Sistem koordinat polar: Perbedaan antara revisi
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JohnThorne (bicara | kontrib) ←Membuat halaman berisi 'Image:Examples of Polar Coordinates.svg|thumb|250px|Titik-titik dalam sistem koordinat polar dengan kutub/''pole'' ''O'' dan aksis polar ''L''. Warna hijau: titik de...' |
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Titik yang telah ditetapkan (analog dengan titik origin dalam [[sistem koordinat Kartesius]]) disebut ''pole'' atau "kutub", dan [[:en:ray (geometry)|''ray'' atau "sinar"]] dari kutub pada arah yang telah ditetapkan disebut "aksis polar" (''polar axis''). Jarak dari suatu kutub disebut ''radial coordinate'' atau ''radius'', dan sudutnya disebut ''angular coordinate'', ''polar angle'', atau ''[[azimuth]]''.<ref name="brown">{{Cite book| last = Brown| first = Richard G.| editor = Andrew M. Gleason| year = 1997| title = Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis| publisher = McDougal Littell| location = Evanston, Illinois| isbn = 0-395-77114-5}}</ref>
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[[Image:Hipparchos 1.jpeg|thumb|190px|Hipparchus]]
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| url= http://books.google.com.au/books?id=AMOQZfrZq-EC&pg=PA161#v=onepage&f=false
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</ref> <!--From the 9th century onward they were using [[spherical trigonometry]] and [[map projection]] methods to determine these quantities accurately. The calculation is essentially the conversion of the [[Geodetic coordinates#Coordinates|equatorial polar coordinates]] of Mecca (i.e. its [[longitude]] and [[latitude]]) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its [[antipodal point]].<ref>King ([[#CITEREFKing2005|2005]], [http://books.google.com.au/books?id=AMOQZfrZq-EC&pg=PA169#v=onepage&f=false p. 169]). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.</ref>
There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in [[Harvard University|Harvard]] professor [[Julian Lowell Coolidge]]'s ''Origin of Polar Coordinates.''<ref name="coolidge">{{Cite journal| last = Coolidge| first = Julian| authorlink = Julian Lowell Coolidge| title = The Origin of Polar Coordinates| journal = American Mathematical Monthly| volume = 59| pages = 78–85| year = 1952| url = http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html| doi = 10.2307/2307104| issue = 2| publisher = Mathematical Association of America| jstor = 2307104}}</ref> [[Grégoire de Saint-Vincent]] and [[Bonaventura Cavalieri]] independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an [[Archimedean spiral]]. [[Blaise Pascal]] subsequently used polar coordinates to calculate the length of [[parabola|parabolic arcs]].
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The actual term ''polar coordinates'' has been attributed to [[Gregorio Fontana]] and was used by 18th-century Italian writers. The term appeared in [[English language|English]] in [[George Peacock]]'s 1816 translation of [[Sylvestre François Lacroix|Lacroix]]'s ''Differential and Integral Calculus''.<ref>{{Cite web| last = Miller| first = Jeff| title = Earliest Known Uses of Some of the Words of Mathematics| url = http://members.aol.com/jeff570/p.html| accessdate = 2006-09-10}}</ref><ref>{{Cite book| last = Smith| first = David Eugene| title = History of Mathematics, Vol II| publisher = Ginn and Co.| year = 1925| location = Boston| pages = 324}}</ref> [[Alexis Clairaut]] was the first to think of polar coordinates in three dimensions, and [[Leonhard Euler]] was the first to actually develop them.<ref name="coolidge" />
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== Kaidah ==
[[Image:Polar graph paper.svg|thumb|right|300px|
▲[[Image:Polar graph paper.svg|thumb|right|300px|A polar grid with several angles labeled in degrees]]
▲The radial coordinate is often denoted by ''r'', and the angular coordinate by [[phi|''φ'']], [[theta|''θ'']], or ''t''. The angular coordinate is specified as ''φ'' by [[International Organisation for Standardisation|ISO]] standard [[ISO 31-11|31-11]].
Dalam banyak konteks, suatu koordinat angular positif berarti sudut ''φ'' diukur [[berlawanan dengan jarum jam]] dari aksis.
▲Angles in polar notation are generally expressed in either [[degree (angle)|degree]]s or [[radian]]s (2[[pi|π]] rad being equal to 360°). Degrees are traditionally used in [[navigation]], [[surveying]], and many applied disciplines, while radians are more common in mathematics and mathematical [[physics]].<ref>{{Cite book| last = Serway| first = Raymond A.| coauthors = Jewett, Jr., John W.| title = Principles of Physics| publisher = Brooks/Cole—Thomson Learning| year = 2005| isbn = 0-534-49143-X}}</ref>
Dalam literatur matematika, aksis polar sering digambar horizontal dan mengarah ke kanan.
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===Uniqueness of polar coordinates===
Adding any number of full [[turn (geometry)|turn]]s (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates {{nowrap|(''r'', ''φ'' ± ''n''×360°)}} or {{nowrap|(−''r'', ''φ'' ± (2''n'' + 1)180°)}}, where ''n'' is any [[integer]].<ref>{{Cite web| url = http://www.fortbendisd.com/campuses/documents/Teacher/2006%5Cteacher_20060413_0948.pdf| title = Polar Coordinates and Graphing| accessdate = 2006-09-22| date = 2006-04-13| format = PDF}}</ref> Moreover, the pole itself can be expressed as (0, ''φ'') for any angle ''φ''.<ref>{{Cite book|title=Precalculus: With Unit-Circle Trigonometry|last=Lee|first=Theodore|author2=David Cohen |author3=David Sklar |year=2005|publisher=Thomson Brooks/Cole|edition=Fourth|isbn=0-534-40230-5}}</ref>
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