Unit Circle

The unit circle is a fundamental concept in mathematics, particularly in trigonometry and complex number theory. It is a circle with a radius of 1 and its center at the origin of the coordinate plane (0,0).

Definition

The unit circle is defined by the equation x² + y² = 1. Any point (x, y) that satisfies this equation lies on the unit circle.

Trigonometry and the Unit Circle

The unit circle is closely related to trigonometry. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle formed by the positive x-axis and the line segment from the origin to the point, while the y-coordinate corresponds to the sine of this angle.

This relationship is often written as:

cos(θ) = x

sin(θ) = y

where θ (theta) is the angle in radians.

Complex Numbers and the Unit Circle

In the complex plane, the unit circle can also be described using complex numbers. Every point on the unit circle corresponds to a complex number of the form e^(iθ) = cos(θ) + i*sin(θ), where i is the imaginary unit.

Applications

The unit circle is a fundamental tool in many areas of mathematics, including trigonometry, calculus, geometry, and complex analysis. It's also used in physics and engineering to study periodic phenomena, waveforms, and rotations.