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Unit Circle


UnitCircle

A unit circle is a circle of unit radius, i.e., of radius 1.

TrigonometryUnitCircle

The unit circle plays a significant role in a number of different areas of mathematics. For example, the functions of trigonometry are most simply defined using the unit circle. As shown in the figure above, a point P on the terminal side of an angle theta in angle standard position measured along an arc of the unit circle has as its coordinates (costheta,sintheta) so that costheta is the horizontal coordinate of P and sintheta is its vertical component.

As a result of this definition, the trigonometric functions are periodic with period 2pi.

UnitCircleTrigValues

Another immediate result of this definition is the ability to explicitly write the coordinates of a number of points lying on the unit circle with very little computation. In the figure above, for example, points A, B, C, and D correspond to angles of pi/3, 3pi/4, 7pi/6, and 11pi/6 radians, respectively, whereby it follows that A=(1/2,sqrt(3)/2), B=(-1/sqrt(2),1/sqrt(2)), C=(-sqrt(3)/2,-1/2), and D=(sqrt(3)/2,-1/2). Similarly, this method can be used to find trigonometric values associated to integer multiples of pi/2, plus a number of other angles obtained by half-angle, double-angle, and other multiple-angle formulas.

UnitCircleComplexPlane

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 H^2 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 H^2.


See also

Circle, Complex Plane, Discrete Fourier Transform, Double-Angle Formulas, Fourier Transform, Half-Angle Formulas, Laplace Transform, Multiple-Angle Formulas, Trigonometry, Unit Disk, Unit Square, Z-Transform Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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References

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 3, 1996.Oppenheim, A. V. "The z-Transform." 6.003--Signals and Systems. MIT OpenCourseWare, 2011. http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec23.pdf.

Referenced on Wolfram|Alpha

Unit Circle

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Unit Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnitCircle.html

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