unitcircle

The unit circle is the set of all points in the Euclidean plane that are at distance one from the origin. It is centered at the origin and has radius 1, and its equation is x^2 + y^2 = 1.

For any real angle θ, measured in radians from the positive x-axis, the corresponding point on the unit circle has coordinates (cos θ, sin θ). The angle is taken to increase counterclockwise, with negative angles measured clockwise. Thus the trigonometric functions are defined as the coordinates of points on the circle: cos θ is the x-coordinate, and sin θ is the y-coordinate. The unit circle can be described parametrically by (cos t, sin t) as t varies over the real numbers.

Special positions on the circle correspond to familiar points: θ = 0 gives (1, 0); θ = π/2 gives (0, 1); θ = π gives (−1, 0); θ = 3π/2 gives (0, −1). Angles such as π/6, π/4, and π/3 yield coordinates that involve square roots (for example, (√3/2, 1/2) and (√2/2, √2/2)).

Key properties include the Pythagorean identity cos^2 θ + sin^2 θ = 1, which reflects the circle’s equation. The arc length along the circle corresponding to angle θ is exactly θ when θ is measured in radians, and both sine and cosine have a fundamental period of 2π.

The unit circle provides the geometric foundation for trigonometric functions, unit vectors in the direction of θ, and a compact representation of complex numbers on the unit circle through e^{iθ} = cos θ + i sin θ.