Unit circle

In mathematics, a unit circle is a special circle with a radius of exactly one unit. This shape is very important, especially in 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 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. The x and y coordinates of any point on the unit circle always fit the equation x² + y² = 1 thanks to a math rule called the Pythagorean theorem.

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.

In the complex plane

In the complex plane, numbers that have 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.

Trigonometric functions on the unit circle

The trigonometric functions cosine and sine of an angle θ are defined using the unit circle. Here, the angle θ is made by two rays: one fixed along the positive x-axis (the initial arm) and another that goes from the origin to a point (x, y) on the circle (the terminal arm). The value of θ shows how much the ray has turned from the initial arm, with turning counterclockwise being positive and turning 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 shows us 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 helps us learn about how math rules change and make patterns.