正矢

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單位圓上兩種正矢函數(Versin和Vercos)和兩種餘矢函數(coversin和covercos)的位置

正矢(英文:VersineVersed sine),在三角函数之中被定義為,值域在0~2之間。

概述[编辑]

正矢函數(versine[1][2][3][4][5]versed sine[6][7][8][9][10])是一個三角函數,出現在一些早期的三角表中。 其常計為versin(θ)sinver(θ)[11][12] vers(θ)ver(θ)[13]siv(θ)[14][15]拉丁語中,其被稱為sinus versus (翻轉的正弦), versinusversussagitta (箭頭)。[16]

其等價定義為

相關函數[编辑]

  • 餘的正矢(英文:versed cosinevercosine[17],寫為vercosin(θ)vercos(θ)vcs(θ)
  • 餘矢(英文:coversed sinecoversine[18],寫為,有時亦縮寫為
  • 餘的餘矢(英文:coversed cosine[19]covercosine),寫為covercosin(θ)covercos(θ)cvc(θ)

與上述四個函數類似,還存在四個“半值”函數:

  • 半正矢(英文:haversed sine,[20] haversinesemiversus[21][22]),寫為haversin(θ)semiversin(θ)semiversinus(θ)havers(θ)hav(θ)[23][24] hvs(θ)[註 1] sem(θ)hv(θ)[25],因半正矢公式出名,且曾用於導航術
  • 餘的半正矢(英文:haversed cosine[26] or havercosine),寫為havercosin(θ), havercos(θ), hac(θ)hvc(θ)
  • 半餘矢(英文:hacoversed sinehacoversine[27]cohaversine),寫為hacoversin(θ)semicoversin(θ)hacovers(θ)hacov(θ)[28]hcv(θ)。
  • 餘的半餘矢(英文:hacoversed cosine[29]hacovercosinecohavercosine),寫為hacovercosin(θ)hacovercos(θ)hcc(θ)

定義[编辑]

正矢 [2] Versin plot 2.svg
餘矢 [2] Coversin plot 2.svg
餘的正矢 [17] Vercosin plot 2.svg
餘的餘矢 [19] Covercosin plot 2.svg
半正矢 [2] Haversin plot 2.svg
半餘矢 [27] Hacoversin plot 2.svg
餘的半正矢 [26] Havercosin plot 2.svg
餘的半餘矢 [29] Hacovercosin plot 2.svg
角θ的所有三角函数在几何上可以依据以O點為圓心的单位圓来构造。

微分與積分[编辑]

參見[编辑]

註釋[编辑]

  1. ^ 在訊號分析中,hvs有時用於半正矢函數(haversine function),也有時代表单位阶跃函数

參考文獻[编辑]

  1. ^ Inman, James. Navigation and Nautical Astronomy: For the Use of British Seamen 3. London, UK: W. Woodward, C. & J. Rivington. 1835 [1821] [2015-11-09]. (原始内容存档于2022-05-27).  (Fourth edition: [1]页面存档备份,存于互联网档案馆).)
  2. ^ 2.0 2.1 2.2 2.3 Zucker, Ruth. Chapter 4.3.147: Elementary Transcendental Functions - Circular functions. Abramowitz, Milton; Stegun, Irene Ann (编). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. 1983: 78. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.  已忽略未知参数|orig-date= (帮助)
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  5. ^ Beebe, Nelson H. F. Chapter 11.1. Sine and cosine properties. The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library 1. Salt Lake City, UT, USA: Springer International Publishing AG. 2017-08-22: 301. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721. doi:10.1007/978-3-319-64110-2. 
  6. ^ Hall, Arthur Graham; Frink, Fred Goodrich. Review Exercises [100] Secondary Trigonometric Functions. 写于Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. January 1909: 125–127 [2017-08-12]. 
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  9. ^ Korn, Grandino Arthur; Korn, Theresa M. Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function. Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review有限度免费查阅,超限则需付费订阅 3. Mineola, New York, USA: Dover Publications, Inc. 2000: 892–893 [1961]. ISBN 978-0-486-41147-7.  (See errata.)
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  11. ^ Edler von Braunmühl, Anton. Vorlesungen über Geschichte der Trigonometrie - Von der Erfindung der Logarithmen bis auf die Gegenwart [Lectures on history of trigonometry - from the invention of logarithms up to the present] 2. Leipzig, Germany: B. G. Teubner. 1903: 231 [2015-12-09]. (原始内容存档于2022-05-26) (德语). 
  12. ^ Cajori, Florian. A History of Mathematical Notations 2 2 (3rd corrected printing of 1929 issue). Chicago, USA: Open court publishing company. 1952: 172 [March 1929] [2015-11-11]. ISBN 978-1-60206-714-1. 1602067147. The haversine first appears in the tables of logarithmic versines of José de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). See J. D. White in Nautical Magazine (February and July 1926).  (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
  13. ^ Shaneyfelt, Ted V. 德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?. Hilo, Hawaii: University of Hawaii. [2015-11-08]. (原始内容存档于2015-09-19). 
  14. ^ Cauchy, Augustin-Louis. Analyse Algébrique. Cours d'Analyse de l'Ecole royale polytechnique 1. L'Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi. 1821 (法语). access-date=2015-11-07--> (reissued by Cambridge University Press, 2009; ISBN 978-1-108-00208-0)
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  19. ^ 19.0 19.1 Weisstein, Eric W. (编). Covercosine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. (原始内容存档于2014-03-28) (英语). 
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  27. ^ 27.0 27.1 Weisstein, Eric W. (编). Hacoversine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. (原始内容存档于2014-03-29) (英语). 
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外部連結[编辑]