Solve n_{1}sin(θ_{1})=n_{2}sin(θ_{2})

$$n _ { 1 } \sin ( \theta _ { 1 } ) = n _ { 2 } \sin ( \theta _ { 2 } )$$

Solve for n_1

$$\left\{\begin{matrix}n_{1}=\frac{n_{2}\sin(\theta _{2})}{\sin(\theta _{1})}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta _{1}=\pi n_{1}\\n_{1}\in \mathrm{R}\text{, }&\left(n_{2}=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{1}=\pi n_{1}\right)\text{ or }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{2}=\pi n_{1}\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{1}=\pi n_{1}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\sin(\theta _{1})n_{1}=n_{2}\sin(\theta _{2})$$

Divide both sides by \sin(\theta _{1}).

$$\frac{\sin(\theta _{1})n_{1}}{\sin(\theta _{1})}=\frac{n_{2}\sin(\theta _{2})}{\sin(\theta _{1})}$$

Dividing by \sin(\theta _{1}) undoes the multiplication by \sin(\theta _{1}).

$$n_{1}=\frac{n_{2}\sin(\theta _{2})}{\sin(\theta _{1})}$$

Solve for n_2

$$\left\{\begin{matrix}n_{2}=\frac{n_{1}\sin(\theta _{1})}{\sin(\theta _{2})}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta _{2}=\pi n_{1}\\n_{2}\in \mathrm{R}\text{, }&\left(n_{1}=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{2}=\pi n_{1}\right)\text{ or }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{1}=\pi n_{1}\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta _{2}=\pi n_{1}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$n_{2}\sin(\theta _{2})=n_{1}\sin(\theta _{1})$$

The equation is in standard form.

$$\sin(\theta _{2})n_{2}=n_{1}\sin(\theta _{1})$$

Divide both sides by \sin(\theta _{2}).

$$\frac{\sin(\theta _{2})n_{2}}{\sin(\theta _{2})}=\frac{n_{1}\sin(\theta _{1})}{\sin(\theta _{2})}$$

Dividing by \sin(\theta _{2}) undoes the multiplication by \sin(\theta _{2}).

$$n_{2}=\frac{n_{1}\sin(\theta _{1})}{\sin(\theta _{2})}$$