求解 y=sin(x^2)

$$y = \sin ( { x }^{ 2 } )$$

求解 x 的值 (复数求解)

$$x=\left(-1\right)\left(\left(-i\right)\ln(iy+\left(-1\right)\left(\left(-1\right)y^{2}+1\right)^{\frac{1}{2}})+2\pi n_{1}\right)^{\frac{1}{2}}\text{, }n_{1}\in \mathrm{Z}$$

$$x=\left(\left(-i\right)\ln(iy+\left(-1\right)\left(\left(-1\right)y^{2}+1\right)^{\frac{1}{2}})+2\pi n_{1}\right)^{\frac{1}{2}}\text{, }n_{1}\in \mathrm{Z}$$

$$x=\left(-1\right)\left(\left(-i\right)\ln(iy+\left(\left(-1\right)y^{2}+1\right)^{\frac{1}{2}})+2\pi n_{5}\right)^{\frac{1}{2}}\text{, }n_{5}\in \mathrm{Z}$$

$$x=\left(\left(-i\right)\ln(iy+\left(\left(-1\right)y^{2}+1\right)^{\frac{1}{2}})+2\pi n_{5}\right)^{\frac{1}{2}}\text{, }n_{5}\in \mathrm{Z}$$

求解 x 的值

$$\left\{\begin{matrix}\\x=-\sqrt{\arcsin(y)+2\pi n_{1}}\text{, }n_{1}\in \mathrm{Z}\text{, }\left(n_{1}\geq 1\text{ and }|y|\leq 1\right)\text{ or }\left(n_{1}>-1\text{ and }y\leq 1\text{ and }y\geq 0\right)\text{; }x=\sqrt{\arcsin(y)+2\pi n_{1}}\text{, }n_{1}\in \mathrm{Z}\text{, }\left(n_{1}\geq 1\text{ and }|y|\leq 1\right)\text{ or }\left(n_{1}>-1\text{ and }y\leq 1\text{ and }y\geq 0\right)\text{, }&\text{unconditionally}\\x=\sqrt{-\arcsin(y)+2\pi n_{2}+\pi }\text{, }n_{2}\in \mathrm{Z}\text{, }n_{2}>-1\text{; }x=-\sqrt{-\arcsin(y)+2\pi n_{2}+\pi }\text{, }n_{2}\in \mathrm{Z}\text{, }n_{2}>-1\text{, }&|y|\leq 1\end{matrix}\right.$$

求解 y 的值

$$y=\sin(x^{2})$$