查看“︁外餘割”︁的源代码

{{函數
|name =外餘割
|image =Excosecant plot.svg

|heading1 =1
|parity =非奇非偶
|domain = <math>\left\{x\in\mathbb{R}|x\neq k\pi,\,k\in\mathbb{Z}\right\}</math><br/><math>\left\{x\in\mathbb{R}|x\neq 180^\circ k,\,k\in\mathbb{Z}\right\}</math>
|codomain = <math>-2 \geq \operatorname{excsc} x \geq 0</math>
|period = <math>2\pi</math><br/>（360°）

|heading2 = 1
|zero = ∞
|plusinf = N/A
|minusinf = N/A
|max = +∞
|min = -∞
|vr1 =
|f1 =
|vr2 =
|f2 =
|vr3 =
|f3 =
|vr4 =
|f4 =
|vr5 =
|f5 =

|heading3 = 1
|asymptote = <math>x=k\pi</math><br/>（{{math|1=x=180°''k''}}）
|root = <math>\frac\pi{2}+2k\pi</math><br/>（<math>90^\circ+360^\circ k</math>）
|critical = <math>k\pi-\tfrac{\pi}{2}</math><br/>（{{math|180°''k''-90°}}）
|inflection = 
|notes = k是一個[[整數]]。
}}
'''外餘割'''（{{lang|en|'''excosecant'''}}<ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |author-first1=Arthur Graham |author-last1=Hall |author-first2=Fred Goodrich |author-last2=Frink |date=January 1909 |location=Ann Arbor, Michigan, USA |chapter=Review Exercises [100] Secondary Trigonometric Functions |publisher={{link-en|亨利·霍爾特公司|Henry Holt and Company|Henry Holt and Company}} / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA |publication-place=New York, USA |page=125 |url=https://archive.org/stream/planetrigonometr00hallrich#page/125/mode/1up |access-date=2017-08-12 }}</ref><ref name="Weisstein_excosec"/>）又稱'''餘外割'''（{{lang|en|'''coexsecant'''}}<ref name="Bohannan_1903">{{cite book |title=Plane Trigonometry |author-first=Rosser Daniel |author-last=Bohannan |location=[[Ohio State University]] |publisher=Allyn and Bacon, Boston, USA / J. S. Cushing & Co. — Berwick & Smith Co., Norwood, MA |date=1904 |orig-year=1903 |chapter=$131. The Versed Sine, Exsecant and Coexsecant. §132. Exercises |pages=235–236 |chapter-url=https://archive.org/stream/planetrigonometr00boharich/planetrigonometr00boharich_djvu.txt |access-date=2017-07-09 }}</ref><ref name="Frye_1918">{{cite book |title=Civil engineer's pocket-book: a reference-book for engineers, contractors and students containing rules, data, methods, formulas and tables |author-first=Albert I. |author-last=Frye |date=1918 |orig-year=1913 |edition=2 (corrected) |publisher={{link-en|大衛·範·諾斯特蘭德|D. Van Nostrand Company|D. Van Nostrand Company}}; {{link-en|康斯特布爾及羅賓遜有限公司|Constable and Company, Ltd.|Constable and Company, Ltd.}} |location=New York, USA; London, UK |url=https://archive.org/details/civilengineerspo00frye |access-date=2015-11-16}}</ref><ref name="Vlijmen_2005">{{cite web|title=Goniology |work=Eenheden, constanten en conversies<!-- |trans-work=Units of measurement, constants and conversions --> |author-first=Oscar |author-last=van Vlijmen |date=2005-12-28 |orig-year=2003 |url=http://home.kpn.nl/vanadovv/Gonio.html |access-date=2015-11-28 |url-status=live |archive-url=https://web.archive.org/web/20091028065825/http://home.hetnet.nl/~vanadovv/Gonio.html |archive-date=2009-10-28}}</ref>）是一種可以根據[[餘割]]定義的[[三角函數]]，现很少使用。
其符号通常表示为<math>\operatorname{excosec} x</math>或<math>\operatorname{exc} x</math><ref name="Shaneyfelt">{{cite web |title=德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine? |author-first=Ted V. |author-last=Shaneyfelt |publisher=[[University of Hawaii]] |location=Hilo, Hawaii |url=http://www2.hawaii.edu/~tvs/trig.html |access-date=2015-11-08 |url-status=live |archive-url=https://web.archive.org/web/20150919053929/http://www2.hawaii.edu/~tvs/trig.html |archive-date=2015-09-19}}</ref>。
