The Angles (Latin: Angli; German: Angeln) were one of the main Germanic peoples who settled in Great Britain in the post-Roman period. They founded several of the kingdoms of Anglo-Saxon England, and their name is the root of the name England. The name comes from Anglia, a peninsula located on the Baltic shore of what is now Schleswig-Holstein.
The name of the Angles may have been first recorded in Latinised form, as Anglii, in the Germania of Tacitus. It is thought to derive from the name of the area they originally inhabited, the Anglia Peninsula (Angeln in modern German, Angel in Danish). This name has been hypothesised to originate from the Germanic root for "narrow" (compare German and Dutch eng = "narrow"), meaning "the Narrow [Water]", i.e., the Schlei estuary; the root would be *h₂enǵʰ, "tight". Another theory is that the name meant "hook" (as in angling for fish), in reference to the shape of the peninsula; Indo-European linguist Julius Pokorny derives it from Proto-Indo-European *h₂enk-, "bend" (see ankle).
During the fifth century, all Germanic tribes who invaded Britain were referred to as Englisc, who were speakers of Old English (which was known as Englisc, Ænglisc, or Anglisc). Englisc and its descendant, English, also goes back to Proto-Indo-European *h₂enǵʰ-, meaning narrow. In any case, the Angles may have been called such because they were a fishing people or were originally descended from such, so England would mean "land of the fishermen", and English would be "the fishermen's language".
Gregory the Great, in an epistle, simplified the Latinised name Anglii to Angli, the latter form developing into the preferred form of the word. The country remained Anglia in Latin. Alfred the Great's translation of Orosius's history of the world uses Angelcynn (-kin) to describe the English people; Bede used Angelfolc (-folk); also such forms as Engel, Englan (the people), Englaland, and Englisc occur, all showing i-mutation.
The earliest recorded mention of the Angles may be in chapter 40 of Tacitus's Germania written around AD 98. Tacitus describes the "Anglii" as one of the more remote Suebic tribes compared to the Semnones and Langobardi, who lived on the Elbe and were better known to the Romans. He grouped the Angles with several other tribes in that region, the Reudigni, Aviones, Varini, Eudoses, Suarini, and Nuitones. These were all living behind ramparts of rivers and woods, and therefore inaccessible to attack.
He gives no precise indication of their geographical situation, but states that, together with the six other tribes, they worshiped Nerthus, or Mother Earth, whose sanctuary was located on "an island in the Ocean". The Eudoses are the Jutes; these names probably refer to localities in Jutland or on the Baltic coast. The coast contains sufficient estuaries, inlets, rivers, islands, swamps, and marshes to have been then inaccessible to those not familiar with the terrain, such as the Romans, who considered it unknown, inaccessible, with a small population and of little economic interest.
The majority of scholars believe that the Anglii lived on the coasts of the Baltic Sea, probably in the southern part of the Jutish peninsula. This view is based partly on Old English and Danish traditions regarding persons and events of the fourth century, and partly because striking affinities to the cult of Nerthus as described by Tacitus are to be found in pre-Christian Scandinavian religion.
Ptolemy, writing in around 150 AD, in his atlas Geography (2.10), describes the Sueboi Angeilloi, Latinised to Suevi Angili, further south, living in a stretch of land between the northern Rhine and central Elbe, but apparently not touching either river, with the Suebic Langobardi on the Rhine to their west, and the Suebic Semnones on the Elbe stretching to their east.
These Suevi Angili would have been in Lower Saxony or near it, but they are not coastal. The three Suebic peoples are separated from the coastal Chauci (between the Ems and the Elbe), and Saxones (east of the Elbe mouth), by a series of tribes including, between the Weser and Elbe, the Angrivarii, "Laccobardi" (probably another reference to the Langobardi, but taken by Ptolemy from another source), and the Dulgubnii. South of the Saxons, and east of the Elbe, Ptolemy lists the "Ouirounoi" (Latinised as Viruni, and probably the Varini) and Teutonoari, which either denotes "the Teuton men", or else it denotes people living in the area where the Teutons had previously lived (whom Ptolemy attests as still living to the east of the Teutonoari). Ptolemy describes the coast to the east of the Saxons as inhabited by the Farodini, a name not known from any other sources.
