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52-661: GCS may refer to: Cartography [ edit ] Galactic coordinate system Geographic coordinate system Computing [ edit ] Game creation system Gauss Centre for Supercomputing , in Germany Google Cloud Storage Group communication system Group Control System , an IBM VM Operating system component Education [ edit ] Gadsden County School District , in Florida, United States Gallantry Cross, Silver , an honour of

104-540: A Canadian Forces medal Grand Central Station , in New York City Satellite ground control station UAV ground control station Global Civic Sharing , a South Korean charity Global Combat Ship , of the Royal Navy Gold Coast Suns , an Australian Football League team Green Cove Springs, Florida See also [ edit ] GC (disambiguation) Topics referred to by

156-1347: A constant η equal to 1 inverse radian (1 rad ) in a fashion similar to the introduction of the constant ε 0 . With this change the formula for the angle subtended at the center of a circle, s = rθ , is modified to become s = ηrθ , and the Taylor series for the sine of an angle θ becomes: Sin ⁡ θ = sin ⁡   x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = η θ − ( η θ ) 3 3 ! + ( η θ ) 5 5 ! − ( η θ ) 7 7 ! + ⋯ , {\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,} where x = η θ = θ / rad {\displaystyle x=\eta \theta =\theta /{\text{rad}}}

208-412: A deviation from a straight line ; the second, angle as quantity, by Carpus of Antioch , who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship. In mathematical expressions , it is common to use Greek letters ( α , β , γ , θ , φ , . . . ) as variables denoting the size of some angle (the symbol π

260-451: A full turn are not equivalent. To measure an angle θ , a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses . The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle: θ = s r r a d . {\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .} Conventionally, in mathematics and

312-527: A galactic north pole at RA 12  40 , dec +28° (in the B1900.0 epoch convention) and a 0° longitude at the point where the galactic plane and equatorial plane intersected. In 1958, the International Astronomical Union (IAU) defined the galactic coordinate system in reference to radio observations of galactic neutral hydrogen through the hydrogen line , changing the definition of

364-419: A point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of

416-414: A straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x , the measure of angle C would be 180° − x . Similarly, the measure of angle D would be 180° − x . Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D , either of these angle measures may be used to determine

468-502: A triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle. The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure . In some contexts, such as identifying

520-421: A two-dimensional Cartesian coordinate system , an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis , while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward

572-427: Is ⁠ 1 / 256 ⁠ of a turn. Plane angle may be defined as θ = s / r , where θ is the magnitude in radians of the subtended angle, s is circular arc length, and r is radius. One radian corresponds to the angle for which s = r , hence 1 radian = 1 m/m = 1. However, rad is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using

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624-407: Is declination . NGP refers to the coordinate values of the north galactic pole and NCP to those of the north celestial pole. The reverse (galactic to equatorial) can also be accomplished with the following conversion formulas. Where: In some applications use is made of rectangular coordinates based on galactic longitude and latitude and distance. In some work regarding the distant past or future

676-484: Is "pedagogically unsatisfying". In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s ), and torsional stiffness (N⋅m/rad), and not in

728-401: Is a right-handed system (positive towards the east and towards the north galactic pole); in the other, the U axis is directed toward the galactic anticenter ( l = 180°), and it is a left-handed system (positive towards the east and towards the north galactic pole). The galactic equator runs through the following constellations : Angle In Euclidean geometry , an angle is

780-496: Is clear that the complete form is meant. Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1 . This radian convention allows the omission of η in mathematical formulas. It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference. In

832-493: Is different from Wikidata All article disambiguation pages All disambiguation pages Galactic coordinate system Longitude (symbol l ) measures the angular distance of an object eastward along the galactic equator from the Galactic Center. Analogous to terrestrial longitude , galactic longitude is usually measured in degrees (°). Latitude (symbol b ) measures the angle of an object northward of

884-410: Is in the interior of angle AOC, then m ∠ A O C = m ∠ A O B + m ∠ B O C {\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} } I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. Three special angle pairs involve

936-511: 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. Throughout history, angles have been measured in various units . These are known as angular units , with the most contemporary units being the degree ( ° ), the radian (rad), and the gradian (grad), though many others have been used throughout history . Most units of angular measurement are defined such that one turn (i.e.,

