\section*{Exercise 9 - V01009}

For a non-right triangle, the angles and side \(c\) are given. Derive a formula for calculating the height \(h_{c}\)!
In the right-angled triangles \(A F C\) and \(B F C\), on the one hand, \(h_{c}=a \sin \alpha\), and on the other hand, \(h_{c}=b \sin \beta\). For the area \(F\), substituting the two equations gives

\[
F=\frac{1}{2} c \cdot h_{c}=\frac{1}{2} a b \sin \gamma=\frac{1}{2} \frac{h_{c}}{\sin \alpha} \frac{h_{c}}{\sin \beta} \sin \gamma
\]

Solving the equation for \(h_{c}\) yields the desired relationship

\[
h_{c}=\frac{c \cdot \sin \alpha \cdot \sin \beta}{\sin \gamma}
\]