<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

Determine whether or not there exists a number $\alpha&gt;0$, so that
there exists a hyperbolic triangle T whose interior angles are
$\frac{\pi}{3},\ \frac{\pi}{5}$, and $\alpha$, and whose
hyperbolic area area(T) is $\frac{\pi}{25}$. If such an $\alpha$
exists, determine its value (or values).
\medskip

{\bf Answer}

By the Gauss-Bonnet Formula:

${\rm{area}}(\tau)=\pi-$\ (sum of interior angles)

and so \begin{eqnarray*} {\rm{area}}(\tau) &amp; = &amp;
\pi-\left(\ds\frac{\pi}{3}+\ds\frac{\pi}{5}+\alpha\right)\\ &amp; = &amp;
\pi\left(1-\ds\frac{1}{3}-\ds\frac{1}{5}\right)-\alpha\\ &amp; = &amp;
\pi\ds\frac{7}{15}-\alpha \end{eqnarray*}

Since the only requirement is that area$(\tau)&gt;0$, there is such
an $\alpha$, namely
$\alpha=\left(\ds\frac{7}{15}-\ds\frac{1}{25}\right)\pi =
\un{\ds\frac{32}{75}\pi}$

\end{document}
</pre></body></html>