In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1.[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.[2][note 1]
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
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
editIn 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 form a group called the circle group, usually denoted In quantum mechanics, a unit complex number is called a phase factor.
Trigonometric functions on the unit circle
editThe trigonometric functions cosine and sine of angle
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.
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OP from the origin O to a point P(x1,y1) on the unit circle such that an angle t with 0 < t <
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than
Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be easily calculated by hand using the angle sum and difference formulas.
Complex dynamics
editThe Julia set of discrete nonlinear dynamical system with evolution function: is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems.
See also
editNotes
edit- ^ For further discussion, see the technical distinction between a circle and a disk.[2]
References
edit- ^ Weisstein, Eric W. "Unit Circle". mathworld.wolfram.com. Retrieved 2020-05-05.
- ^ a b Weisstein, Eric W. "Hypersphere". mathworld.wolfram.com. Retrieved 2020-05-06.