# 正矢

## 概述

${\displaystyle \operatorname {versin} (\theta )=1-\cos(\theta )=2\sin ^{2}\left({\frac {\theta }{2}}\right)}$

### 相關函數

• 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${\displaystyle \operatorname {coversin} (\theta )}$，有時亦縮寫為${\displaystyle \operatorname {cvs} (\theta )}$
• 餘的餘矢（英文：coversed cosine[19]covercosine），寫為covercosin(θ)covercos(θ)cvc(θ)

• 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

正矢 ${\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}$[2] ${\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}$[2] ${\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}$[17] ${\displaystyle {\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,}$[19] ${\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}=\sin ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}\,}$[2] ${\displaystyle {\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,}$[27] ${\displaystyle {\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}=\cos ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1+\cos(\theta )}{2}}\,}$[26] ${\displaystyle {\textrm {hacovercosin}}(\theta ):={\frac {{\textrm {covercosin}}(\theta )}{2}}={\frac {1+\sin(\theta )}{2}}\,}$[29]

## 微分與積分

 ${\displaystyle {\frac {d}{dx}}\mathrm {versin} (x)=\sin {x}}$ ${\displaystyle \int \mathrm {versin} (x)\,dx=x-\sin {x}+C}$ ${\displaystyle {\frac {d}{dx}}\mathrm {coversin} (x)=-\cos {x}}$ ${\displaystyle \int \mathrm {coversin} (x)\,dx=x+\cos {x}+C}$ ${\displaystyle {\frac {d}{dx}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}}$ ${\displaystyle \int \mathrm {haversin} (x)\,dx={\frac {x-\sin {x}}{2}}+C}$ ${\displaystyle {\frac {d}{dx}}\mathrm {hacoversin} (x)={\frac {-\cos {x}}{2}}}$ ${\displaystyle \int \mathrm {hacoversin} (x)\,dx={\frac {x+\cos {x}}{2}}+C}$

## 註釋

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

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. 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)
3. ^ Tapson, Frank. Background Notes on Measures: Angles. 1.4. Cleave Books. 2004 [2015-11-12]. （原始内容存档于2007-02-09）.
4. ^ Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome. 32.13. The Cosine cos(x) and Sine sin(x) functions - Cognate functions. An Atlas of Functions: with Equator, the Atlas Function Calculator 2. Springer Science+Business Media, LLC. 2009: 322 [1987]. ISBN 978-0-387-48806-6. LCCN 2008937525. doi:10.1007/978-0-387-48807-3.
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].
7. ^ Boyer, Carl Benjamin. 5: Commentary on the Paper of E. J. Dijksterhuis (The Origins of Classical Mechanics from Aristotle to Newton). Clagett, Marshall (编). Critical Problems in the History of Science 3. Madison, Milwaukee, and London: University of Wisconsin Press, Ltd. 1969: 185–190 [1959] [2015-11-16]. ISBN 0-299-01874-1. LCCN 59-5304. 9780299018740.
8. ^ Swanson, Todd; Andersen, Janet; Keeley, Robert. 5 (Trigonometric Functions) (PDF). Precalculus: A Study of Functions and Their Applications. Harcourt Brace & Company. 1999: 344 [2015-11-12]. （原始内容存档 (PDF)于2003-06-17）.
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 3. Mineola, New York, USA: Dover Publications, Inc. 2000: 892–893 [1961]. ISBN 978-0-486-41147-7. (See errata.)
10. ^ Calvert, James B. Trigonometry. 2007-09-14 [2004-01-10] [2015-11-08]. （原始内容存档于2007-10-02）.
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)
15. ^ Bradley, Robert E.; Sandifer, Charles Edward. Buchwald, J. Z. , 编. Cauchy's Cours d'analyse: An Annotated Translation. Sources and Studies in the History of Mathematics and Physical Sciences. Cauchy, Augustin-Louis (Springer Science+Business Media, LLC). 2010-01-14: 10, 285 [2009] [2015-11-09]. ISBN 978-1-4419-0548-2. LCCN 2009932254. doi:10.1007/978-1-4419-0549-9. 1441905499, 978-1-4419-0549-9. （原始内容存档于2016-06-24）. (See errata.)
16. ^ van Brummelen, Glen Robert. Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. 2013 [2015-11-10]. ISBN 9780691148922. 0691148929.
17. Weisstein, Eric W. (编). Vercosine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. （原始内容存档于2014-03-24） （英语）.
18. ^ Weisstein, Eric W. (编). Coversine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. （原始内容存档于2005-11-27） （英语）.
19. 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）.
20. ^ Weisstein, Eric W. (编). Haversine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. （原始内容存档于2005-03-10） （英语）.
21. ^ Fulst, Otto. 17, 18. Lütjen, Johannes; Stein, Walter; Zwiebler, Gerhard (编). Nautische Tafeln 24. Bremen, Germany: Arthur Geist Verlag. 1972 （德语）.
22. ^ Sauer, Frank. Semiversus-Verfahren: Logarithmische Berechnung der Höhe. Hotheim am Taunus, Germany: Astrosail. 2015 [2004] [2015-11-12]. （原始内容存档于2013-09-17） （德语）.
23. ^ Rider, Paul Reece; Davis, Alfred. Plane Trigonometry. New York, USA: D. Van Nostrand Company. 1923: 42 [2015-12-08]. （原始内容存档于2022-05-28）.
24. ^ Haversine. Wolfram Language & System: Documentation Center. 7.0. 2008 [2015-11-06]. （原始内容存档于2014-09-01）.
25. ^ Rudzinski, Greg. Ix, Hanno. Ultra compact sight reduction. Ocean Navigator (Portland, ME, USA: Navigator Publishing LLC). July 2015, (227): 42–43 [2015-11-07]. ISSN 0886-0149.
26. Weisstein, Eric W. (编). Havercosine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. （原始内容存档于2014-03-29） （英语）.
27. 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）.
28. ^ van Vlijmen, Oscar. Goniology. Eenheden, constanten en conversies. 2005-12-28 [2003] [2015-11-28]. （原始内容存档于2009-10-28） （英语）.
29. Weisstein, Eric W. (编). Hacovercosine. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-11-06]. （原始内容存档于2014-03-29） （英语）.