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This article is about angles in geometry. For other uses, see Angle (disambiguation).
"∠", the angle symbol.
"∠", the angle symbol.

In geometry and trigonometry, an angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle (Sidorov 2001). Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Geometry, a vertex (plural "vertices" is a special kind of point. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).

The word angle comes from the Latin word angulus, meaning "a corner". Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Latin angere, meaning "to compress into a bend" or "to strangle", the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle. Cognates in Linguistics are words that have a common origin They may occur within a language such as shirt and skirt as two English words descended from Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States In Human anatomy, the ankle Joint is formed where the Foot and the leg meet " All three are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow" (Slocum 2007).

Contents

History

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. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry According to Proclus an angle must be either a quality or a quantity, or a relationship. Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos 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. Eudemus is also the name of a general of Alexander the Great (died 316 BC Carpus of Antioch was an ancient Greek Mathematician. It is not certain when he lived he may have lived any time between the 2nd century BC and the

Measuring angles

The angle θ is the quotient of s and r.
The angle θ is the quotient of s and r.

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e. In Geometry, an arc is a closed segment of a Differentiable Curve in the two-dimensional plane; for example a circular g. with a pair of compasses. A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):

\theta = \frac{s}{r}(k).

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.

In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral using polar coordinates, an extra full turn gives rise to a quite different point on the curve. In Mathematics, a spiral is a Curve which emanates from a central point getting progressively farther away as it revolves around the point In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

Units

Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above. Of these units, treated in more detail below, the degree and the radian are by far the most common.

With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i. e. one revolution) is equal to n units, for some whole number n. For example, in the case of degrees, n = 360. A full circle of n units is obtained by setting k = n/(2π) in the formula above. (Proof. The formula above can be rewritten as k = θr/s. One full circle, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Substituting n for θ and 2πr for s in the formula, results in k = nr/(2πr) = n/(2π). )

θ = s/r rad = 1 rad.
θ = s/r rad = 1 rad.

Positive and negative angles

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. A clockwise motion is one that proceeds 'like the Clock 's hands' from the top to the right then down and then to the left and back to the top A clockwise motion is one that proceeds 'like the Clock 's hands' from the top to the right then down and then to the left and back to the top If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In many geometrical situations a negative angle of −θ is effectively equivalent to a positive angle of "one full rotation less θ". For example, a clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to a counterclockwise rotation of 360° − 45° (that is, an angle of 315°).

In three dimensional geometry, "clockwise" and "counterclockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Navigation is the process of reading and controlling the movement of a craft or vehicle from one place to another In Navigation, a bearing is the direction one object is from another object Negative bearings are not used in navigation, so north-west is 315 degrees.

Approximations

Identifying angles

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early . . ) to serve as variables standing for the size of some angle. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. (To avoid confusion with its other meaning, the symbol π is not used for this purpose. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems ) Lower case roman letters (a, b, c, . . . ) are also used. See the figures in this article for examples.

In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i. e. the lines from point A to point B and point A to point C) is denoted ∠BAC or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").

Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180° degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.

Types of angles

Right angle.
Right angle.
Reflex angle.
Reflex angle.
The complementary angles a and b (b is the complement of a, and a is the complement of b).
The complementary angles a and b (b is the complement of a, and a is the complement of b). A pair of Angles is complementary if the sum of their measures add up to 90 degrees.
Acute (a), obtuse (b), and straight (c) angles. Here, a and b are supplementary angles.
Acute (a), obtuse (b), and straight (c) angles. Here, a and b are supplementary angles. A pair of Angles is supplementary if their measurements add up to 180 degrees If the two supplementary angles are adjacent (i

A formal definition

Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if θ is a Euclidean angle, it is true that

\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}

and

\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}

for two numbers x and y. So an angle in the Euclidean plane can be legitimately given by two numbers x and y.

To the ratio \frac{y}{x} there correspond two angles in the geometric range 0 < θ < 2π, since

\frac{\sin \theta }{\cos \theta } = \frac{\frac{y}{\sqrt{x^2 + y^2}}}{\frac{x}{\sqrt{x^2 + y^2}}} = \frac{y}{x} = \frac{-y}{-x} = \frac{\sin (\theta + \pi)}{\cos (\theta + \pi) }.

