# 反正弦

 反正弦 性質 奇偶性 奇 定義域 [-1, 1] 到達域 ${\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$ 周期 N/A 特定值 當x=0 0 當x=+∞ N/A 當x=-∞ N/A 最大值 ${\displaystyle {\frac {\pi }{2}}}$ 最小值 ${\displaystyle -{\frac {\pi }{2}}}$ 其他性質 渐近线 N/A 根 0 拐點 原點 不動點 0

## 定義

${\displaystyle \arcsin :\left[-1,1\right]\rightarrow \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$

${\displaystyle \arcsin x=-{\mathrm {i} }\ln \left({\mathrm {i} }x+{\sqrt {1-x^{2}}}\right)\,}$

## 運算

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

${\displaystyle \arcsin x=\sum _{k=0}^{\infty }{-{\frac {1}{2}} \choose k}(-1)^{k}{\frac {x^{2k+1}}{2k+1}}=x+{\frac {1}{6}}x^{3}+{\frac {3}{40}}x^{5}+{\frac {5}{112}}x^{7}+\cdots }$ .

${\displaystyle \arcsin \left(-x\right)=-\arcsin x}$

${\displaystyle \arcsin x_{1}\pm \arcsin x_{2}={\begin{cases}X&\pm x_{1}x_{2}\leq 0\lor x_{1}^{2}+x_{2}^{2}\leq 1\\\pi -X&x_{1}>0\land \pm x_{2}>0\land x_{1}^{2}+x_{2}^{2}>1\\-\pi -X&x_{1}<0\land \pm x_{2}<0\land x_{1}^{2}+x_{2}^{2}>1\end{cases}}}$

${\displaystyle \arcsin(x\pm y)=\arcsin \left({\sqrt {\frac {1+x^{2}-y^{2}-{\sqrt {1+x^{4}+y^{4}-2x^{2}y^{2}-2x^{2}-2y^{2}}}}{2}}}\right)\pm \arcsin \left({\sqrt {\frac {1-x^{2}+y^{2}-{\sqrt {1+x^{4}+y^{4}-2x^{2}y^{2}-2x^{2}-2y^{2}}}}{2}}}\right)}$

${\displaystyle \arcsin(2x)=2\arcsin \left({\sqrt {\frac {1-{\sqrt {1-4x^{2}}}}{2}}}\right)}$

${\displaystyle \arcsin \left({\frac {x}{2}}\right)=2\arcsin \left({\sqrt {\frac {1-{\sqrt {1-{\frac {x^{2}}{4}}}}}{2}}}\right)}$

${\displaystyle \arcsin(kx)=2\arcsin \left({\sqrt {\frac {1-{\sqrt {1-k^{2}x^{2}}}}{2}}}\right)}$ （对0 ≤ kx ≤ 1）

${\displaystyle \arcsin(sinx)={\begin{cases}-(X+\pi )&x\in [-\pi ,-{\frac {\pi }{2}}]\\X&x\in (-{\frac {\pi }{2}},{\frac {\pi }{2}})\\\pi -X&x\in [{\frac {\pi }{2}},\pi ]\end{cases}}}$