| = 1. When broken into real and imaginary components z = x + iy, this condition is |z|^2 = z\bar{z} = x^2 + y^2 = 1. The complex unit circle can be parametrized by angle measure \theta from the positive real axis using the complex exponential function, z = e^{i\theta} = \cos \theta + i \sin \theta. (See Euler's formula.) Under the complex multiplication operation, the unit complex numbers form a group called the circle group, usually denoted \mathbb{T}. In quantum mechanics, a unit complex number is called a phase factor. ==Trigonometric functions on the unit circle==

The trigonometric functions cosine and sine of angle are defined using the unit circle. In this geometric construction, the angle is formed by two rays: the initial arm, which remains fixed along the positive -axis, and the terminal arm, a ray extending from the origin to a point on the circumference of the unit circle. The value of represents the measure of rotation from the initial arm to the terminal arm, where counterclockwise rotation is designated as positive and clockwise rotation as negative. Consequently, the trigonometric functions are defined by the coordinates of the point where the terminal arm intersects the circle: \cos \theta = x \quad\text{and}\quad \sin \theta = y. The equation gives the relation \cos^2\theta + \sin^2\theta = 1. The unit circle also demonstrates that sine and cosine are periodic functions, with the identities \cos \theta = \cos(2\pi k+\theta) \sin \theta = \sin(2\pi k+\theta) for any integer . Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius from the origin to a point on the unit circle such that an angle with is formed with the positive arm of the -axis. Now consider a point and line segments . The result is a right triangle with . Because has length , length , and has length 1 as a radius on the unit circle, and . Having established these equivalences, take another radius from the origin to a point on the circle such that the same angle is formed with the negative arm of the -axis. Now consider a point and line segments . The result is a right triangle with . It can hence be seen that, because , is at in the same way that P is at . The conclusion is that, since is the same as and is the same as , it is true that and . It may be inferred in a similar manner that , since and . A simple demonstration of the above can be seen in the equality . When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than . However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2. 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. ==Complex dynamics==

The Julia set of discrete nonlinear dynamical system with evolution function: f_0(x) = x^2 is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems. ==See also==