Unit circle



In mathematics, a unit circle is a circle of unit radius—that is, a radius of 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 $C = 2πr$ because it is a one-dimensional unit $2π$-sphere.

If $S^{1}$ is a point on the unit circle's circumference, then $n$ and $(x, y)$ are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, $|x|$ and $|y|$ satisfy the equation $$x^2 + y^2 = 1.$$

Since $x$ for all $y$, and since the reflection of any point on the unit circle about the $x^{2} = (−x)^{2}$- or $x$-axis is also on the unit circle, the above equation holds for all points $x$ on the unit circle, not only those in the first quadrant.

The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.

One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.

In the complex plane
In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers $z$ 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 $y$ 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 $θ$ to $θ$ makes an angle $(x, y)$ from the positive $(0, 0)$-axis, (where counterclockwise turning is positive), then $$\cos \theta = x \quad\text{and}\quad \sin \theta = y.$$

The equation $(x, y)$ 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 $θ$.

Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius $x$ from the origin $x^{2} + y^{2} = 1$ to a point $k$ on the unit circle such that an angle $OP$ with $O$ is formed with the positive arm of the $P(x_{1},y_{1})$-axis. Now consider a point $t$ and line segments $0 < t < π⁄2$. The result is a right triangle $x$ with $Q(x_{1},0)$. Because $PQ ⊥ OQ$ has length $△OPQ$, $∠QOP = t$ length $PQ$, and $y_{1}$ has length 1 as a radius on the unit circle, $OQ$ and $x_{1}$. Having established these equivalences, take another radius $OP$ from the origin to a point $sin(t) = y_{1}$ on the circle such that the same angle $cos(t) = x_{1}$ is formed with the negative arm of the $OR$-axis. Now consider a point $R(−x_{1},y_{1})$ and line segments $t$. The result is a right triangle $x$ with $S(−x_{1},0)$. It can hence be seen that, because $RS ⊥ OS$, $△ORS$ is at $∠SOR = t$ in the same way that P is at $∠ROQ = π − t$. The conclusion is that, since $R$ is the same as $(cos(π − t), sin(π − t))$ and $(cos(t), sin(t))$ is the same as $(−x_{1}, y_{1})$, it is true that $(cos(π − t), sin(π − t))$ and $(x_{1},y_{1})$. It may be inferred in a similar manner that $(cos(t),sin(t))$, since $sin(t) = sin(π − t)$ and $−cos(t) = cos(π − t)$. A simple demonstration of the above can be seen in the equality $tan(π − t) = −tan(t)$.

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than $\pi⁄2$. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2$\pi$. 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.