Owing to the uncertainty of this passage, much speculation existed regarding the original home of the Anglii. One theory is that they or part of them dwelt or moved among other coastal people, perhaps confederated up to the basin of the Saale (in the neighbourhood of the ancient canton of Engilin) on the Unstrut valleys below the Kyffhäuserkreis, from which region the Lex Anglorum et Werinorum hoc est Thuringorum is believed by many to have come. The ethnic names of Frisians and Warines are also attested in these Saxon districts.
A second possible solution is that these Angles of Ptolemy are not those of Schleswig at all. According to Julius Pokorny, the Angri- in Angrivarii, the -angr in Hardanger and the Angl- in Anglii all come from the same root meaning "bend", but in different senses. In other words, the similarity of the names is strictly coincidental and does not reflect any ethnic unity beyond Germanic.
However, Gudmund Schütte, in his analysis of Ptolemy, believes that the Angles have simply been moved by an error coming from Ptolemy's use of imperfect sources. He points out that Angles are placed correctly just to the northeast of the Langobardi, but that these have been duplicated, so that they appear once, correctly, on the lower Elbe, and a second time, incorrectly, at the northern Rhine.
Bede states that the Anglii, before coming to Great Britain, dwelt in a land called Angulus, "which lies between the province of the Jutes and the Saxons, and remains unpopulated to this day." Similar evidence is given by the Historia Brittonum. King Alfred the Great and the chronicler Æthelweard identified this place with Anglia, in the province of Schleswig (Slesvig) (though it may then have been of greater extent), and this identification agrees with the indications given by Bede.
In the Norwegian seafarer Ohthere of Hålogaland's account of a two-day voyage from the Oslo fjord to Schleswig, he reported the lands on his starboard bow, and Alfred appended the note "on these islands dwelt the Engle before they came hither".[n 1] Confirmation is afforded by English and Danish traditions relating to two kings named Wermund and Offa of Angel, from whom the Mercian royal family claimed descent and whose exploits are connected with Anglia, Schleswig, and Rendsburg. Danish tradition has preserved record of two governors of Schleswig, father and son, in their service, Frowinus (Freawine) and Wigo (Wig), from whom the royal family of Wessex claimed descent. During the fifth century, the Anglii invaded Great Britain, after which time their name does not recur on the continent except in the title of the legal code issued to the Thuringians: Lex Anglorum et Werinorum hoc est Thuringorum.
The Angles are the subject of a legend about Pope Gregory I, who happened to see a group of Angle children from Deira for sale as slaves in the Roman market. As the story would later be told by the Anglo-Saxon monk and historian Bede, Gregory was struck by the unusual appearance of the slaves and asked about their background. When told they were called "Anglii" (Angles), he replied with a Latin pun that translates well into English: “Bene, nam et angelicam habent faciem, et tales angelorum in caelis decet esse coheredes” ("It is well, for they have an angelic face, and such people ought to be co-heirs of the angels in heaven"). Supposedly, this encounter inspired the pope to launch a mission to bring Christianity to their countrymen.
The province of Schleswig has proved rich in prehistoric antiquities that date apparently from the fourth and fifth centuries. A large cremation cemetery has been found at Borgstedt, between Rendsburg and Eckernförde, and it has yielded many urns and brooches closely resembling those found in pagan graves in England. Of still greater importance are the great deposits at Thorsberg moor (in Anglia) and Nydam, which contained large quantities of arms, ornaments, articles of clothing, agricultural implements, etc., and in Nydam, even ships. By the help of these discoveries, Angle culture in the age preceding the invasion of Britannia can be pieced together.
