外正割

外正割（exsecant， 拉丁文：secans exterior） 是一種可以根據正割定義的三角函數，符号通常表示为 exsec ⁡ x {\displaystyle \operatorname {exsec} x} 或 exs ⁡ x {\displaystyle \operatorname {exs} x} ， 曾經在应用在鐵道工程學（英语：Railway_engineering）、测量学、土木工程、天文學和球面几何学等学科，可以幫助提高精度，但如今除了簡化一些計算外很少使用。外正割的函數值比正割函數少1，換句話說，其與正割的關係可以用下列等式表達： exsec ⁡ x = sec ⁡ ( x ) − 1 {\displaystyle \operatorname {exsec} x=\sec(x)-1} 。 在單位圓上，外正割位於正割線上單位圓的外側，因此稱為外正割。 此外，外正割也有exterior secant、 external secant、 outward secant或outer secant等稱呼。 在單位圓上，角 θ {\displaystyle \theta } 的外正割可以定義為，在x軸上，從單位圓圓周沿x軸到「角 θ {\displaystyle \theta } 的終邊與單位圓交點的切線」的長度。 由於從角的頂點沿x軸到「角 θ {\displaystyle \theta } 的終邊與單位圓交點的切線」的長度為正割，因此正割與外正割相差1，即外正割為正割扣掉單位圓半徑。 外正割也可以定義為： exsec ⁡ ( θ ) = sec ⁡ ( θ ) − 1 = 1 cos ⁡ ( θ ) − 1 {\displaystyle \operatorname {exsec} (\theta )=\sec(\theta )-1={\frac {1}{\cos(\theta )}}-1} 直到20世紀80年代，外正割函數在測量學、鐵路工程（英语：Railway engineering）（例如佈置鐵路曲線和橫向高度（英语：Cant_(road/rail)））、土木工程、天文學和球面三角學等領域都很重要，但現在很少使用。這主要是因為计算器和计算机的廣泛使用已經消除了對諸如此類的專門函數的三角表的需求。 定義一個特殊的外正割函數的原因與定義正矢函數的理據相似： 對於小角度 θ，sec(θ) 函數值接近於1，因此使用上述的外正割公式涉及減法時會幾乎會出現兩個幾乎相等的數量的相減，這個過程可能導致導致嚴重的灾难性抵消或數值誤差。 因此，正割函數表需要非常高的精確度才能用於外正割，這使得專用的外正割函數表非常有用。 即使使用计算机，如果使用基於餘弦的定義，對於小角度的外正割來說，浮點誤差可能會成為問題。 在這種極限情況下，更準確的公式應該使用這個恒等式： exsec ⁡ ( θ ) = 1 − cos ⁡ ( θ ) cos ⁡ ( θ ) = versin ⁡ ( θ ) cos ⁡ ( θ ) = versin ⁡ ( θ ) sec ⁡ ( θ ) = 2 ( sin ⁡ ( θ 2 ) ) 2 sec ⁡ ( θ ) {\displaystyle \operatorname {exsec} (\theta )={\frac {1-\cos(\theta )}{\cos(\theta )}}={\frac {\operatorname {versin} (\theta )}{\cos(\theta )}}=\operatorname {versin} (\theta )\sec(\theta )=2\left(\sin \left({\frac {\theta }{2}}\right)\right)^{2}\sec(\theta )} 在計算機出現之前，這需要相當費時的乘法運算。 伽利略·伽利莱早在 1632 年就使用了外正割函數，但當時他將其稱為segante（意為正割）。 其拉丁語名稱secans external至少可以追朔到1745年左右才開始有使用紀錄。 而其英語名稱external secant和其縮寫縮寫ex的用法：ex.sec，最早可以追溯到1855 年。 當時查爾斯·哈斯萊特（Charles Haslett）發表了第一個已知最早的外正割函數表。 ex secant和exsec等變體於則需要到1880年才開始有使用紀錄。1894年後，exsecant一術語的使用則比較少。 d d θ exsec ⁡ ( θ ) = tan ⁡ ( θ ) sec ⁡ ( θ ) = sin ⁡ ( θ ) cos 2 ⁡ ( θ ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \theta }}\operatorname {exsec} (\theta )=\tan(\theta )\sec(\theta )={\frac {\sin(\theta )}{\cos ^{2}(\theta )}}} ∫ exsec ⁡ ( θ ) d θ = ln ⁡ [ cos ⁡ ( θ 2 ) + sin ⁡ ( θ 2 ) ] − ln ⁡ [ cos ⁡ ( θ 2 ) − sin ⁡ ( θ 2 ) ] − θ + C {\displaystyle \int \operatorname {exsec} (\theta )\,\mathrm {d} \theta =\ln \left[\cos \left({\frac {\theta }{2}}\right)+\sin \left({\frac {\theta }{2}}\right)\right]-\ln \left[\cos \left({\frac {\theta }{2}}\right)-\sin \left({\frac {\theta }{2}}\right)\right]-\theta +C} 外正割可由單位圓推導： exsec ⁡ ( θ ) = sec ⁡ ( θ ) − cos ⁡ ( θ ) − versin ⁡ ( θ ) . {\displaystyle \operatorname {exsec} (\theta )=\sec(\theta )-\cos(\theta )-\operatorname {versin} (\theta ).} 外正割函数可由正切函数表示为： exsec ⁡ ( θ ) = tan ⁡ ( θ ) tan ⁡ ( θ 2 ) . {\displaystyle \operatorname {exsec} (\theta )=\tan(\theta )\tan \left({\frac {\theta }{2}}\right).} 外正割函數與正弦函數的關係為： exsec ⁡ ( θ ) = 1 1 − ( sin ⁡ ( θ ) ) 2 − 1. {\displaystyle \operatorname {exsec} (\theta )={\frac {1}{\sqrt {1-(\sin(\theta ))^{2}}}}-1.} 外正割函數可以擴展到複平面。 