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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 *S*^{1} because it is a one-dimensional unit *n*-sphere.^{ [2] }^{ [note 1] }

- In the complex plane
- Trigonometric functions on the unit circle
- Circle group
- Complex dynamics
- Notes
- References
- See also

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 *x*^{2} = (−*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.

The unit circle can be considered as the unit complex numbers, i.e., the set of complex numbers *z* of the form

for all *t* (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as 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 *x*^{2} + *y*^{2} = 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 OA from the origin to a point P(*x*_{1},*y*_{1}) on the unit circle such that an angle *t* with 0 < *t* < π/2 is formed with the positive arm of the *x*-axis. Now consider a point Q(*x*_{1},0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = *t*. Because PQ has length *y*_{1}, OQ length *x*_{1}, and OA length 1, sin(*t*) = *y*_{1} and cos(*t*) = *x*_{1}. Having established these equivalences, take another radius OR from the origin to a point R(−*x*_{1},*y*_{1}) on the circle such that the same angle *t* is formed with the negative arm of the *x*-axis. Now consider a point S(−*x*_{1},0) and line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = *t*. It can hence be seen that, because ∠ROQ = π − *t*, R is at (cos(π − *t*),sin(π − *t*)) in the same way that P is at (cos(*t*),sin(*t*)). The conclusion is that, since (−*x*_{1},*y*_{1}) is the same as (cos(π − *t*),sin(π − *t*)) and (*x*_{1},*y*_{1}) is the same as (cos(*t*),sin(*t*)), it is true that sin(*t*) = sin(π − *t*) and −cos(*t*) = cos(π − *t*). It may be inferred in a similar manner that tan(π − *t*) = −tan(*t*), since tan(*t*) = *y*_{1}/*x*_{1} and tan(π − *t*) = *y*_{1}/−*x*_{1}. A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/√2.

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right.

Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the angle sum and difference formulas.

Complex numbers can be identified with points in the Euclidean plane, namely the number *a* + *bi* is identified with the point (*a*, *b*). Under this identification, the unit circle is a group under multiplication, called the *circle group*; it is usually denoted On the plane, multiplication by cos *θ* + *i* sin *θ* gives a counterclockwise rotation by *θ*. This group has important applications in mathematics and science.^{[ example needed ]}

The 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.

- ↑ Confusingly, in geometry a unit circle is often considered to be a 2-sphere—not a 1-sphere. The unit circle is "embedded" in a 2-dimensional plane that contains both height and width—hence why it is called a 2-sphere in geometry. However, the surface of the circle itself is one-dimensional, which is why topologists classify it as a 1-sphere. For further discussion, see the technical distinction between a circle and a disk.
^{ [2] }

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle from that point.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

The **haversine formula** determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the **law of haversines**, that relates the sides and angles of spherical triangles.

**Spherical trigonometry** is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons defined by a number of intersecting great circles on the sphere. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

In mathematics, a **rose** or **rhodonea curve** is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.

In trigonometry, **tangent half-angle formulas** relate the tangent of half of an angle to trigonometric functions of the entire angle. Among these are the following

The **Pythagorean trigonometric identity**, also called simply the **Pythagorean identity**, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

In geometry, the area enclosed by a circle of radius r is π*r*^{2}. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416.

The **small-angle approximations** can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:

In spherical trigonometry, the **law of cosines** is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

In astronomy, **position angle** is the convention for measuring angles on the sky. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images, this is a counterclockwise measure relative to the axis into the direction of positive declination.

In mathematics, the **sine** is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

The main **trigonometric identities** between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

The **differentiation of trigonometric functions** is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(*a*) = cos(*a*), meaning that the rate of change of sin(*x*) at a particular angle *x = a* is given by the cosine of that angle.

**Trigonometry** is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine.

In trigonometry, the **law of cosines** relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

- ↑ Weisstein, Eric W. "Unit Circle".
*mathworld.wolfram.com*. Retrieved 2020-05-05. - ↑ Weisstein, Eric W. "Hypersphere".
*mathworld.wolfram.com*. Retrieved 2020-05-06.

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