Solve I_{m}=I_0^2sinθ_{0}

$$I _ { m } = I _ { 0 } ^ { 2 } \sin \theta _ { 0 }$$

Solve for I_0 (complex solution)

$$\left\{\begin{matrix}I_{0}=-i\sqrt{I_{m}}\left(-\sin(\theta _{0})\right)^{-\frac{1}{2}}\text{; }I_{0}=i\sqrt{I_{m}}\left(-\sin(\theta _{0})\right)^{-\frac{1}{2}}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta _{0}=\pi n_{1}\\I_{0}\in \mathrm{C}\text{, }&I_{m}=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{0}=\pi n_{1}\end{matrix}\right.$$

Solve for I_0

$$\left\{\begin{matrix}I_{0}=\sqrt{\frac{I_{m}}{\sin(\theta _{0})}}\text{; }I_{0}=-\sqrt{\frac{I_{m}}{\sin(\theta _{0})}}\text{, }&\left(I_{m}\geq 0\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\theta _{0}>2\pi n_{2}\text{ and }\theta _{0}<2\pi n_{2}+\pi \right)\right)\text{ or }\left(I_{m}\leq 0\text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(\theta _{0}>2\pi n_{3}+\pi \text{ and }\theta _{0}<2\pi n_{3}+2\pi \right)\right)\\I_{0}\in \mathrm{R}\text{, }&I_{m}=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{0}=\pi n_{1}\end{matrix}\right.$$

Solve for I_m

$$I_{m}=I_{0}^{2}\sin(\theta _{0})$$