Unit circle

In mathematics, a unit circle is a special circle with a radius of exactly one unit. This simple shape is very important, especially in a part of math called trigonometry. The unit circle is usually drawn with its center at the origin (the point (0, 0)) on a grid called the Cartesian coordinate system in something called the Euclidean plane.

If you pick any point on the edge of the unit circle, the distances from that point to the x-axis and y-axis create the two shorter sides of a right triangle. The longest side, or hypotenuse, of that triangle is always 1, because it matches the radius of the circle. Because of a famous math rule called the Pythagorean theorem, the x and y coordinates of any point on the unit circle always fit the equation x² + y² = 1.

The space inside the unit circle is called the open unit disk, and if you include the edge of the circle too, it’s called the closed unit disk. Mathematicians can also define other kinds of “unit circles” using different ideas about distance, but the most common one is the circle of radius 1 we’ve been talking about.

In the complex plane

In the complex plane, numbers with a size of exactly one are called the unit complex numbers. These numbers can be written as ( z = x + iy ), where ( x ) and ( y ) are real numbers, and they satisfy the condition ( x^2 + y^2 = 1 ).

The complex unit circle can be described using an angle ( \theta ) from the positive real axis. Using the exponential function, any point on the unit circle can be written as ( z = e^{i\theta} = \cos \theta + i \sin \theta ). This relationship is known as Euler's formula and is important in many areas of mathematics, including quantum mechanics, where such numbers are called phase factors. Under complex multiplication, these unit complex numbers form a group called the circle group, usually denoted ( \mathbb{T} ).

Trigonometric functions on the unit circle

The trigonometric functions cosine and sine of an angle θ are defined using the unit circle. In this setup, the angle θ is formed by two rays: one fixed along the positive x-axis (the initial arm) and another extending from the origin to a point (x, y) on the circle (the terminal arm). The value of θ shows how much the ray has rotated from the initial arm, with counterclockwise being positive and clockwise being negative. The coordinates of the point where the terminal arm meets the circle give us the values of cosine and sine: cos θ = x and sin θ = y.

The unit circle also helps us understand that sine and cosine repeat their values in a predictable way. This means that for any whole number k, cos θ = cos(2πk + θ) and sin θ = sin(2πk + θ). Using the unit circle, we can find the values of trigonometric functions for any angle, not just those between 0 and π/2. This includes all six main trigonometric functions and some older ones as well.

Complex dynamics

Main article: Complex dynamics

The Julia set of a special math rule, where each point is squared, creates a perfect circle with a radius of 1. This simple example is very useful for studying how math rules change over time and repeat patterns.