988-459: Is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, ∠BAC always refers to the anticlockwise (positive) angle from B to C about A and ∠CAB the anticlockwise (positive) angle from C to B about A. There is some common terminology for angles, whose measure is always non-negative (see § Signed angles ): The names, intervals, and measuring units are shown in

1040-462: Is the angle in radians. The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while sin is the traditional function on pure numbers which assumes its argument is a dimensionless number in radians. The capitalised symbol Sin {\displaystyle \operatorname {Sin} } can be denoted sin {\displaystyle \sin } if it

1092-404: Is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In navigation , bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so

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1144-447: Is typically not used for this purpose to avoid confusion with the constant denoted by that symbol ). Lower case Roman letters ( a ,  b ,  c , . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example,

1196-563: 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 that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus , an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes , who regarded an angle as

1248-678: The SI , the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted. The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form ⁠ k / 2 π ⁠ , where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians ): θ = k 2 π ⋅ s r . {\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.} The value of θ thus defined

1300-484: The area of a circle , π r . The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use. In particular, Quincey identifies Torrens' proposal to introduce

1352-446: The cotangent of its complement, and its secant equals the cosecant of its complement.) The prefix " co- " in the names of some trigonometric ratios refers to the word "complementary". If the two supplementary angles are adjacent (i.e., have a common vertex and share just one side), their non-shared sides form a straight line . Such angles are called a linear pair of angles . However, supplementary angles do not have to be on

1404-464: The equatorial pole . The galactic longitude increases in the same direction as right ascension. Galactic latitude is positive towards the north galactic pole, with a plane passing through the Sun and parallel to the galactic equator being 0°, whilst the poles are ±90°. Based on this definition, the galactic poles and equator can be found from spherical trigonometry and can be precessed to other epochs ; see

1456-475: The Galactic longitude by 32° and the latitude by 1.5°. In the equatorial coordinate system , for equinox and equator of 1950.0 , the north galactic pole is defined at right ascension 12  49 , declination +27.4°, in the constellation Coma Berenices , with a probable error of ±0.1°. Longitude 0° is the great semicircle that originates from this point along the line in position angle 123° with respect to

1508-1764: The Republic of Venda Gaston Christian School , in Lowell, North Carolina, United States German Church School , in Addis Ababa, Ethiopia Glenelg Country School , in Ellicott City, Maryland, United States Gorey Community School , in County Wexford, Ireland Government College of Science, Lahore , Pakistan Grace Christian School (Florida) , in Valrico, Florida, United States Grace Church School , in New York City Granville County Schools , in North Carolina, United States Greenfield Community School , in Dubai Greenville Christian School , in Mississippi, United States Greenwood College School , in Toronto, Ontario, Canada Guadalupe Catholic School , in Makati, Philippines Guildford County School , in England Medicine [ edit ] Gamma-glutamylcysteine synthetase Gender confirming surgery Glasgow Coma Scale Glucocorticosteroids Glycine cleavage system Other uses [ edit ] General Campaign Star (Canada) ,

1560-468: The adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: When two adjacent angles form

1612-409: The angle of the rays lying tangent to the respective curves at their point of intersection. The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius , and may be a negative number . In the case of a geometric angle, the arc is centered at the vertex and delimited by

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1664-558: The angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n . Two exceptions are the radian (and its decimal submultiples) and the diameter part. In the International System of Quantities , an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis . The following table lists some units used to represent angles. It

1716-457: The angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or B A C ^ {\displaystyle {\widehat {\rm {BAC}}}} . Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). In other ways, an angle denoted as, say, ∠BAC might refer to any of four angles:

1768-494: The area of a circular sector θ = 2 A / r gives 1 radian as 1 m /m = 1. The key fact is that the radian is a dimensionless unit equal to 1 . In SI 2019, the SI radian is defined accordingly as 1 rad = 1 . It is a long-established practice in mathematics and across all areas of science to make use of rad = 1 . Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of

1820-407: The clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see § Signed angles ). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees

1872-400: 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 are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles . Two intersecting curves may also define an angle, which is

