Unit circle
A circle of radius 1, central to trigonometry and complex analysis.
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The unit circle is any circle with a radius of exactly 1. In trigonometry, it's usually drawn centered at the origin (0, 0) of the standard Cartesian plane. Topologists often call it S¹, since it's the one-dimensional version of a unit n-sphere.
For any point (x, y) on its circumference, the absolute values |x| and |y| are the legs of a right triangle whose hypotenuse is 1. By the Pythagorean theorem, this gives the equation x² + y² = 1. Because squaring a number removes its sign, reflecting a point across the x- or y-axis keeps it on the circle, so the equation holds for all quadrants.
The area inside the circle is the open unit disk; if you include the boundary, it's the closed unit disk. Different definitions of "distance" produce other unit circles, like the Riemannian circle.
In the complex plane, numbers with a magnitude of 1 are called unit complex numbers. For a complex number z = x + iy, the condition |z| = 1 becomes |z|² = z·z̄ = x² + y² = 1. You can describe these points using an angle θ from the positive real axis: z = e^(iθ) = cos θ + i sin θ (Euler's formula). Under multiplication, these numbers form a group called the circle group, often written as 𝕋. In quantum mechanics, a unit complex number is called a phase factor.
The trigonometric functions cosine and sine of an angle θ are defined using the unit circle. The angle is formed by a fixed ray along the positive x‑axis (the initial arm) and a rotating ray (the terminal arm) that goes from the origin to a point (x, y) on the circle. Counterclockwise rotation is positive; clockwise is negative. Then cos θ = x and sin θ = y. From x² + y² = 1, we get cos²θ + sin²θ = 1.
The unit circle also shows that sine and cosine are periodic: for any integer k, cos θ = cos(2πk + θ) and sin θ = sin(2πk + θ). You can see this by constructing triangles. Draw a radius OP to a point P(x₁, y₁) on the circle, making an angle t (0 < t < π/2) with the positive x‑axis. Drop a perpendicular from P to the x‑axis at Q(x₁, 0). Triangle △OPQ is right‑angled, with PQ = y₁, OQ = x₁, and OP = 1, so sin(t) = y₁ and cos(t) = x₁. Now draw another radius OR to point R(–x₁, y₁), making the same angle t with the *negative* x‑axis. Drop a perpendicular from R to the x‑axis at S(–x₁, 0). Triangle △ORS is also
- field
- Mathematics
- known_for
- Defining trigonometric functions, complex unit circle, circle group, and Julia set of f₀(x)=x²
- definition
- Circle of radius 1 centered at the origin (0,0) in the Euclidean plane
- equation
- x² + y² = 1
- topology_notation
- S¹
Lore & Background
In the complex plane, numbers of magnitude one are called unit complex numbers, forming the set of complex numbers z such that |z| = 1. When broken into real and imaginary components z = x + iy, this condition becomes |z|² = z·z̄ = x² + y² = 1. The complex unit circle can be parametrized by angle measure θ from the positive real axis using the complex exponential function, z = e^(iθ) = cos θ + i sin θ, via Euler's formula. Under complex multiplication, the unit complex numbers form a group called the circle group, usually denoted 𝕋. In quantum mechanics, a unit complex number is called a phase factor.
Reader's Guide
The unit circle is fundamental to trigonometry, where the trigonometric functions cosine and sine of angle θ are defined using the unit circle. The angle θ is formed by two rays: the initial arm along the positive x-axis and the terminal arm from the origin to a point (x, y) on the circumference. Cosine and sine are defined as cos θ = x and sin θ = y, leading to the identity cos²θ + sin²θ = 1. The unit circle demonstrates that sine and cosine are periodic functions with period 2π, and it allows trigonometric functions to produce meaningful values for any real-valued angle measure, even those greater than 2π. All six standard trigonometric functions, as well as archaic functions like versine and exsecant, can be defined geometrically in terms of a unit circle. In complex dynamics, the Julia set of the discrete nonlinear dynamical system with evolution function f₀(x) = x² is a unit circle, making it a widely used case in the study of dynamical systems.
Did You Know?
- The interior of the unit circle is called the open unit disk, while the interior combined with the circle itself is called the closed unit disk.
- In topology, the unit circle is often denoted as S¹ because it is a one-dimensional unit n-sphere.
- The unit circle can be parametrized by angle measure θ using the complex exponential function z = e^(iθ) = cos θ + i sin θ.
- The Julia set of the discrete nonlinear dynamical system f₀(x) = x² is a unit circle.
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