其[[函數值]]比[[餘割函數]]少1，換句話說，其與餘割的關係可以用下列等式表達：<ref name="Weisstein_excosec">{{cite mathworld |urlname = Excosecant |title = Excosecant}}</ref>
:<math>\operatorname{excosec} x=\csc(x)-1</math>。
在[[單位圓]]上，外餘割位於[[餘割]]線上[[單位圓]]的外側，因此稱為外餘割。此外，外餘割也有{{lang|en|exterior cosecant}}<ref name="article eng2017trigonometric">{{cite journal
  |title=Trigonometric Functions
  |author=Eng, M and Schwarz, René
  |journal=Image
  |volume=3
  |number=2
  |pages=4
  |url=https://downloads.rene-schwarz.com/download/M003-Trigonometric_Functions.pdf
  |year=2017
  |access-date=2023-10-31
  |archive-date=2023-10-31
  |archive-url=https://web.archive.org/web/20231031094609/https://downloads.rene-schwarz.com/download/M003-Trigonometric_Functions.pdf
  |dead-url=no
}}</ref>、external cosecant<ref name="Gottschalk_2002">{{cite book |title=Some Quaint & Curious & Almost Forgotten Trig Functions |author-first=Walter Helbig |author-last=Gottschalk  |date=2002 |volume=80 |work=Gottschalk's Gestalts - A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics |publisher=Infinite Vistas Press |location=Providence, Rhode Island, USA |id=PVD RI, GG80 |url=http://gottschalksgestalts.org/pdf/GG80.pdf |access-date=2015-11-17 |url-status=live |archive-url=https://web.archive.org/web/20130925164327/http://gottschalksgestalts.org/pdf/GG80.pdf |archive-date=2013-09-25}}</ref>、outward cosecant和outer cosecant等稱呼。在數學表達式中，外餘割除了表示為<math>\operatorname{excosec} x</math>或<math>\operatorname{exc} x</math>之外，在不同文獻中，外餘割也有<math>\operatorname{coexsec} x</math><ref name="Searles_1880">{{cite book |title=Field Engineering - A Hand-book of the Theory and Practice of Railway Surveying, Location, and Construction, designed for the Class-room, Field and Office, and containing a large number of useful tables, original and selected |author-first=William Henry |author-last=Searles |date=1880-03-01 |location=New York, USA |publisher=[[John Wiley & Sons]] |url=https://www.forgottenbooks.com/en/download/FieldEngineering_10268502.pdf |access-date=2017-08-13 |url-status=live |archive-url=https://web.archive.org/web/20170813134233/https://www.forgottenbooks.com/en/download/FieldEngineering_10268502.pdf |archive-date=2017-08-13}} [https://archive.org/stream/fieldengineerin00seargoog#page/n7/mode/1up 8th revised edition, 1887] [http://www.survivorlibrary.com/library/field_engineering-railway_surveying_location_and_construction-1910.pdf<!-- https://web.archive.org/web/20170813133914/http://www.survivorlibrary.com/library/field_engineering-railway_surveying_location_and_construction-1910.pdf --> 16th edition, 1910]</ref><ref name="Frye_1918"/><ref name="Vlijmen_2005"/>、<math>\operatorname{excsc} x</math><ref name="Hall_1909"/><ref name="Weisstein_excosec"/>、<math>\operatorname{xcs} x</math><ref>{{cite web | url=https://midimagic.sgc-hosting.com/anitrig.htm | title=Animated Trigonometry Construction | access-date=2023-10-31 | archive-date=2023-10-31 | archive-url=https://web.archive.org/web/20231031101237/https://midimagic.sgc-hosting.com/anitrig.htm | dead-url=no }}</ref>等表示方式。

