反三角函数的的相互关系

arcsin⁡x=−arcsin⁡(−x)=π2−arccos⁡x=arctan⁡x1−x2=arccos⁡1−x2=arccot⁡1−x2x(1)\begin{align} \arcsin x&=-\arcsin(-x)\\ &=\frac\pi 2-\arccos x\\&=\arctan\frac{x}{\sqrt{1-x^2}} \\&=\arccos\sqrt{1-x^2} \\ &=\operatorname{arccot}\frac{\sqrt{1-x^2}}x \end{align}\tag 1

最后两个等号只在 x>0 时成立，下同

arccos⁡x=π−arccos⁡(−x)=π2−arcsin⁡x=arccot⁡x1−x2=arcsin⁡1−x2=arctan⁡1−x2x(2)\begin{align} \arccos x&=\pi-\arccos(-x)\\ &=\frac\pi 2-\arcsin x\\ &=\operatorname{arccot}\frac{x}{\sqrt{1-x^2}}\\ &=\arcsin\sqrt{1-x^2}\\ &=\arctan\frac{\sqrt{1-x^2}}x \end{align}\tag2

arctan⁡x=−arctan⁡(−x)=π2−arccot⁡x=arcsin⁡x1+x2=arccos⁡11+x2=arccot⁡1x(3)\begin{align} \arctan x&=-\arctan(-x)\\ &=\frac\pi 2-\operatorname{arccot}x\\ &=\arcsin\frac{x}{\sqrt{1+x^2}}\\ &=\arccos\frac 1{\sqrt{1+x^2}}\\ &=\operatorname{arccot}\frac 1x \end{align}\tag 3

arccot⁡x=π−arccot⁡(−x)=π2−arctan⁡x=arccos⁡x1+x2=arcsin⁡11+x2=arctan⁡1x(4)\begin{align} \operatorname{arccot}x&=\pi-\operatorname{arccot}(-x)\\&=\frac\pi2-\arctan x\\&=\arccos \frac x{\sqrt{1+x^2}}\\&=\arcsin\frac1{\sqrt{1+x^2}}\\&=\arctan\frac 1x\\ \end{align}\tag 4

反三角函数的和差

反正弦：

arcsin⁡x+arcsin⁡y=arcsin⁡(x1−y2+y1−x2)(xy≤0orx2+y2≤1)=π−arcsin⁡(x1−y2+y1−x2)(x>0,y>0,x2+y2>1)=−π−arcsin⁡(x1−y2+y1−x2)(x<0,y<0,x2+y2>1)(5)\begin{align} \arcsin x+\arcsin y&=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})\quad(xy\le0\quad or\quad x^2+y^2\le1)\\ &=\pi-\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})\quad(x>0,y>0,x^2+y^2>1)\\ &=-\pi-\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})\quad(x<0,y<0,x^2+y^2>1)\\ \end{align}\tag5

arcsin⁡x−arcsin⁡y=arcsin⁡(x1−y2−y1−x2)(xy≥0orx2+y2≤1)=π−arcsin⁡(x1−y2−y1−x2)(x>0,y<0,x2+y2>1)=−π−arcsin⁡(x1−y2−y1−x2)(x<0,y>0,x2+y2>1)(6)\begin{align} \arcsin x-\arcsin y&=\arcsin(x\sqrt{1-y^2}-y\sqrt{1-x^2})\quad(xy\ge0\quad or\quad x^2+y^2\le1)\\ &=\pi-\arcsin(x\sqrt{1-y^2}-y\sqrt{1-x^2})\quad(x>0,y<0,x^2+y^2>1)\\ &=-\pi-\arcsin(x\sqrt{1-y^2}-y\sqrt{1-x^2})\quad(x<0,y>0,x^2+y^2>1)\\ \end{align}\tag6

