\section*{Problem 4 - 041224}

Solve the system of equations

\[
\begin{array}{r}
\frac{\sin x+\sin y}{\sin x-\sin y}=\frac{5}{3} \\
x+y=90^{\circ}
\end{array}
\]

A solution with integer degree approximations should be provided.
}

It is

\[
\frac{\sin x+\sin y}{\sin x-\sin y}=\frac{2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}}{2 \sin \frac{x-y}{2} \cos \frac{x+y}{2}}=\cot \frac{x-y}{2}
\]

since due to \(x+y=90^{\circ}\) here \(\sin \frac{x+y}{2}=\cos \frac{x+y}{2}=\frac{1}{2} \sqrt{2}\).

Furthermore, \(\frac{x-y}{2}=\frac{x}{2}-\left(45^{\circ}-\frac{x}{2}\right)=x-45^{\circ}\).

Thus, the equality sign is satisfied for all \(x\) and \(y\) such that \(\cot \left(x-45^{\circ}\right)=\frac{5}{3}\) and \(y=90^{\circ}-x\).

This yields \(x-45^{\circ} \approx 31^{\circ}+k \cdot 180^{\circ}\) (k integer).

\[
x \approx 76^{\circ}+k \cdot 180^{\circ} \quad ; \quad y \approx 14^{\circ}-k \cdot 180^{\circ}
\]

Adapted from \([5]\)