From the fact that \(\beta = \frac{\pi}{5}\text{,}\) it follows that \(\alpha = \frac{\pi}{2} - \frac{\pi}{5} = \frac{3\pi}{10}\text{.}\) In addition, we know that
\begin{equation}
\sin\left(\frac{\pi}{5}\right) = \frac{4}{h}\tag{4.1.5}
\end{equation}
and
\begin{equation}
\cos\left(\frac{\pi}{5}\right) = \frac{x}{h}\tag{4.1.6}
\end{equation}
\begin{equation}
h = \frac{4}{\sin\left(\frac{\pi}{5}\right)}\text{,}\tag{4.1.7}
\end{equation}
which is the exact numerical value of
\(h\text{.}\) Substituting this result in
Equation (4.1.6), we find that
\begin{equation}
\cos\left(\frac{\pi}{5}\right) = \frac{x}{\frac{4}{\sin\left(\frac{\pi}{5}\right)}}\text{.}\tag{4.1.8}
\end{equation}
Solving this equation for the single unknown \(x\) shows that
\begin{equation*}
x = \frac{4 \cos\left(\frac{\pi}{5}\right)}{\sin\left(\frac{\pi}{5}\right)}\text{.}
\end{equation*}
The approximate values of \(x\) and \(h\) are \(x \approx 5.506\) and \(h \approx 6.805\text{.}\)