According to sources such as the History of Bede, after the invasion of Britannia, the Angles split up and founded the kingdoms of Northumbria, East Anglia, and Mercia. H.R. Loyn has observed in this context that "a sea voyage is perilous to tribal institutions", and the apparently tribe-based kingdoms were formed in England. Early times had two northern kingdoms (Bernicia and Deira) and two midland ones (Middle Anglia and Mercia), which had by the seventh century resolved themselves into two Angle kingdoms, viz., Northumbria and Mercia. Northumbria held suzerainty amidst the Teutonic presence in the British Isles in the seventh century, but was eclipsed by the rise of Mercia in the eighth century. Both kingdoms fell in the great assaults of the Danish Viking armies in the 9th century. Their royal houses were effectively destroyed in the fighting, and their Angle populations came under the Danelaw. Further south, the Saxon kings of Wessex withstood the Danish assaults. Then in the late 9th and early 10th centuries, the kings of Wessex defeated the Danes and liberated the Angles from the Danelaw. They united their house in marriage with the surviving Angle royalty, and were accepted by the Angles as their kings. This marked the passing of the old Anglo-Saxon world and the dawn of the "English" as a new people. The regions of East Anglia and Northumbria are still known by their original titles. Northumbria once stretched as far north as what is now southeast Scotland, including Edinburgh, and as far south as the Humber Estuary.
The rest of that people stayed at the centre of the Angle homeland in the northeastern portion of the modern German Bundesland of Schleswig-Holstein, on the Jutland Peninsula. There, a small peninsular area is still called Anglia today and is formed as a triangle drawn roughly from modern Flensburg on the Flensburger Fjord to the City of Schleswig and then to Maasholm, on the Schlei inlet.
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In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.
Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
The word angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.
In elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects.
Two line segments are congruent if they have the same length.
Two angles are congruent if they have the same measure.
Two circles are congruent if they have the same diameter.In this sense, two plane figures are congruent implies that their corresponding characteristics are "congruent" or "equal" including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters and areas.
The related concept of similarity applies if the objects have the same shape but do not necessarily have the same size. (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.)England
England is a country that is part of the United Kingdom. It shares land borders with Wales to the west and Scotland to the north-northwest. The Irish Sea lies west of England and the Celtic Sea lies to the southwest. England is separated from continental Europe by the North Sea to the east and the English Channel to the south. The country covers five-eighths of the island of Great Britain, which lies in the North Atlantic, and includes over 100 smaller islands, such as the Isles of Scilly and the Isle of Wight.
The area now called England was first inhabited by modern humans during the Upper Palaeolithic period, but takes its name from the Angles, a Germanic tribe deriving its name from the Anglia peninsula, who settled during the 5th and 6th centuries. England became a unified state in the 10th century, and since the Age of Discovery, which began during the 15th century, has had a significant cultural and legal impact on the wider world. The English language, the Anglican Church, and English law – the basis for the common law legal systems of many other countries around the world – developed in England, and the country's parliamentary system of government has been widely adopted by other nations. The Industrial Revolution began in 18th-century England, transforming its society into the world's first industrialised nation.England's terrain is chiefly low hills and plains, especially in central and southern England. However, there is upland and mountainous terrain in the north (for example, the Lake District and Pennines) and in the west (for example, Dartmoor and the Shropshire Hills). The capital is London, which has the largest metropolitan area in both the United Kingdom and the European Union. England's population of over 55 million comprises 84% of the population of the United Kingdom, largely concentrated around London, the South East, and conurbations in the Midlands, the North West, the North East, and Yorkshire, which each developed as major industrial regions during the 19th century.The Kingdom of England – which after 1535 included Wales – ceased being a separate sovereign state on 1 May 1707, when the Acts of Union put into effect the terms agreed in the Treaty of Union the previous year, resulting in a political union with the Kingdom of Scotland to create the Kingdom of Great Britain. In 1801, Great Britain was united with the Kingdom of Ireland (through another Act of Union) to become the United Kingdom of Great Britain and Ireland. In 1922 the Irish Free State seceded from the United Kingdom, leading to the latter being renamed the United Kingdom of Great Britain and Northern Ireland.Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas.Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra.
