反三角函数

符号${\displaystyle \sin ^{-1},\cos ^{-1}}$等常用于${\displaystyle \arcsin ,\arccos }$等。但是这种符号有时在${\displaystyle \sin ^{-1}x}$和${\displaystyle {\frac {1}{\sin x}}}$之间易造成混淆。

在编程中，函数${\displaystyle \arcsin }$, ${\displaystyle \arccos }$, ${\displaystyle \arctan }$通常叫做${\displaystyle \mathrm {asin} }$, ${\displaystyle \mathrm {acos} }$, ${\displaystyle \mathrm {atan} }$。很多编程语言提供两自变量atan2函数，它计算给定${\displaystyle y}$和${\displaystyle x}$的${\displaystyle {\frac {y}{x}}}$的反正切，但是值域为${\displaystyle [-\pi ,\pi ]}$。

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  在笛卡尔平面上${\displaystyle f(x)=\arcsin x}$(紅)和${\displaystyle f(x)=\arccos x}$(綠)函数的常用主值的图像。
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  在笛卡尔平面上${\displaystyle f(x)=\arctan x}$(紅)和${\displaystyle f(x)=\operatorname {arccot} x}$(綠)函数的常用主值的图像。
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  在笛卡尔平面上${\displaystyle f(x)=\operatorname {arcsec} x}$(紅)和${\displaystyle f(x)=\operatorname {arccsc} x}$(綠)函数的常用主值的图像。

下表列出基本的反三角函数。

名称	常用符号	定义	定义域	值域
反正弦	${\displaystyle y=\arcsin x}$	${\displaystyle x=\sin y}$	${\displaystyle [-1,1]}$	${\displaystyle [-{\frac {\pi }{2}},{\frac {\pi }{2}}]}$
反余弦	${\displaystyle y=\arccos x}$	${\displaystyle x=\cos y}$	${\displaystyle [-1,1]}$	${\displaystyle [0,\pi ]}$
反正切	${\displaystyle y=\arctan x}$	${\displaystyle x=\tan y}$	${\displaystyle \mathbb {R} }$	${\displaystyle (-{\frac {\pi }{2}},{\frac {\pi }{2}})}$
反余切	${\displaystyle y=\operatorname {arccot} x}$	${\displaystyle x=\cot y}$	${\displaystyle \mathbb {R} }$	${\displaystyle (0,\pi )}$
反正割	${\displaystyle y=\operatorname {arcsec} x}$	${\displaystyle x=\sec y}$	${\displaystyle (-\infty ,-1]\cup [1,+\infty )}$	${\displaystyle [0,{\frac {\pi }{2}})\cup ({\frac {\pi }{2}},\pi ]}$
反余割	${\displaystyle y=\operatorname {arccsc} x}$	${\displaystyle x=\csc y}$	${\displaystyle (-\infty ,-1]\cup [1,+\infty )}$	${\displaystyle [-{\frac {\pi }{2}},0)\cup (0,{\frac {\pi }{2}}]}$

（注意：某些數學教科書的作者將${\displaystyle \operatorname {arcsec} }$的值域定為${\displaystyle [0,{\frac {\pi }{2}})\cup [\pi ,{\frac {3\pi }{2}})}$因為當${\displaystyle \tan }$的定義域落在此區間時，${\displaystyle \tan }$的值域${\displaystyle \geq 0}$，如果${\displaystyle \operatorname {arcsec} }$的值域仍定為${\displaystyle [0,{\frac {\pi }{2}})\cup ({\frac {\pi }{2}},\pi ]}$，將會造成${\displaystyle \tan(\operatorname {arcsec} x)=\pm {\sqrt {x^{2}-1}}}$，如果希望${\displaystyle \tan(\operatorname {arcsec} x)={\sqrt {x^{2}-1}}}$，那就必須將${\displaystyle \operatorname {arcsec} }$的值域定為${\displaystyle [0,{\frac {\pi }{2}})\cup [\pi ,{\frac {3\pi }{2}})}$，基於類似的理由${\displaystyle \operatorname {arccsc} }$的值域定為${\displaystyle (-\pi ,-{\frac {\pi }{2}}]\cup (0,{\frac {\pi }{2}}]}$）

