Chapter 1.2, Problem 33E

Show that the following nonlinear system has 18 solutions if $0\le \alpha \le 2\pi ,0\le \beta \le 2\pi$, and $0\le \gamma \le 2\pi$

$\begin{array}{l}\sin \alpha +2\cos \beta +3\tan \gamma =0\\ 2\sin \alpha +5\cos \beta +3\tan \gamma =0\\ -\sin \alpha -5\cos \beta +5\tan \gamma =0\end{array}$

[Hint: Begin by making the substitutions x = sin α, y = cos β, and z = tan γ.]