Any orientation can be achieved by composing three elemental rotations, i.e. rotations about the axes of a coordinate system. Euler angles can be defined by three of these rotations. They can also be defined by elemental geometry and the geometrical definition demonstrates that three rotations are always sufficient to reach any frame.
The three elemental rotations may be extrinsic (rotations about the axes xyz of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system XYZ, solidary with the moving body, which changes its orientation after each elemental rotation).
Euler angles are typically denoted as α, β, γ, or φ, θ, ψ. Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles should always be preceded by their definition.
Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups:
Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)
Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z).Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called proper or classic Euler angles.Hells Angels
The Hells Angels Motorcycle Club (HAMC) is a worldwide one-percenter motorcycle club whose members typically ride Harley-Davidson motorcycles. The organization is predominantly white males and considered to be an organized crime syndicate by the United States Department of Justice. In the United States and Canada, the Hells Angels are incorporated as the Hells Angels Motorcycle Corporation. Common nicknames for the club are the "H.A.", "Red & White", "HAMC" and "81".Kingdom of East Anglia
The Kingdom of the East Angles (Old English: Ēast Engla Rīce; Latin: Regnum Orientalium Anglorum), today known as the Kingdom of East Anglia, was a small independent kingdom of the Angles comprising what are now the English counties of Norfolk and Suffolk and perhaps the eastern part of the Fens. The kingdom formed in the 6th century in the wake of the Anglo-Saxon settlement of Britain. It was ruled by the Wuffingas in the 7th and 8th centuries, but fell to Mercia in 794, and was conquered by the Danes in 869, forming part of the Danelaw. It was conquered by Edward the Elder and incorporated into the Kingdom of England in 918.Les Angles, Hautes-Pyrénées
Les Angles is a commune in the Hautes-Pyrénées department in southwestern France.List of trigonometric identities
In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.Molecular geometry
Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determine the position of each atom.
Molecular geometry influences several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of molecule, i.e. they can be understood as approximately local and hence transferable properties.Parallelogram
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.Perpendicular
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.
A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.
Perpendicularity easily extends to segments and rays. For example, a line segment is perpendicular to a line segment if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, means line segment AB is perpendicular to line segment CD. For information regarding the perpendicular symbol see Up tack.
A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.
Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle (90 degrees).
Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal.Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.Rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.Right angle
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.
Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry.Scapula
In anatomy, the scapula (plural scapulae or scapulas), also known as shoulder bone, shoulder blade, wing bone or blade bone, is the bone that connects the humerus (upper arm bone) with the clavicle (collar bone). Like their connected bones the scapulae are paired, with the scapula on either side of the body being roughly a mirror image of the other. The name derives from early Roman times when it was thought that the bone resembled a trowel or small shovel.
In compound terms, the prefix omo- is used for the shoulder blade in Latin medical terminology. The prefix is derived from ὦμος (ōmos), the Ancient Greek word for shoulder, and is cognate with the Latin (h)umerus.
The scapula forms the back of the shoulder girdle. In humans, it is a flat bone, roughly triangular in shape, placed on a posterolateral aspect of the thoracic cage.Square
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .
In Euclidean geometry any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted.Trigonometry
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") 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. In particular, 3rd-century astronomers first noted that the ratio of the lengths of two sides of a right-angled triangle depends only of one acute angles of the triangle. These dependencies are now called trigonometric functions.
Trigonometry is the foundation of all applied geometry, including geodesy, surveying, celestial mechanics, solid mechanics, navigation.
Trigonometric functions have been extended as functions of a real or complex variable, which are today pervasive in all mathematics.
History of the Germanic peoples
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