外正割

外正割（exsecant^{[1][2][3][4][5][6][7][8]}， 拉丁文：secans exterior^{[9][10][11][12]}） 是一种可以根据正割定义的三角函数，符号通常表示为${\displaystyle \operatorname {exsec} x}$^{[13][14][15][16]}或${\displaystyle \operatorname {exs} x}$^[17]， 曾经在应用在铁道工程学（英语：Railway_engineering）、测量学、土木工程、天文学和球面几何学等学科，可以帮助提高精度，但如今除了简化一些计算外很少使用。外正割的函数值比正割函数少1，换句话说，其与正割的关系可以用下列等式表达：^[16]

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

外正割
性质
奇偶性	偶
定义域	${\displaystyle \left\{x\in \mathbb {R} |x\neq k\pi +{\tfrac {\pi }{2}},\,k\in \mathbb {Z} \right\}}$ ${\displaystyle \left\{x\in \mathbb {R} |x\neq 180^{\circ }k+90^{\circ },\,k\in \mathbb {Z} \right\}}$
到达域	${\displaystyle -2\geq \operatorname {exsec} x\geq 0}$
周期	${\displaystyle 2\pi }$ （360°）
特定值
当x=0	0
当x=+∞	N/A
当x=-∞	N/A
最大值	+∞
最小值	-∞
其他性质
渐近线	${\displaystyle x=\left(2k+1\right){\tfrac {\pi }{2}}}$ （x=180°k+90°）
根	${\displaystyle 2k\pi }$ （${\displaystyle 360^{\circ }k}$）
临界点	${\displaystyle k\pi }$ （180°k）
k是一个整数。

在单位圆上，外正割位于正割线上单位圆的外侧，因此称为外正割。 此外，外正割也有exterior secant、 external secant^{[18][19][20][21]}、 outward secant或outer secant等称呼。

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定义

在单位圆上表示的三角函数

在单位圆上，角${\displaystyle \theta }$的外正割可以定义为，在x轴上，从单位圆圆周沿x轴到“角${\displaystyle \theta }$的终边与单位圆交点的切线”的长度。 由于从角的顶点沿x轴到“角${\displaystyle \theta }$的终边与单位圆交点的切线”的长度为正割，因此正割与外正割相差1，即外正割为正割扣掉单位圆半径。

外正割也可以定义为：^[22]

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

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用途

直到20世纪80年代，外正割函数在测量学^[7]、铁路工程（英语：Railway engineering）^[4]（例如布置铁路曲线和横向高度（英语：Cant_(road/rail)））、土木工程、天文学和球面三角学等领域都很重要，但现在很少使用。^[7][22]这主要是因为计算器和电脑的广泛使用已经消除了对诸如此类的专门函数的三角表的需求。^[7]

定义一个特殊的外正割函数的原因与定义正矢函数的理据相似： 对于小角度 θ，sec(θ) 函数值接近于1，因此使用上述的外正割公式涉及减法时会几乎会出现两个几乎相等的数量的相减，这个过程可能导致导致严重的灾难性抵消或数值误差。 因此，正割函数表需要非常高的精确度才能用于外正割，这使得专用的外正割函数表非常有用。 即使使用电脑，如果使用基于余弦的定义，对于小角度的外正割来说，浮点误差可能会成为问题。 在这种极限情况下，更准确的公式应该使用这个恒等式：^[2][3][21]

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

在计算机出现之前，这需要相当费时的乘法运算。

伽利略·伽利莱早在 1632 年就使用了外正割函数，但当时他将其称为segante（意为正割）^{[23][24][25][26]}。 其拉丁语名称secans external至少可以追朔到1745年左右才开始有使用纪录^{[9][10][11][12]}。 而其英语名称external secant和其缩写缩写ex的用法：ex.sec，最早可以追溯到1855 年。 当时查尔斯·哈斯莱特（Charles Haslett）发表了第一个已知最早的外正割函数表^[27][28]。 ex secant和exsec等变体于则需要到1880年才开始有使用纪录^[18]。1894年后，exsecant一术语的使用则比较少。^[1]

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数学性质

导数

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

积分

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

其他性质

单位圆上的外正割等价于正割减去余弦再减去正矢

外正割可由单位圆推导：

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

外正割函数可由正切函数表示为：^[22]

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

外正割函数与正弦函数的关系为：

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

外正割函数可以扩展到复平面。^[16]

其他恒等式

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

外正割的反函数

反外正割（arcexsecant^[29]）是外正割函数的反函数，符号通常表示为arcexsec^[4][29] 、aexsec^[30][31]、aexs或exsec⁻¹。

反外正割可以定义为：

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

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

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参见

• 外余割：角的余割值减去1
• 正矢：1减去角的余弦值
• 三角函数

参考文献

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外部链接

• 埃里克·韦斯坦因. 外正割. MathWorld.