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${\displaystyle f(x)=\arcsin x}$ (红)和${\displaystyle f(x)=\arccos x}$ (绿)函数的常用主值的图像。

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  在笛卡尔平面上${\displaystyle f(x)=\arctan x}$ (红)和${\displaystyle f(x)=\operatorname {arccot} x}$ (绿)函数的常用主值的图像。

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  在笛卡尔平面上${\displaystyle f(x)=\operatorname {arcsec} x}$ (红)和${\displaystyle f(x)=\operatorname {arccsc} x}$ (绿)函数的常用主值的图像。

下表列出基本的反三角函数。

（注意：某些数学教科书的作者将${\displaystyle \operatorname {arcsec} }$ 的值域定为${\displaystyle [0,{\frac {\pi }{2}})\cup [\pi ,{\frac {3\pi }{2}})}$ 因为当${\displaystyle \tan }$ 的定义域落在此区间时，${\displaystyle \tan }$ 的值域${\displaystyle \geq 0}$ ，如果${\displaystyle \operatorname {arcsec} }$ 的值域仍定为${\displaystyle [0,{\frac {\pi }{2}})\cup ({\frac {\pi }{2}},\pi ]}$ ，将会造成${\displaystyle \tan(\operatorname {arcsec} x)=\pm {\sqrt {x^{2}-1}}}$ ，如果希望${\displaystyle \tan(\operatorname {arcsec} x)={\sqrt {x^{2}-1}}}$ ，那就必须将${\displaystyle \operatorname {arcsec} }$ 的值域定为${\displaystyle [0,{\frac {\pi }{2}})\cup [\pi ,{\frac {3\pi }{2}})}$ ，基于类似的理由${\displaystyle \operatorname {arccsc} }$ 的值域定为${\displaystyle (-\pi ,-{\frac {\pi }{2}}]\cup (0,{\frac {\pi }{2}}]}$ ）

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：

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

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

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

通过定义可知：

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${\displaystyle x}$ 的反三角函数的导函数如下：

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

举例说明，设${\displaystyle \theta =\arcsin x\!}$ ，得到：

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

因为要使根号内部恒为正，所以在条件加上${\displaystyle |x|<1}$ ，其他导数公式同理可证^[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${\displaystyle x=0}$ 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。

举例

使用${\displaystyle \int u\,\mathrm {d} v=uv-\int v\,\mathrm {d} u}$ ，设

${\displaystyle {\begin{aligned}u&{}=&\arcsin x&\quad \quad \mathrm {d} v=\mathrm {d} x\\\mathrm {d} u&{}=&{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}&\quad \quad {}v=x\end{aligned}}}$ 

则

${\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin x-\int {\frac {x}{\sqrt {1-x^{2}}}}\,\mathrm {d} x}$ 

换元

${\displaystyle k=1-x^{2}.\,}$ 

则

${\displaystyle \mathrm {d} k=-2x\,\mathrm {d} x}$ 

且

${\displaystyle \int {\frac {x}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=-{\frac {1}{2}}\int {\frac {\mathrm {d} k}{\sqrt {k}}}=-{\sqrt {k}}}$ 

换元回x得到

${\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin x+{\sqrt {1-x^{2}}}+C}$ 

arcsin x + arcsin y

${\displaystyle \arcsin x+\arcsin y=\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),xy\leq 0\lor x^{2}+y^{2}\leq 1}$ 

${\displaystyle \arcsin x+\arcsin y=\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),x>0,y>0,x^{2}+y^{2}>1}$ 

${\displaystyle \arcsin x+\arcsin y=-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),x<0,y<0,x^{2}+y^{2}>1}$ 

arcsin x - arcsin y

${\displaystyle \arcsin x-\arcsin y=\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right),xy\geq 0\lor x^{2}+y^{2}\leq 1}$ 

${\displaystyle \arcsin x-\arcsin y=\pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right),x>0,y<0,x^{2}+y^{2}>1}$ 

${\displaystyle \arcsin x-\arcsin y=-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),x<0,y>0,x^{2}+y^{2}>1}$ 

arccos x + arccos y

${\displaystyle \arccos x+\arccos y=\arccos \left(xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x+y\geq 0}$ 

${\displaystyle \arccos x+\arccos y=2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x+y<0}$ 

arccos x - arccos y

${\displaystyle \arccos x-\arccos y=-\arccos \left(xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x\geq y}$ 

${\displaystyle \arccos x-\arccos y=\arccos \left(xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x<y}$ 

arctan x + arctan y

${\displaystyle \arctan \,x+\arctan \,y=\arctan \,{\frac {x+y}{1-xy}},xy<1}$ 

${\displaystyle \arctan \,x+\arctan \,y=\pi +\arctan \,{\frac {x+y}{1-xy}},x>0,xy>1}$ 

${\displaystyle \arctan \,x+\arctan \,y=-\pi +\arctan \,{\frac {x+y}{1-xy}},x<0,xy>1}$ 

arctan x - arctan y

${\displaystyle \arctan x-\arctan y=\arctan {\frac {x-y}{1+xy}},xy>-1}$ 

${\displaystyle \arctan x-\arctan y=\pi +\arctan {\frac {x-y}{1+xy}},x>0,xy<-1}$ 

${\displaystyle \arctan x-\arctan y=-\pi +\arctan {\frac {x-y}{1+xy}},x<0,xy<-1}$ 

arccot x + arccot y

${\displaystyle \operatorname {arccot} x+\operatorname {arccot} y=\operatorname {arccot} {\frac {xy-1}{x+y}},x>-y}$ 

${\displaystyle \operatorname {arccot} x+\operatorname {arccot} y=\operatorname {arccot} {\frac {xy-1}{x+y}}+\pi ,x<-y}$ 

arcsin x + arccos y

${\displaystyle \arcsin x+\arccos x={\frac {\pi }{2}},|x|\leq 1}$ 

arctan x + arccot y

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

• 正切半角公式
• 平方根

• 埃里克·韦斯坦因. Inverse Trigonometric Functions. MathWorld. 
• http://mathworld.wolfram.com/InverseTangent.html（页面存档备份，存于互联网档案馆）