反三角函数的的相互关系

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)egin{align} arcsin x&=-arcsin(-x)\ &=fracpi 2-arccos x\&=arctanfrac{x}{sqrt{1-x^2}} \&=arccossqrt{1-x^2} \ &=operatorname{arccot}frac{sqrt{1-x^2}}x end{align} ag 1arcsinx​=−arcsin(−x)=2π​−arccosx=arctan1−x2​x​=arccos1−x2​=arccotx1−x2​​​(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)egin{align} arccos x&=pi-arccos(-x)\ &=fracpi 2-arcsin x\ &=operatorname{arccot}frac{x}{sqrt{1-x^2}}\ &=arcsinsqrt{1-x^2}\ &=arctanfrac{sqrt{1-x^2}}x end{align} ag2arccosx​=π−arccos(−x)=2π​−arcsinx=arccot1−x2​x​=arcsin1−x2​=arctanx1−x2​​​(2)

arctan⁡x=−arctan⁡(−x)=π2−arccot⁡x=arcsin⁡x1+x2=arccos⁡11+x2=arccot⁡1x(3)egin{align} arctan x&=-arctan(-x)\ &=fracpi 2-operatorname{arccot}x\ &=arcsinfrac{x}{sqrt{1+x^2}}\ &=arccosfrac 1{sqrt{1+x^2}}\ &=operatorname{arccot}frac 1x end{align} ag 3arctanx​=−arctan(−x)=2π​−arccotx=arcsin1+x2​x​=arccos1+x2​1​=arccotx1​​(3)

arccot⁡x=π−arccot⁡(−x)=π2−arctan⁡x=arccos⁡x1+x2=arcsin⁡11+x2=arctan⁡1x(4)egin{align} operatorname{arccot}x&=pi-operatorname{arccot}(-x)\&=fracpi2-arctan x\&=arccos frac x{sqrt{1+x^2}}\&=arcsinfrac1{sqrt{1+x^2}}\&=arctanfrac 1x\ end{align} ag 4arccotx​=π−arccot(−x)=2π​−arctanx=arccos1+x2​x​=arcsin1+x2​1​=arctanx1​​(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)(x0,y>0,x^2+y^2>1)\ &=-pi-arcsin(xsqrt{1-y^2}+ysqrt{1-x^2})quad(x0,y>0,x2+y2>1)=−π−arcsin(x1−y2​+y1−x2​)(x0,y1)=−π−arcsin⁡(x1−y2−y1−x2)(x0,x2+y2>1)(6)egin{align} arcsin x-arcsin y&=arcsin(xsqrt{1-y^2}-ysqrt{1-x^2})quad(xyge0quad orquad x^2+y^2le1)\ &=pi-arcsin(xsqrt{1-y^2}-ysqrt{1-x^2})quad(x>0,y1)\ &=-pi-arcsin(xsqrt{1-y^2}-ysqrt{1-x^2})quad(x0,x^2+y^2>1)\ end{align} ag6arcsinx−arcsiny​=arcsin(x1−y2​−y1−x2​)(xy≥0orx2+y2≤1)=π−arcsin(x1−y2​−y1−x2​)(x>0,y1)=−π−arcsin(x1−y2​−y1−x2​)(x0,x2+y2>1)​(6)

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