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Concept
Unit circle
Summary
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.
If (x, y) is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation
Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.
The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.
One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.
unit complex numbers
In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers z such that When broken into real and imaginary components this condition is
The complex unit circle can be parametrized by angle measure from the positive real axis using the complex exponential function, (See Euler's formula.)
Under the complex multiplication operation, the unit complex numbers are group called the circle group, usually denoted In quantum mechanics, a unit complex number is called a phase factor.
The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then
The equation x2 + y2 = 1 gives the relation
The unit circle also demonstrates that sine and cosine are periodic functions, with the identities
for any integer k.
Official source
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