Solve nlambda=esinθ

$$n \lambda = e \sin \theta$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{e\sin(\theta )}{\lambda }\text{, }&\lambda \neq 0\\n\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\text{ and }\lambda =0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\lambda n=e\sin(\theta )$$

Divide both sides by \lambda .

$$\frac{\lambda n}{\lambda }=\frac{e\sin(\theta )}{\lambda }$$

Dividing by \lambda undoes the multiplication by \lambda .

$$n=\frac{e\sin(\theta )}{\lambda }$$

Solve for n

$$\left\{\begin{matrix}n=\frac{e\sin(\theta )}{\lambda }\text{, }&\lambda \neq 0\\n\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\text{ and }\lambda =0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\lambda n=e\sin(\theta )$$

Divide both sides by \lambda .

$$\frac{\lambda n}{\lambda }=\frac{e\sin(\theta )}{\lambda }$$

Dividing by \lambda undoes the multiplication by \lambda .

$$n=\frac{e\sin(\theta )}{\lambda }$$

Solve for θ (complex solution)

$$\theta =i+\left(-i\right)\ln(in\lambda +\left(-i\right)\left(n^{2}\lambda ^{2}+\left(-1\right)e^{2}\right)^{\frac{1}{2}})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =i+\left(-i\right)\ln(in\lambda +i\left(n^{2}\lambda ^{2}+\left(-1\right)e^{2}\right)^{\frac{1}{2}})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}$$

Solve for θ

$$\theta =arcSin(n\lambda e^{-1})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =\pi +2\pi n_{2}+\left(-1\right)arcSin(n\lambda e^{-1})\text{, }n_{2}\in \mathrm{Z}$$