如果${\displaystyle x}$允许是复数，则${\displaystyle y}$的值域只适用它的实部。

余角：

${\displaystyle \arccos x={\frac {\pi }{2}}-\arcsin x}$

${\displaystyle \operatorname {arccot} x={\frac {\pi }{2}}-\arctan x}$

${\displaystyle \operatorname {arccsc} x={\frac {\pi }{2}}-\operatorname {arcsec} x}$

负数参数：

${\displaystyle \arcsin(-x)=-\arcsin x\!}$

${\displaystyle \arccos(-x)=\pi -\arccos x\!}$

${\displaystyle \arctan(-x)=-\arctan x\!}$

${\displaystyle \operatorname {arccot}(-x)=\pi -\operatorname {arccot} x\!}$

${\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}$

${\displaystyle \operatorname {arccsc}(-x)=-\operatorname {arccsc} x\!}$

倒数参数：

${\displaystyle \arccos {\frac {1}{x}}\,=\operatorname {arcsec} x}$

${\displaystyle \arcsin {\frac {1}{x}}\,=\operatorname {arccsc} x}$

${\displaystyle \arctan {\frac {1}{x}}={\frac {\pi }{2}}-\arctan x=\operatorname {arccot} x,\ }$ ${\displaystyle \ x>0}$

${\displaystyle \arctan {\frac {1}{x}}=-{\frac {\pi }{2}}-\arctan x=-\pi +\operatorname {arccot} x,\ }$ ${\displaystyle \ x<0}$

${\displaystyle \operatorname {arccot} {\frac {1}{x}}={\frac {\pi }{2}}-\operatorname {arccot} x=\arctan x,\ }$ ${\displaystyle \ x>0}$

${\displaystyle \operatorname {arccot} {\frac {1}{x}}={\frac {3\pi }{2}}-\operatorname {arccot} x=\pi +\arctan x,\ }$ ${\displaystyle \ x<0}$

${\displaystyle \operatorname {arcsec} {\frac {1}{x}}=\arccos x}$

${\displaystyle \operatorname {arccsc} {\frac {1}{x}}=\arcsin x}$

如果有一段正弦表：

${\displaystyle \arccos x=\arcsin {\sqrt {1-x^{2}}},}$ ${\displaystyle \ 0\leq x\leq 1}$

${\displaystyle \arctan x=\arcsin {\frac {x}{\sqrt {x^{2}+1}}}}$

注意只要在使用了复数的平方根的时候，我们选择正实部的平方根(或者正虚部，如果是负实数的平方根的话)。

从半角公式${\displaystyle \tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}}$，可得到：

${\displaystyle \arcsin x=2\arctan {\frac {x}{1+{\sqrt {1-x^{2}}}}}}$

${\displaystyle \arccos x=2\arctan {\frac {\sqrt {1-x^{2}}}{1+x}},}$ ${\displaystyle -1<x\leq +1}$