反余弦：

arccos⁡x+arccos⁡y=arccos⁡[xy−(1−x2)(1−y2)](x+y≥0)=2π−arccos⁡[xy−(1−x2)(1−y2)](x+y<0)(7)\begin{align} \arccos x+\arccos y&=\arccos[xy-\sqrt{(1-x^2)(1-y^2})]\quad(x+y\ge0)\\ &=2\pi-\arccos[xy-\sqrt{(1-x^2)(1-y^2})]\quad(x+y<0)\\ \end{align}\tag7

arccos⁡x−arccos⁡y=−arccos⁡[xy+(1−x2)(1−y2)](x≥y)=arccos⁡[xy+(1−x2)(1−y2)](x<y)(8)\begin{align} \arccos x-\arccos y&=-\arccos[xy+\sqrt{(1-x^2)(1-y^2})]\qquad\qquad(x\ge y)\\ &=\arccos[xy+\sqrt{(1-x^2)(1-y^2})]\qquad\qquad(x<y)\\ \end{align}\tag8

反正切：

arctan⁡x+arctan⁡y=arctan⁡x+y1−xy(xy<1)=π+arctan⁡x+y1−xy(x>0,xy>1)=−π+arctan⁡x+y1−xy(x<0,xy>1)(9)\begin{align} \arctan x+\arctan y&=\arctan\frac{x+y}{1-xy}\quad(xy<1)\\ &=\pi+\arctan\frac{x+y}{1-xy}\quad(x>0,xy>1)\\ &=-\pi+\arctan\frac{x+y}{1-xy}\quad(x<0,xy>1)\\ \end{align}\tag9

arctan⁡x−arctan⁡y=arctan⁡x−y1+xy(xy>−1)=π+arctan⁡x−y1+xy(x>0,xy<−1)=−π+arctan⁡x−y1+xy(x<0,xy<−1)(10)\begin{align} \arctan x-\arctan y&=\arctan\frac{x-y}{1+xy}\quad(xy>-1)\\ &=\pi+\arctan\frac{x-y}{1+xy}\quad(x>0,xy<-1)\\ &=-\pi+\arctan\frac{x-y}{1+xy}\quad(x<0,xy<-1)\\ \end{align}\tag{10}

反三角函数的二倍

反正弦：

2arcsin⁡x=arcsin⁡(2x1−x2)(∣x∣≤22)=π−arcsin⁡(2x1−x2)(22<x≤1)=−π−arcsin⁡(2x1−x2)(−1≤x<−22)(11)\begin{align} 2\arcsin x&=\arcsin(2x\sqrt{1-x^2})\quad(|x|\le\frac{\sqrt{2}}{ 2})\\ &=\pi-\arcsin(2x\sqrt{1-x^2})\quad(\frac{\sqrt2}2<x\le1)\\ &=-\pi-\arcsin(2x\sqrt{1-x^2})\quad(-1\le x<-\frac{\sqrt2}2)\\ \end{align}\tag{11}

反余弦：

2arccos⁡x=arccos⁡(2x2−1)(0≤x≤1)=2π−arccos⁡(2x2−1)(−1≤x<0)(12)\begin{align} 2\arccos x&=\arccos(2x^2-1)\quad\quad\quad\quad\quad(0\le x\le 1)\\ &=2\pi-\arccos(2x^2-1)\quad\quad\quad\quad\quad(-1\le x<0)\\ \end{align}\tag{12}

反正切：

2arctan⁡x=arctan⁡2x1−x2(∣x∣<1)=π+arctan⁡2x1−x2(∣x∣>1)=−π+arctan⁡2x1−x2(x<−1)(13)\begin{align} 2\arctan x&=\arctan\frac{2x}{1-x^2}\quad\qquad\qquad\qquad(|x|<1)\\ &=\pi+\arctan\frac{2x}{1-x^2}\quad\qquad\qquad\qquad(|x|>1)\\ &=-\pi+\arctan\frac{2x}{1-x^2}\qquad\qquad\qquad(x<-1) \end{align}\tag{13}

余弦反余弦复合的重要关系公式：

cos⁡(narccos⁡x)=(x+x2−1)n+(x−x2−1)n2(n≥1)(14)\cos(n\arccos x)=\frac{(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n}2\quad(n\ge 1)\tag{14}