正矢

正矢（英文：Versine、Versed sine），在三角函数之中被定義為${\displaystyle {\textrm {versin}}\theta =1-\cos \theta \,}$，值域在0～2之間。

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

概述

正矢函數（versine^{[1][2][3][4][5]}或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]或 siv(θ)。^[14][15] 在拉丁語中，其被稱為sinus versus (翻轉的正弦), versinus、 versus或 sagitta (箭頭)。^[16]

其等價定義為

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

相關函數

• 餘的正矢（英文：versed cosine、vercosine）^[17]，寫為vercosin(θ)、 vercos(θ)或 vcs(θ)。
• 餘矢（英文：coversed sine、coversine）^[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]或 covercosine），寫為covercosin(θ)、 covercos(θ)或 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] haversine或 semiversus^[21][22]），寫為haversin(θ)、 semiversin(θ)、 semiversinus(θ)、 havers(θ)、 hav(θ)、^[23][24] hvs(θ)、^{[註 1]} sem(θ)或 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)或 hvc(θ)。
• 半餘矢（英文：hacoversed sine、hacoversine^[27]或 cohaversine），寫為hacoversin(θ)、 semicoversin(θ)、 hacovers(θ)、 hacov(θ)^[28]或 hcv(θ)。
• 餘的半餘矢（英文：hacoversed cosine^[29]、 hacovercosine或 cohavercosine），寫為hacovercosin(θ)、 hacovercos(θ)或 hcc(θ)。

定義

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

 

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

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

參見

• 三角函數

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

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外部連結

• Sagitta, Apothem, and Chord （页面存档备份，存于互联网档案馆） by Ed Pegg, Jr., The Wolfram Demonstrations Project.