Unit circle
Definition
The unit circle is a circle with a radius of one. Often used in mathematics, it is centered at the origin (0,0) in the Cartesian coordinate system. The unit circle provides a visual representation of the trigonometric functions, such as sine, cosine, and tangent.
Mathematical Description
The unit circle is described by the equation x² + y² = 1 in the Cartesian coordinate system. Here, x and y are the coordinates of any point on the circle, and the right side of the equation (1) represents the radius squared.
The unit circle can also be described in terms of the polar coordinates r and θ, where r is the radius of the circle (always equal to 1 for the unit circle) and θ is the angle subtended at the origin by a point on the circle. For the unit circle, the polar equation is simply r = 1.
Trigonometric Functions
The unit circle is fundamental to the study of trigonometry. The x- and y-coordinates of a point on the unit circle provide the cosine and sine of the angle θ, respectively. This relationship is often used to define these trigonometric functions for all real numbers:
- Cosine: cos(θ) = x
- Sine: sin(θ) = y
The tangent function can also be defined using the unit circle: tan(θ) = y/x = sin(θ)/cos(θ). However, this definition is only valid when cos(θ) ≠ 0.
Complex Plane
The unit circle also plays a significant role in the field of complex numbers. A complex number z can be represented as a point in the complex plane, where the horizontal axis represents the real part of z and the vertical axis represents the imaginary part. The absolute value (or modulus) of z is the distance from the origin to the point representing z, and the argument (or angle) of z is the angle θ between the positive real axis and the line segment connecting the origin and the point representing z.
When a complex number lies on the unit circle, its absolute value is 1. The unit circle in the complex plane is often used to visualize and prove properties of complex numbers.