Using rotations

Suppose we have two unit vectors \vec{u} and \vec{v} in the euclidean plane \mathbb{R}^2. Then there exists one positive isometry (a rotation), and one only, from \mathbb{R}^2 to \mathbb{R}^2 that maps u onto v. For the Mechanical engineering and Architecture usage see Isometric projection. Let r be such a rotation. Then the relation \vec{a}\mathcal{R}\vec{b} defined by \vec{b}=r(\vec{a}) is an equivalence relation and we call angle of the rotation r the equivalence class \mathbb{T}/\mathcal{R}, where \mathbb{T} denotes the unit circle of \mathbb{R}^2. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector (1,0), then for any point M on \mathbb{T} at distance θ from (1,0) (on the circle), let \vec{u}=\overrightarrow{OM}. If we call rθ the rotation that transforms (1,0) into \vec{u}, then \left[r_\theta\right]\mapsto\theta is a bijection, which means we can identify any angle with a number between 0 and 2π.

Angles between curves

The angle between the two curves is defined as the angle between the tangents A and B at P
The angle between the two curves is defined as the angle between the tangents A and B at P

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.

The dot product and generalisation

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

\mathbf{u} \cdot \mathbf{v} = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. This article assumes some familiarity with Analytic geometry and the concept of a limit.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Elliptic geometry is also sometimes called Riemannian geometry. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Where U and V are tangent vectors and gij are the components of the metric tensor G,

\cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}.

Angles in geography and astronomy

In geography we specify the location of any point on the Earth using a Geographic coordinate system. Geography (from Greek γεωγραφία - geografia) is the study of the Earth and its lands features inhabitants and phenomena A geographic coordinate system enables every location on the Earth to be specified in three coordinates using mainly a spherical coordinate system. This system specifies the latitude and longitude of any location, in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references. Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the The Prime Meridian is the meridian (line of Longitude) at which longitude is defined to be 0°

In astronomy, we similarly specify a given point on the celestial sphere using any of several Astronomical coordinate systems, where the references vary according to the particular system. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study In Astronomy and Navigation, the celestial sphere is an imaginary rotating Sphere of "gigantic Radius " Astronomical coordinate systems are Coordinate systems used in astronomy to describe the location of objects in the sky and in the universe

Astronomers can also measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The angle between those lines can be measured, and is the angular separation between the two stars.

Astronomers also measure the apparent size of objects. The angular diameter of an object as seen from a given position is the "visual diameter" of the object measured as an angle For example, the full moon has an angular measurement of approximately 0. Full moon is a Lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun. 5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree. " The small-angle formula can be used to convert such an angular measurement into a distance/size ratio. Small-angle approximation is a useful simplification of the laws of Trigonometry which is only approximately true for finite angles but correct in the limit as the

References

See also

External links

Dictionary

angle

-noun

  1. (geometry) A figure formed by two rays which start from a common point (a plane angle) or by three planes that intersect (a solid angle).
  2. (geometry) The measure of such a figure. In the case of a plane angle, this is the ratio (or proportional to the ratio) of the arc length to the radius of a section of a circle cut by the two rays, centered at their common point. In the case of a solid angle, this is the ratio of the surface area to the square of the radius of the section of a sphere.
  3. A corner where two walls intersect.
  4. A change in direction.
  5. A viewpoint.
  6. (media) The focus of a news story, either in print or broadcasting.
  7. (slang, professional wrestling) A storyline between two wrestlers, providing the background for and approach to a feud.
  8. (slang) A scheme; a means of benefitting from a situation, usually hidden, possibly illegal.

-verb

  1. (transitive, often in the passive) To place (something) at an angle.
  2. (intransitive)To try to catch fish with a hook and line.
  3. (intransitive, informal) To change direction rapidly.

Angle

-noun

  1. A member of an ancient Germanic tribe, one of several which invaded Britain and merged to become the Anglo-Saxons.
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