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 |x| and |y| 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 x 2 + y 2 = 1. {\textstyle x^{2}+y^{2}=1.}
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.
In the complex plane
In the complex plane, numbers of magnitude one are called the unit complex numbers. This is the set of complex numbers z such that | z | = 1. {\displaystyle |z|=1.} When broken into real and imaginary components z = x + i y , {\displaystyle z=x+iy,} this condition is | z | 2 = z z ¯ = x 2 + y 2 = 1. {\displaystyle |z|^{2}=z{\bar {z}}=x^{2}+y^{2}=1.}
The complex unit circle can be parametrized by angle measure θ {\displaystyle \theta } from the positive real axis using the complex exponential function, z = e i θ = cos θ + i sin θ . {\displaystyle z=e^{i\theta }=\cos \theta +i\sin \theta .} (See Euler's formula.) Under the complex multiplication operation, the unit complex numbers form a group called the circle group, usually denoted T . {\displaystyle \mathbb {T} .} In quantum mechanics, a unit complex number is called a phase factor.
Trigonometric functions on the unit circle
The trigonometric functions cosine and sine of angle θ are defined using the unit circle. In this geometric construction, the angle θ is formed by two rays: the initial arm, which remains fixed along the positive x-axis, and the terminal arm, a ray extending from the origin to a point (x, y) on the circumference of the unit circle. The value of θ represents the measure of rotation from the initial arm to the terminal arm, where counterclockwise rotation is designated as positive and clockwise rotation as negative. Consequently, the trigonometric functions are defined by the coordinates of the point where the terminal arm intersects the circle:
cos θ = x and sin θ = y . {\displaystyle \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}
The equation x2 + y2 = 1 gives the relation
cos 2 θ + sin 2 θ = 1. {\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}
The unit circle also demonstrates that sine and cosine are periodic functions, with the identities
cos θ = cos ( 2 π k + θ ) {\displaystyle \cos \theta =\cos(2\pi k+\theta )}
sin θ = sin ( 2 π k + θ ) {\displaystyle \sin \theta =\sin(2\pi k+\theta )}
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