# 正矢

（重定向自半正矢

## 相關函數

• 餘矢（英文：coversed sinecoversine），寫為${\displaystyle \operatorname {coversin} (\theta )}$，有時亦縮寫為${\displaystyle \operatorname {cvs} (\theta )}$
• 半正矢（英文：haversed sinehaversine），寫為${\displaystyle \operatorname {haversin} (\theta )}$，因半正矢公式出名，且曾用於導航術
• 半餘矢（英文：hacoversed sinehacoversinecohaversine），寫為${\displaystyle \operatorname {hacoversin} (\theta )}$

## 定義

 正矢 ${\displaystyle {\textrm {versin}}(\theta )=2\sin ^{2}\left({\frac {\theta }{2}}\right)=1-\cos \theta \,}$ 餘矢 ${\displaystyle {\textrm {coversin}}\theta ={\textrm {versin}}\left({\frac {\pi }{2}}-\theta \right)=1-\sin \theta \,}$ 半正矢 ${\displaystyle {\textrm {haversin}}\theta ={\frac {{\textrm {versin}}\theta }{2}}={\frac {1-\cos \theta }{2}}\,}$ 半餘矢 ${\displaystyle {\textrm {hacoversin}}\theta ={\frac {{\textrm {coversin}}\theta }{2}}={\frac {1-\sin \theta }{2}}\,}$

## 微分與積分

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

## 參考文獻

• Cites coinage by Prof. Jas. Inman, D. D., in his Navigation and Nautical Astronomy, 3rd ed. (1835).
• Nair, Bhaskaran. Track measurement systems—concepts and techniques. Rail International. 1972, 3 (3): 159–166. ISSN 0020-8442.