Generated by the Physics Derivation Graph.
\cite{2001_HRW}; see figure 34-27 on page 824
}})
Eq. \ref{eq:5563180} is an initial equation.
\begin{equation}
\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}
\label{eq:5563180}
\end{equation}
Subtract \(\theta_{\rm Brewster}\) from both sides of Eq. \ref{eq:5563180}; yields Eq. \ref{eq:3893026}.
\begin{equation}
\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}
\label{eq:3893026}
\end{equation}
Eq. \ref{eq:9932375} is an initial equation.
\begin{equation}
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
\label{eq:9932375}
\end{equation}
Change variable \(\theta_2\) to \(\theta_{\rm refracted}\) and \(\theta_1\) to \(\theta_{\rm Brewster}\) in Eq. \ref{eq:9932375}; yields Eq. \ref{eq:4176694}.
\begin{equation}
n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )
\label{eq:4176694}
\end{equation}
Substitute LHS of Eq. \ref{eq:3893026} into Eq. \ref{eq:4176694}; yields Eq. \ref{eq:4962698}.
\begin{equation}
n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )
\label{eq:4962698}
\end{equation}
Eq. \ref{eq:3940135} is an identity.
\begin{equation}
\sin( 90^{\circ} - x ) = \cos( x )
\label{eq:3940135}
\end{equation}
Change variable \(\theta_{\rm Brewster}\) to \(x\) in Eq. \ref{eq:3940135}; yields Eq. \ref{eq:7426234}.
\begin{equation}
\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )
\label{eq:7426234}
\end{equation}
Substitute LHS of Eq. \ref{eq:7426234} into Eq. \ref{eq:4962698}; yields Eq. \ref{eq:9701820}.
\begin{equation}
n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )
\label{eq:9701820}
\end{equation}
Divide both sides of Eq. \ref{eq:9701820} by \(n_1\); yields Eq. \ref{eq:9314305}.
\begin{equation}
\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )
\label{eq:9314305}
\end{equation}
Divide both sides of Eq. \ref{eq:9314305} by \(\cos( \theta_{\rm Brewster} )\); yields Eq. \ref{eq:8585856}.
\begin{equation}
\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}
\label{eq:8585856}
\end{equation}
Eq. \ref{eq:2621708} is an identity.
\begin{equation}
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
\label{eq:2621708}
\end{equation}
Change variable \(\theta_{\rm Brewster}\) to \(x\) in Eq. \ref{eq:2621708}; yields Eq. \ref{eq:1898054}.
\begin{equation}
\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}
\label{eq:1898054}
\end{equation}
Substitute LHS of Eq. \ref{eq:1898054} into Eq. \ref{eq:8585856}; yields Eq. \ref{eq:5179630}.
\begin{equation}
\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }
\label{eq:5179630}
\end{equation}
Apply function \(\arctan{ x }\) with argument \(x\) to Eq. \ref{eq:5179630}; yields Eq. \ref{eq:8186016}
\begin{equation}
\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
\label{eq:8186016}
\end{equation}
Eq. \ref{eq:8186016} is one of the final equations.