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A chord of a curve is a geometric line segment whose endpoints both lie on the curve. A secant or a secant line is the line extension of a chord. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...
A secant line of a curve is a line that intersects two or more points on the curve. ...
Image File history File links Chord_in_mathematics. ...
Image File history File links Chord_in_mathematics. ...
Chords of a circle
- Further information: Chord properties
Among properties of chords of a circle are the following: Circle illustration This article is about the shape and mathematical concept of circle. ...
Circle illustration This article is about the shape and mathematical concept of circle. ...
- Chords are equidistant from the center if and only if their lengths are equal.
- A chord's perpendicular bisector passes the center.
- 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).
For the numerical analysis algorithm, see bisection method. ...
The power of a point A (circle power,power of a circle) with respect to a circle with center 0 and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. ...
Chords in trigonometry 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. Image File history File links TrigonometricChord. ...
Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek Trigona = three angles and metron = measure [1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...
For the Athenian tyrant, see Hipparchus (son of Pisistratus). ...
The chord function is defined geometrically as in the picture to the left. The chord of an angle is the length of the chord between two points on a unit circle separated by that angle. By taking one of the angles to be zero, it can easily be related to the modern sine function: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
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, not extant, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...
| Name | Sine-based | Chord-based | | Pythagorean | sin2θ + cos2θ = 1 |  | | Half-angle |  |  | The half-angle identity greatly expedites the creation of chord tables. Ancient chord tables typically used a large value for the radius of the circle, and reported the chords for this circle. It was then a simple matter of scaling to determine the necessary chord for any circle. According to G. J. Toomer, Hipparchus used a circle of radius 3438' (=3438/60=57.3). This value is extremely close to 180 / π (=57.29577951...). One advantage of this choice of radius was that he could very accurately approximate the chord of a small angle as the angle itself. In modern terms, it allowed a simple linear approximation: Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ...
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