1924-450: The final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation , which

1976-403: The galactic coordinate system does not rotate with time, Sgr A* is actually decreasing in longitude at the rate of galactic rotation at the sun, Ω , approximately 5.7 milliarcseconds per year (see Oort constants ). An object's location expressed in the equatorial coordinate system can be transformed into the galactic coordinate system. In these equations, α is right ascension , δ

2028-438: The galactic coordinate system is taken as rotating so that the x -axis always goes to the centre of the galaxy. There are two major rectangular variations of galactic coordinates, commonly used for computing space velocities of galactic objects. In these systems the xyz -axes are designated UVW , but the definitions vary by author. In one system, the U axis is directed toward the Galactic Center ( l = 0°), and it

2080-444: The galactic equator (or midplane) as viewed from Earth. Analogous to terrestrial latitude , galactic latitude is usually measured in degrees (°). The first galactic coordinate system was used by William Herschel in 1785. A number of different coordinate systems, each differing by a few degrees, were used until 1932, when Lund Observatory assembled a set of conversion tables that defined a standard galactic coordinate system based on

2132-565: The measure of Angle B . Using the measure of either angle C or angle D , we find the measure of angle B to be 180° − (180° − x ) = 180° − 180° + x = x . Therefore, both angle A and angle B have measures equal to x and are equal in measure. A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles , interior angles , alternate exterior angles , alternate interior angles , corresponding angles , and consecutive interior angles . The angle addition postulate states that if B

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2184-467: The negative y -axis. When Cartesian coordinates are represented by standard position , defined by the x -axis rightward and the y -axis upward, positive rotations are anticlockwise , and negative cycles are clockwise . In many contexts, an angle of − θ is effectively equivalent to an angle of "one full turn minus θ ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although

2236-437: The quantities of torque (N⋅m) and angular momentum (kg⋅m /s). At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for

2288-404: The radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimetres, where r is the magnitude of the radius of the pulley in centimetres and θ is the magnitude of the angle through which the pulley turns in radians. When multiplying r by θ , the unit radian does not appear in the product, nor does

2340-582: The same line and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. In Euclidean geometry, any sum of two angles in

2392-524: The same number of digits as the table, 17  45.7 , −29.01° (J2000), there is an offset of about 0.07° from the defined coordinate center, well within the 1958 error estimate of ±0.1°. Due to the Sun's position, which currently lies 56.75 ± 6.20  ly north of the midplane, and the heliocentric definition adopted by the IAU, the galactic coordinates of Sgr A* are latitude +0° 07′ 12″ south, longitude 0° 04′ 06″ . Since as defined

2444-403: The same term [REDACTED] This disambiguation page lists articles associated with the title GCS . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=GCS&oldid=1202482214 " Category : Disambiguation pages Hidden categories: Short description

2496-588: 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 include the Greek ἀγκύλος ([ankylοs] Error: {{Lang}}: Non-latn text/Latn script subtag mismatch ( help ) ) meaning "crooked, curved" and the English word " ankle ". Both are connected with

2548-976: The summation of angles: The adjective complementary is from the Latin complementum , associated with the verb complere , "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed the complement of the angle. If angles A and B are complementary, the following relationships hold: sin 2 ⁡ A + sin 2 ⁡ B = 1 cos 2 ⁡ A + cos 2 ⁡ B = 1 tan ⁡ A = cot ⁡ B sec ⁡ A = csc ⁡ B {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (The tangent of an angle equals

2600-422: The table below: When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other. The equality of vertically opposite angles is called the vertical angle theorem . Eudemus of Rhodes attributed the proof to Thales of Miletus . The proposition showed that since both of a pair of vertical angles are supplementary to both of

2652-463: The table. The IAU recommended that during the transition period from the old, pre-1958 system to the new, the old longitude and latitude should be designated l and b while the new should be designated l and b . This convention is occasionally seen. Radio source Sagittarius A* , which is the best physical marker of the true Galactic Center , is located at 17  45  40.0409 , −29° 00′ 28.118″ (J2000). Rounded to

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2704-489: The unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the angular velocity of a rolling wheel, ω = v / r , radians appear in the units of ω but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge

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