外餘割曾被用來描述[[費米子]]的[[动能]]。<ref name="Hawking_2002"/><ref name="Stávek_2017"/>

== 定義 ==
[[Image:Circle-trig6.svg|left|thumb|200px|在[[单位圆]]上表示的[[三角函数]]]]
在[[單位圓]]上，角<math>\theta</math>的外餘割可以定義為，在{{math|''y''}}軸上，從單位圓[[圓周]]沿y軸到「角<math>\theta</math>的終邊與單位圓交點的切線」的長度。由於從角的[[頂點 (幾何)#角的頂點|頂點]]沿y軸到「角<math>\theta</math>的終邊與單位圓交點的切線」的長度為[[餘割]]，因此[[餘割]]與外餘割相差1，即外餘割為餘割扣掉單位圓[[半徑]]。

外餘割也可以定義為：
:<math>\operatorname{excsc}(\theta) = \operatorname{exsec}\left(\frac{\pi}{2} - \theta\right)</math><math>= \csc(\theta) - 1 = \frac{1}{\sin(\theta)} - 1.</math>

== 歷史 ==
直到20世紀80年代，外餘割函數與外正割函數都在數個有高精度計算需求的領域中有著重要的作用。<ref name="Calvert_2004">{{cite web |author-first=James B. |author-last=Calvert |title=Trigonometry |orig-year=2004-01-10 |date=2007-09-14 |url-status=dead |archive-url=https://web.archive.org/web/20071002214133/http://mysite.du.edu/~jcalvert/math/trig.htm |archive-date=2007-10-02 |url=http://www.du.edu/~jcalvert/math/trig.htm |access-date=2015-11-08 }}</ref><ref name="Atlas_2009">{{cite book
 |title=An Atlas of Functions: with Equator, the Atlas Function Calculator
 |url=https://archive.org/details/atlasfunctions00oldh
 |url-access=limited
 |author-first1=Keith B. |author-last1=Oldham
 |author-first2=Jan C. |author-last2=Myland
 |author-first3=Jerome |author-last3=Spanier
 |publisher=[[施普林格科學+商業媒體|Springer Science+Business Media, LLC]]
 |edition=2
 |date=2009
 |chapter=33.13. The Secant sec(x) and Cosecant csc(x) functions - Cognate functions
 |page=[https://archive.org/details/atlasfunctions00oldh/page/n348 336]
 |orig-year=1987 |isbn=978-0-387-48806-6 |doi=10.1007/978-0-387-48807-3
 |lccn=2008937525 }}</ref>由於在[[角度]]接近<math>\frac{\pi}2</math>（90度）時，[[餘割]]函數的值會接近於1，引此使用上述公式來計算外餘割的話，會在這些角度的函數值上出現嚴重的[[灾难性抵消]]或數值誤差。因此這時對餘割函數表的精確度要求將非常高，而若定義了外餘割函數，則使用外餘割函數的函數表則能一定程度上的避免上述問題。但後來隨著-{zh-cn:计算器; zh-tw:計算機; zh-hk:計數機;}-和-{zh-cn:计算机; zh-hk:計算機; zh-tw:電腦; zh-sg:电脑;}-的發展與廣泛使用，因此外餘割函數的需求已經逐漸變的不明顯，因此現在只有非常少數的情況會使用到外餘割函數。<ref name="Calvert_2004"/>