lim θ → 0 exsec ⁡ ( θ ) θ = 0 {\displaystyle \lim _{\theta \to 0}{\frac {\operatorname {exsec} (\theta )}{\theta }}=0} versin ⁡ ( θ ) + coversin ⁡ ( θ ) versin ⁡ ( θ ) − coversin ⁡ ( θ ) − exsec ⁡ ( θ ) + excsc ⁡ ( θ ) exsec ⁡ ( θ ) − excsc ⁡ ( θ ) = 2 versin ⁡ ( θ ) coversin ⁡ ( θ ) versin ⁡ ( θ ) − coversin ⁡ ( θ ) {\displaystyle {\frac {\operatorname {versin} (\theta )+\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}-{\frac {\operatorname {exsec} (\theta )+\operatorname {excsc} (\theta )}{\operatorname {exsec} (\theta )-\operatorname {excsc} (\theta )}}={\frac {2\operatorname {versin} (\theta )\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}} ( exsec ⁡ ( θ ) + versin ⁡ ( θ ) ) ( excsc ⁡ ( θ ) + coversin ⁡ ( θ ) ) = sin ⁡ ( θ ) cos ⁡ ( θ ) {\displaystyle (\operatorname {exsec} (\theta )+\operatorname {versin} (\theta ))\,(\operatorname {excsc} (\theta )+\operatorname {coversin} (\theta ))=\sin(\theta )\cos(\theta )} exsec ⁡ ( 2 θ ) = 2 sin 2 ⁡ ( θ ) 1 − 2 sin 2 ⁡ ( θ ) {\displaystyle \operatorname {exsec} (2\theta )={\frac {2\sin ^{2}(\theta )}{1-2\sin ^{2}(\theta )}}} exsec ⁡ ( 2 θ ) cos ⁡ ( 2 θ ) = tan ⁡ ( θ ) {\displaystyle \operatorname {exsec} (2\theta )\,\cos(2\theta )=\tan(\theta )} exsec 2 ⁡ ( θ ) + 2 exsec ⁡ ( θ ) = tan 2 ⁡ ( θ ) {\displaystyle \operatorname {exsec} ^{2}(\theta )+2\operatorname {exsec} (\theta )=\tan ^{2}(\theta )} 反外正割（arcexsecant）是外正割函數的反函數，符号通常表示为arcexsec 、aexsec、aexs或exsec−1。 反外正割可以定義為： arcexsec ⁡ ( y ) = arcsec ⁡ ( y + 1 ) {\displaystyle \operatorname {arcexsec} (y)=\operatorname {arcsec}(y+1)}   = arccos ⁡ ( 1 y + 1 ) = arctan ⁡ ( y 2 + 2 y ) {\displaystyle \ =\arccos \left({\frac {1}{y+1}}\right)=\arctan({\sqrt {y^{2}+2y}})} （對所有的y ≤ −2或y ≥ 0） 外餘割：角的餘割值減去1 正矢：1減去角的餘弦值 三角函数 Allen, Calvin Frank. 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ISSN 1916-9639. doi:10.5539/apr.v9n2p57 . ISSN 1916-9647.  [1] Haslett, Charles. Hackley, Charles W. , 编. The Mechanic's, Machinist's, Engineer's Practical Book of Reference: Containing tables and formulæ for use in superficial and solid mensuration; strength and weight of materials; mechanics; machinery; hydraulics, hydrodynamics; marine engines, chemistry; and miscellaneous recipes. Adapted to and for the use of all classes of practical mechanics. Together with the Engineer's Field Book: Containing formulæ for the various of running and changing lines, locating side tracks and switches, &c., &c. Tables of radii and their logarithms, natural and logarithmic versed sines and external secants, natural sines and tangents to every degree and minute of the quadrant, and logarithms from the natural numbers from 1 to 10,000. New York, USA: James G. Gregory, successor of W. A. Townsend & Co. (Stringer & Townsend). September 1855 [2017-08-13]. […] Still there would be much labor of computation which may be saved by the use of tables of external secants and versed sines, which have been employed with great success recently by the Engineers on the Ohio and Mississippi Railroad（英语：Ohio and Mississippi Railroad）, and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by Mr. Haslett, one of the Engineers of that Road, are now for the first time given to the public. […] In presenting this work to the public, the Author claims for it the adaptation of a new principle in trigonometrical analysis of the formulas generally used in field calculations. Experience has shown, that versed sines and external secants as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and many calculations concerning curves and running lines made less intricate, and results obtained with more accuracy and far less trouble, than by any methods laid down in works of this kind. The examples given have all been suggested by actual practice, and will explain themselves. […] As a book for practical use in field work, it is confidently believed that this is more direct in the application of rules and facility of calculation than any work now in use. 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