Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 117
1.
Acute Angle -- from MathWorld - Weisstein, Eric W.
An angle of less than \pi/2 radians (90°) is called an acute angle. See also: Acute Triangle, Angle, Full Angle, Obtuse Angle, Reflex Angle, Right Angle, Straight Angle
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Angle -- from MathWorld - Weisstein, Eric W.
Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle \theta between them. Angles are usually measured in degrees (denoted {}^\circ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad). One full rotation in these three measures corresponds to 360°, 2\pi rad, or 400 grad. Half a full rotation is called a straight angle, and a...
4.
Angle Bisector -- from MathWorld - Weisstein, Eric W.
The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle into two equal parts. The angle bisectors meet at the incenter I, which has trilinear coordinates 1:1:1. The length t_1 of the bisector A_1T_1 of angle A_1 in the above triangle \Delta A_1A_2A_3 is given by t_1^2=a_2 a_3\left[{1-{a_1^2\over(a_2+a_3)^2}}\right], where t_i\equiv \overline{A_iT_i} and a_i\equiv\overline{A_jA_k}. The...
5.
Angle Bisector Theorem -- from MathWorld - Weisstein, Eric W.
The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle.
6.
Angle Trisection -- from MathWorld - Weisstein, Eric W.
Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836). Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as \pi/2 and \pi radians (90° and 180°, respectively), which can be trisected. Furthermore, some...
7.
Antigonal Points -- from MathWorld - Weisstein, Eric W.
Given \angle AXB+\angle AYB=\pi radians in the above figure, then X and Y are said to be antigonal points with respect to A and B.
8.
Antiparallel -- from MathWorld - Weisstein, Eric W.
Two lines PQ and RS are said to be antiparallel with respect to the sides of an angle A if they make the same angle in the opposite senses with the bisector of that angle. If PQ and RS are antiparallel with respect to PR and QS, then the latter are also antiparallel with respect to the former. Furthermore, if PQ and RS are antiparallel, then the points P, Q, R, and S are concyclic (Johnson 1929, p. 172; Honsberger 1995, pp. 87-88). There are a number of fundamental relationships involving a...
9.
Arc Minute -- from MathWorld - Weisstein, Eric W.
A unit of angular measure equal to 60 arc seconds, or 1/60 of a degree. The arc minute is denoted {}^{\prime} (not to be confused with the symbol for feet ). See also: Arc Second, Degree
10.
Arc Second -- from MathWorld - Weisstein, Eric W.
A unit of angular measure equal to 1/60 of an arc minute, or 1/3600 of a degree. The arc second is denoted {}^{\prime\prime} (not to be confused with the symbol for inches ). See also: Arc Minute, Degree
11.
Central Angle -- from MathWorld - Weisstein, Eric W.
A central angle is an angle \angle AOC with endpoints A and C located on a circle's circumference and vertex O located at the circle's center (Rhoad et al. 1984, p. 420). A central angle in a circle determines an arc {\mathrel{\mathop{AC}\limits^\frown}}. For an inscribed angle \angle ABC and central angle \angle AOC with the same endpoints, \angle AOC = 2\angle ABC (Jurgensen et al. 1963, p. 328). See also: Arc, Inscribed Angle, Vertex Angle
12.
Centroidal Line -- from MathWorld - Weisstein, Eric W.
The three planes determined by the edges of a trihedron and the internal bisectors of the respectively opposite faces are coaxal, and the common line of these planes is called the centroidal line. See also: Trihedron
13.
Colunar Triangle -- from MathWorld - Weisstein, Eric W.
Given a Schwarz triangle (p\ q\ r), replacing each polygon vertex with its antipodes gives the three colunar spherical triangles (p\ q'\ r'), (p'\ q\ r'), (p'\ q'\ r), where {1\over p}+{1\over p'}=1 {1\over q}+{1\over q'}=1 {1\over r}+{1\over r'}=1. See also: Schwarz Triangle, Spherical Triangle
14.
Complementary Angle -- from MathWorld - Weisstein, Eric W.
Two angles \alpha and \pi/2-\alpha are said to be complementary. See also: Angle, Right Angle, Supplementary Angle
15.
Contact Angle -- from MathWorld - Weisstein, Eric W.
The angle \alpha between the normal vector of a sphere (or other geometric object) at a point where a plane is tangent to it and the normal vector of the plane. In the above figure, \alpha = \cos^{-1}\left({a\over R}\right) = \sin^{-1}\left({R-h\over R}\right). See also: Spherical Cap
16.
Degree -- from MathWorld - Weisstein, Eric W.
The word "degree" has many meanings in mathematics. The most common meaning is the unit of angle measure defined such that an entire rotation is 360°. This unit harks back to the Babylonians, who used a base 60 number system. 360° likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is subdivided into 60 minutes per degree, and 60 seconds per minute. In Mathematica, the symbol giving the number of radians in one degree...
17.
Directed Angle -- from MathWorld - Weisstein, Eric W.
The symbol \measuredangle ABC denotes the directed angle from AB to BC, which is the signed angle through which AB must be rotated about B to coincide with BC. Four points ABCD lie on a circle (i.e., are concyclic) iff \measuredangle ABC=\measuredangle ADC. It is also true that \measuredangle l_1l_2+\measuredangle l_2l_1=0^\circ {\rm\ or\ } 360^\circ. Three points A, B, and C are collinear iff \measuredangle ABC=0^\circ or 180°. For any four points, A, B, C, and D, \measuredangle...
18.
Direction Cosine -- from MathWorld - Weisstein, Eric W.
Let a be the angle between v and x, b the angle between v and y, and c the angle between v and z. Then the direction cosines are equivalent to the (x,y,z) coordinates of a unit vector \hat\mathbf{v}, \alpha \equiv \cos a \equiv {\mathbf{v}\cdot \hat\mathbf{x}\over \left\vert{\mathbf{v}}\right\vert} \beta \equiv \cos b \equiv {\mathbf{v}\cdot \hat\mathbf{y}\over \left\vert{\mathbf{v}}\right\vert} \gamma \equiv \cos c \equiv {\mathbf{v}\cdot \hat\mathbf{z}\over...
19.
Double-Angle Formulas -- from MathWorld - Weisstein, Eric W.
Formulas expressing trigonometric functions of an angle 2x in terms of functions of an angle x, \sin(2 x) = 2\sin x\cos x \cos(2 x) = \cos^2 x-\sin^2 x = 2\cos^2 x-1 = 1-2\sin^2 x \tan(2 x) = {2\tan x\over 1-\tan^2 x}. The corresponding hyperbolic function double-angle formulas are \sinh(2x) = 2\sinh x\cosh x \cosh(2x) = 2\cosh^2 x-1 \tanh(2x) = {2\tanh x\over 1+\tanh^2 x}. See also: Half-Angle Formulas, Hyperbolic Functions, Multiple-Angle Formulas, Prosthaphaeresis Formulas, Trigonometric...
20.
Exterior Angle -- from MathWorld - Weisstein, Eric W.
An exterior angle \beta of a polygon is the angle formed externally between two adjacent sides. It is therefore equal to 2\pi-\alpha, where \alpha is the corresponding internal angle between two adjacent sides (Zwillinger 1995, p. 270). Consider the angles \gamma_i formed between a side of a polygon and the extension of an adjacent side. Since there are two directions in which a side can be extended, there are two such angles at each vertex. However, since corresponding angles are opposite,...
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