${\displaystyle \arctan x=2\arctan {\frac {x}{1+{\sqrt {1+x^{2}}}}}}$

通過定義可知：

${\displaystyle \theta }$	${\displaystyle \sin \theta }$	${\displaystyle \cos \theta }$	${\displaystyle \tan \theta }$	圖示
${\displaystyle \arcsin x}$	${\displaystyle \sin(\arcsin x)=x}$	${\displaystyle \cos(\arcsin x)={\sqrt {1-x^{2}}}}$	${\displaystyle \tan(\arcsin x)={\frac {x}{\sqrt {1-x^{2}}}}}$
${\displaystyle \arccos x}$	${\displaystyle \sin(\arccos x)={\sqrt {1-x^{2}}}}$	${\displaystyle \cos(\arccos x)=x}$	${\displaystyle \tan(\arccos x)={\frac {\sqrt {1-x^{2}}}{x}}}$
${\displaystyle \arctan x}$	${\displaystyle \sin(\arctan x)={\frac {x}{\sqrt {1+x^{2}}}}}$	${\displaystyle \cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}$	${\displaystyle \tan(\arctan x)=x}$
${\displaystyle \operatorname {arccot} x}$	${\displaystyle \sin(\operatorname {arccot} x)={\frac {1}{\sqrt {1+x^{2}}}}}$	${\displaystyle \cos(\operatorname {arccot} x)={\frac {x}{\sqrt {1+x^{2}}}}}$	${\displaystyle \tan(\operatorname {arccot} x)={\frac {1}{x}}}$
${\displaystyle \operatorname {arcsec} x}$	${\displaystyle \sin(\operatorname {arcsec} x)={\frac {\sqrt {x^{2}-1}}{x}}}$	${\displaystyle \cos(\operatorname {arcsec} x)={\frac {1}{x}}}$	${\displaystyle \tan(\operatorname {arcsec} x)={\sqrt {x^{2}-1}}}$
${\displaystyle \operatorname {arccsc} x}$	${\displaystyle \sin(\operatorname {arccsc} x)={\frac {1}{x}}}$	${\displaystyle \cos(\operatorname {arccsc} x)={\frac {\sqrt {x^{2}-1}}{x}}}$	${\displaystyle \tan(\operatorname {arccsc} x)={\frac {1}{\sqrt {x^{2}-1}}}}$

每个三角函数都周期于它的参数的实部上，在每个${\displaystyle 2\pi }$区间内通过它的所有值两次。正弦和余割的周期开始于${\displaystyle 2\pi k-{\frac {\pi }{2}}}$结束于${\displaystyle 2\pi k+{\frac {\pi }{2}}}$（这里的${\displaystyle k}$是一个整数），在${\displaystyle 2\pi k+{\frac {\pi }{2}}}$到${\displaystyle 2\pi k+{\frac {3\pi }{2}}}$上倒过来。余弦和正割的周期开始于${\displaystyle 2\pi k}$结束于${\displaystyle 2\pi k+\pi }$，在${\displaystyle 2\pi k+\pi }$到${\displaystyle 2\pi k+2\pi }$上倒过来。正切的周期开始于${\displaystyle 2\pi k-{\frac {\pi }{2}}}$结束于${\displaystyle 2\pi k+{\frac {\pi }{2}}}$，接着(向前)在${\displaystyle 2\pi k+{\frac {\pi }{2}}}$到${\displaystyle 2\pi k+{\frac {3\pi }{2}}}$上重复。余切的周期开始于${\displaystyle 2\pi k}$结束于${\displaystyle 2\pi k+\pi }$，接着（向前）在${\displaystyle 2\pi k+\pi }$到${\displaystyle 2\pi k+2\pi }$上重复。

这个周期性反应在一般反函数上：

${\displaystyle \sin y=x\ \Leftrightarrow \ (\ y=\arcsin x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=\pi -\arcsin x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}$

${\displaystyle \cos y=x\ \Leftrightarrow \ (\ y=\arccos x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=2\pi -\arccos x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}$

${\displaystyle \tan y=x\ \Leftrightarrow \ \ y=\arctan x+k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} }$

${\displaystyle \cot y=x\ \Leftrightarrow \ \ y=\operatorname {arccot} x+k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} }$

${\displaystyle \sec y=x\ \Leftrightarrow \ (\ y=\operatorname {arcsec} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=2\pi -\operatorname {arcsec} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}$

${\displaystyle \csc y=x\ \Leftrightarrow \ (\ y=\operatorname {arccsc} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=\pi -\operatorname {arccsc} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}$

