外正割

外正割（exsecant^{[1][2][3][4][5][6][7][8]}， 拉丁文：secans exterior^{[9][10][11][12]}） 是一種可以根據正割定義的三角函數，符號通常表示為${\displaystyle \operatorname {exsec} x}$^{[13][14][15][16]}或${\displaystyle \operatorname {exs} x}$^[17]， 曾經在應用在鐵道工程學（英語：Railway_engineering）、測量學、土木工程、天文學和球面幾何學等學科，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：^[16]

${\displaystyle \operatorname {exsec} x=\sec(x)-1}$。

外正割
性質
奇偶性	偶
定義域	${\displaystyle \left\{x\in \mathbb {R} |x\neq k\pi +{\tfrac {\pi }{2}},\,k\in \mathbb {Z} \right\}}$ ${\displaystyle \left\{x\in \mathbb {R} |x\neq 180^{\circ }k+90^{\circ },\,k\in \mathbb {Z} \right\}}$
到達域	${\displaystyle -2\geq \operatorname {exsec} x\geq 0}$
周期	${\displaystyle 2\pi }$ （360°）
特定值
當x=0	0
當x=+∞	N/A
當x=-∞	N/A
最大值	+∞
最小值	-∞
其他性質
漸近線	${\displaystyle x=\left(2k+1\right){\tfrac {\pi }{2}}}$ （x=180°k+90°）
根	${\displaystyle 2k\pi }$ （${\displaystyle 360^{\circ }k}$）
臨界點	${\displaystyle k\pi }$ （180°k）
k是一個整數。

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^{[18][19][20][21]}、 outward secant或outer secant等稱呼。

定義

在單位圓上表示的三角函數

在單位圓上，角${\displaystyle \theta }$的外正割可以定義為，在x軸上，從單位圓圓周沿x軸到「角${\displaystyle \theta }$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${\displaystyle \theta }$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：^[22]

${\displaystyle \operatorname {exsec} (\theta )=\sec(\theta )-1={\frac {1}{\cos(\theta )}}-1}$

用途

直到20世紀80年代，外正割函數在測量學^[7]、鐵路工程（英語：Railway engineering）^[4]（例如佈置鐵路曲線和橫向高度（英語：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。^[7][22]這主要是因為計數機和計算機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。^[7]

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災難性抵消或數值誤差。 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。 即使使用計算機，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恆等式：^[2][3][21]

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

在計算機出現之前，這需要相當費時的乘法運算。

伽利略·伽利萊早在 1632 年就使用了外正割函數，但當時他將其稱為segante（意為正割）^{[23][24][25][26]}。 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^{[9][10][11][12]}。 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^[27][28]。 ex secant和exsec等變體於則需要到1880年才開始有使用紀錄^[18]。1894年後，exsecant一術語的使用則比較少。^[1]

數學性質

導數

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \theta }}\operatorname {exsec} (\theta )=\tan(\theta )\sec(\theta )={\frac {\sin(\theta )}{\cos ^{2}(\theta )}}}$^[16]

積分

${\displaystyle \int \operatorname {exsec} (\theta )\,\mathrm {d} \theta =\ln \left[\cos \left({\frac {\theta }{2}}\right)+\sin \left({\frac {\theta }{2}}\right)\right]-\ln \left[\cos \left({\frac {\theta }{2}}\right)-\sin \left({\frac {\theta }{2}}\right)\right]-\theta +C}$^[16]

其他性質

單位圓上的外正割等價於正割減去餘弦再減去正矢

外正割可由單位圓推導：

${\displaystyle \operatorname {exsec} (\theta )=\sec(\theta )-\cos(\theta )-\operatorname {versin} (\theta ).}$

外正割函數可由正切函數表示為：^[22]

${\displaystyle \operatorname {exsec} (\theta )=\tan(\theta )\tan \left({\frac {\theta }{2}}\right).}$

外正割函數與正弦函數的關係為：

${\displaystyle \operatorname {exsec} (\theta )={\frac {1}{\sqrt {1-(\sin(\theta ))^{2}}}}-1.}$

外正割函數可以擴展到複平面。^[16]

其他恆等式

${\displaystyle \lim _{\theta \to 0}{\frac {\operatorname {exsec} (\theta )}{\theta }}=0}$^[4]

${\displaystyle {\frac {\operatorname {versin} (\theta )+\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}-{\frac {\operatorname {exsec} (\theta )+\operatorname {excsc} (\theta )}{\operatorname {exsec} (\theta )-\operatorname {excsc} (\theta )}}={\frac {2\operatorname {versin} (\theta )\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}}$^[4]

${\displaystyle (\operatorname {exsec} (\theta )+\operatorname {versin} (\theta ))\,(\operatorname {excsc} (\theta )+\operatorname {coversin} (\theta ))=\sin(\theta )\cos(\theta )}$^[4]

${\displaystyle \operatorname {exsec} (2\theta )={\frac {2\sin ^{2}(\theta )}{1-2\sin ^{2}(\theta )}}}$^[4]

${\displaystyle \operatorname {exsec} (2\theta )\,\cos(2\theta )=\tan(\theta )}$^[4]

${\displaystyle \operatorname {exsec} ^{2}(\theta )+2\operatorname {exsec} (\theta )=\tan ^{2}(\theta )}$^[4]

反外正割

外正割的反函數

反外正割（arcexsecant^[29]）是外正割函數的反函數，符號通常表示為arcexsec^[4][29] 、aexsec^[30][31]、aexs或exsec⁻¹。

反外正割可以定義為：

${\displaystyle \operatorname {arcexsec} (y)=\operatorname {arcsec}(y+1)}$

${\displaystyle \ =\arccos \left({\frac {1}{y+1}}\right)=\arctan({\sqrt {y^{2}+2y}})}$^[29][30][31]（對所有的y ≤ −2或y ≥ 0）^[29]

參見

• 外餘割：角的餘割值減去1
• 正矢：1減去角的餘弦值
• 三角函數

參考文獻

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外部連結

• 埃里克·韋斯坦因. 外正割. MathWorld.