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Unit Circle Definition
The unit circle is a circle with a radius of 1 which is centered at the origin on the x-y plane. The unit circle plays a significant role in several different areas of mathematics. In particular the functions of trigonometry are most simply defined using the unit circle. As shown in the figure below, a point p on the terminal side of an angle θ in angle standard position measured along an arc of the unit circle has as its coordinates (cos θ, sin θ) so that cos θ is the horizontal coordinate of p and sin θ is its vertical component. As a result of this definition, the trigonometric functions are periodic with period 2π.
Another immediate result of this definition is the ability to explicitly write the coordinates of several points lying on the unit circle with very little computation. In the figure above, for instance, points A, B, C, and D correspond to angles of π⁄3, 3π⁄4, 7π⁄6, and 11π⁄6 radians, respectively, whereby it follows that A = (1⁄2, 3⁄2), B = (-1⁄2, 1⁄2), C = (-3⁄2, -1⁄2), and D = (3⁄2, -1⁄2). Similarly, this method can be used to find trigonometric values associated to integer multiples of π⁄2, plus a number of other angles obtained by half-angle, double-angle, and other multiple-angle formulas.
The unit circle can also be considered to be the contour in the complex plane defined by |z| = 1, where |z| denotes the complex modulus. This role of the unit circle also has a number of significant results, not the least of which occurs in applied complex analysis as the subset of the complex plane where the Z-transform reduces to the discrete Fourier transform.
From yet another perspective, the unit circle is viewed as the so-called ideal boundary of the two-dimensional hyperbolic plane ℍ2 in both the Poincaré hyperbolic disk and Klein-Beltrami models of hyperbolic geometry. In both these models, the hyperbolic plane is viewed as the open unit disk, whereby the unit circle represents the collection of infinite limit points of sequences in ℍ2.
Related Definitions
Sources
“Unit Circle.” From Wolfram MathWorld, mathworld.wolfram.com/UnitCircle.html.