对于实数${\displaystyle x}$的反三角函數的导函数如下：

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin x&{}={\frac {1}{\sqrt {1-x^{2}}}};\qquad |x|<1\\{\frac {\mathrm {d} }{\mathrm {d} x}}\arccos x&{}={\frac {-1}{\sqrt {1-x^{2}}}};\qquad |x|<1\\{\frac {\mathrm {d} }{\mathrm {d} x}}\arctan x&{}={\frac {1}{1+x^{2}}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccot} x&{}={\frac {-1}{1+x^{2}}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsec} x&{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccsc} x&{}={\frac {-1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\\end{aligned}}}$

舉例說明，设${\displaystyle \theta =\arcsin x\!}$，得到：

${\displaystyle {\frac {d\arcsin x}{dx}}={\frac {d\theta }{d\sin \theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}}$

因為要使根號內部恆為正，所以在條件加上${\displaystyle |x|<1}$，其他導數公式同理可證[1]。

积分其导数并固定在一点上的值给出反三角函数作为定积分的表达式：

${\displaystyle {\begin{aligned}\arcsin x&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arccos x&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arctan x&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arccot} x&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcsec} x&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arccsc} x&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\end{aligned}}}$

当${\displaystyle x}$等于1时，在有极限的域上的积分是瑕积分，但仍是良好定义的。

如同正弦和余弦函数，反三角函数可以使用无穷级数计算如下：

${\displaystyle {\begin{aligned}\arcsin z&{}=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \\&{}=\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{2n+1}}{(2n+1)}};\qquad |z|\leq 1\end{aligned}}}$

${\displaystyle {\begin{aligned}\arccos z&{}={\frac {\pi }{2}}-\arcsin z\\&{}={\frac {\pi }{2}}-\left[z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \right]\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{2n+1}}{(2n+1)}};\qquad |z|\leq 1\end{aligned}}}$

${\displaystyle {\begin{aligned}\arctan z&{}=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \\&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {arccot} z&{}={\frac {\pi }{2}}-\arctan z\\&{}={\frac {\pi }{2}}-\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \right)\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {arcsec} z&{}=\arccos \left(z^{-1}\right)\\&{}={\frac {\pi }{2}}-\left[z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots \right]\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{-(2n+1)}}{(2n+1)}};\qquad \left|z\right|\geq -4\end{aligned}}}$

${\displaystyle {\begin{aligned}\operatorname {arccsc} z&{}=\arcsin \left(z^{-1}\right)\\&{}=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots \\&{}=\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{-(2n+1)}}{2n+1}};\qquad \left|z\right|\geq 1\end{aligned}}}$

欧拉发现了反正切的更有效的级数：

${\displaystyle \arctan x=\infty {x}{1+x^{2}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kx^{2}}{(2k+1)(1+x^{2})}}}$。

（注意对x= 0在和中的项是空积1。）

${\displaystyle {\begin{aligned}\int \arcsin x\,dx&{}=x\,\arcsin x+{\sqrt {1-x^{2}}}+C,\qquad x\leq 1\\\int \arccos x\,dx&{}=x\,\arccos x-{\sqrt {1-x^{2}}}+C,\qquad x\leq 1\\\int \arctan x\,dx&{}=x\,\arctan x-{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arccot} x\,dx&{}=x\,\operatorname {arccot} x+{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arcsec} x\,dx&{}=x\,\operatorname {arcsec} x-\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C=x\,\operatorname {arcsec} x+\operatorname {sgn}(x)\ln \left|x-{\sqrt {x^{2}-1}}\right|+C\\\int \operatorname {arccsc} x\,dx&{}=x\,\operatorname {arccsc} x+\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C=x\,\operatorname {arccsc} x-\operatorname {sgn}(x)\ln \left|x-{\sqrt {x^{2}-1}}\right|+C\\\end{aligned}}}$

使用分部积分法和上面的简单导数很容易得出它们。

• 正切半角公式
• 平方根

• 埃里克·韦斯坦因. . MathWorld.
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