A **chord** of a circle is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just *secant*. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word *chord* is from the Latin *chorda* meaning *bowstring*.

Among properties of chords of a circle are the following:

- Chords are equidistant from the center if and only if their lengths are equal.
- Equal chords are subtended by equal angles from the center of the circle.
- A chord that passes through the center of a circle is called a diameter and is the longest chord.
- If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).

The midpoints of a set of parallel chords of an ellipse are collinear.^{ [1] }

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7.5 degrees. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.^{ [2] }

The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle *θ* is taken in the positive sense and must lie in the interval 0 < *θ* ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos *θ*, sin *θ*), and then using the Pythagorean theorem to calculate the chord length:^{ [2] }

The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably a great deal was known about them. In the table below (where is the chord length, and the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones:

Name | Sine-based | Chord-based |
---|---|---|

Pythagorean | ||

Half-angle | ||

Apothem (a) | ||

Angle (θ) |

The inverse function exists as well:^{ [3] }

- Circular segment - the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
- Scale of chords
- Ptolemy's table of chords
- Holditch's theorem, for a chord rotating in a convex closed curve
- Circle graph
- Exsecant and excosecant
- Versine and haversine
- Zindler curve (closed and simple curve in which all chords that divide the arc length into halves have the same length)

A **circle** is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

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, **Thales's theorem** states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's *Elements*. It is generally attributed to Thales of Miletus but it is sometimes attributed to Pythagoras.

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.

In geometry, a **circular segment** is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc of a circle and by the chord connecting the endpoints of the arc.

The **versine** or **versed sine** is a trigonometric function found in some of the earliest trigonometric tables. The versine of an angle is 1 minus its cosine.

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.

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.

The **exsecant** and **excosecant** are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astronomy, and spherical trigonometry and could help improve accuracy, but are rarely used today except to simplify some calculations.

In Euclidean geometry, **Ptolemy's theorem** is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

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 3.14159.

* On Sizes and Distances * is a text by the ancient Greek astronomer Hipparchus in which approximations are made for the radii of the Sun and the Moon as well as their distances from the Earth. It is not extant, but some of its contents have been preserved in the works of Ptolemy and his commentator Pappus of Alexandria. Several modern historians have attempted to reconstruct the methods of Hipparchus using the available texts.

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 .

A **circular arc** is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the **minor arc**, will subtend an angle at the centre of the circle that is less than π radians, and the other arc, the **major arc**, will subtend an angle greater than π radians.

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

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.

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

In geometry, the **sagitta** of a circular arc is the distance from the center of the arc to the center of its base. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin *sagitta*, meaning an arrow.

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 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.

- ↑ Chakerian, G. D. (1979). "7". In Honsberger, R. (ed.).
*A Distorted View of Geometry*.*Mathematical Plums*. Washington, DC, USA: Mathematical Association of America. p. 147. - 1 2 Maor, Eli (1998),
*Trigonometric Delights*, Princeton University Press, pp. 25–27, ISBN 978-0-691-15820-4 - ↑ Simpson, David G. (2001-11-08). "AUXTRIG" (FORTRAN-90 source code). Greenbelt, Maryland, USA: NASA Goddard Space Flight Center. Retrieved 2015-10-26.

- Hawking, Stephen William, ed. (2002).
*On the Shoulders of Giants: The Great Works of Physics and Astronomy*. Philadelphia, USA: Running Press. ISBN 0-7624-1698-X. LCCN 2002100441 . Retrieved 2017-07-31. - Stávek, Jiří (2017-03-10) [2017-02-26]. "On the Hidden Beauty of Trigonometric Functions".
*Applied Physics Research*. Prague, CZ: Canadian Center of Science and Education.**9**(2): 57–64. doi: 10.5539/apr.v9n2p57 . ISSN 1916-9639. ISSN 1916-9647.

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