外餘割的術語coexsecant<ref name="Bohannan_1903"/>和coexsec<ref name="Allen_1894">{{cite book |title=Railroad Curves and Earthwork |author-first=Calvin Frank |author-last=Allen |publisher=Spon & Chamberlain; E. & F. Spon, Ltd. |location=New York, USA; London, UK |orig-year=1889 |date=1894 |url=https://archive.org/details/railroadcurvesea00allerich |access-date=2015-11-16}}</ref>早在1880年就已經有文獻使用了<ref name="Allen_1894"/><ref name="Bohannan_1903"/>，而自1909年開始，外餘割在文獻中則是使用excosecant<ref name="Hall_1909"/>。該函數也被[[阿尔伯特·爱因斯坦]]用來描述[[費米子]]的[[动能]]。<ref name="Hawking_2002">{{cite book |editor-last=Hawking |editor-first=Stephen William |editor-link=Stephen William Hawking |date=2002 |title=On the Shoulders of Giants: The Great Works of Physics and Astronomy |publisher={{link-en|Running Press (出版社)|Running Press|Running Press}} |location=Philadelphia, USA |lccn=2002100441 |isbn=0-7624-1698-X |url=https://archive.org/details/isbn_9780762413485 |url-access=registration |access-date=2017-07-31 }}</ref><ref name="Stávek_2017">{{cite journal |author-first=Jiří |author-last=Stávek |location=Prague, CZ |title=On the Hidden Beauty of Trigonometric Functions |date=2017-03-10 |orig-year=2017-02-26 |journal=Applied Physics Research |publisher=Canadian Center of Science and Education |volume=9 |number=2 |issn=1916-9639 |id={{eISSN|1916-9647}} |doi=10.5539/apr.v9n2p57 |pages=57–64 |doi-access=free }} [http://www.ccsenet.org/journal/index.php/apr/article/download/67093/36393<!-- https://web.archive.org/web/20170731001724/http://www.ccsenet.org/journal/index.php/apr/article/download/67093/36393 -->]</ref>

== 計算 ==
在早期[[電腦|-{zh-cn:计算机; zh-hk:計算機; zh-tw:電腦; zh-sg:电脑;}-]]不普遍的時候，外餘割函數的計算若使用公式<math>\operatorname{excsc}(\theta) = \csc(\theta) - 1</math>來計算的話，會在角度接近<math>\frac{\pi}2+2\pi</math>（90度）及其[[同界角]]時出現嚴重的[[灾难性抵消]]或數值誤差。因此若要更精確地計算外餘割函數的話，需要使用以下等式：<ref name="Gottschalk_2002"/>
:<math display="block">\operatorname{excsc}(\theta) = \frac{1-\sin(\theta)}{\sin(\theta)}
 = \frac{\operatorname{coversin}(\theta)}{\sin(\theta)}
 = \operatorname{coversin}(\theta) \csc(\theta). </math>
但在-{zh-cn:计算机; zh-hk:計算機; zh-tw:電腦; zh-sg:电脑;}-不普遍的的時代，要做這些乘法運算非常耗時，因此專用於外餘割函數的函數表就會變得很有用。

== 恒等式 ==
=== 导数 ===
: <math>\frac{\mathrm{d}}{\mathrm{d}\theta}\operatorname{excsc}(\theta) = -\cot(\theta)\csc(\theta) = \frac{-\cos(\theta)}{\sin^2(\theta)}</math>

=== 積分 ===
: <math>\int\operatorname{excsc}(\theta)\,\mathrm{d}\theta = \ln\left[\tan\left(\frac{\theta}{2}\right)\right] - \theta + C</math>

=== 与其他三角函数的关系 ===
:<math>\operatorname{excsc}(\theta) = \operatorname{csc}(\theta) - \sin(\theta) - \operatorname{coversin}(\theta).</math>
:<math>\operatorname{excsc}(\theta) = 2 \left(\cos\left(\frac{\theta}{2}\right)\right)^2 \csc(\theta).\ </math>
:<math>\operatorname{excsc}(\theta) = \cot(\theta) \cot\left(\frac{\theta}{2}\right).</math>
:<math>\operatorname{excsc}(\theta) = \frac{1}{\sqrt{1 - (\cos(\theta))^2}} - 1.</math><ref name="Stávek_2017"/>

== 反外餘割 ==
[[File:Arcexcsc plot.svg|thumb|外餘割的反函數]]
反外餘割（arcexcosecant）是外餘割的[[反函數]]。符號通常會表示為'''arcexcosec'''、 '''arcexcsc'''<ref name="Hall_1909"/>、 '''aexcsc'''、 '''aexc'''、 '''arccoexsecant'''、 '''arccoexsec'''或'''excsc<sup>−1</sup>'''。其定義為：
:<math>\operatorname{arcexcsc}(y) = \arccsc(y+1) = \arcsin\left(\frac{1}{y+1}\right)</math>

== 參見 ==
*[[外正割]]

== 參考文獻 ==
{{Reflist|2}}

== 外部連結 ==
* {{MathWorld |urlname = Excosecant}}

{{三角函數}}
[[Category:三角函数]]