A point has rectangular coordinates $(10, -18, 35)$ and spherical coordinates $(\rho, \theta, \phi).$  Find $\tan \theta.$
Dividing the equations $x = \rho \sin \phi \cos \theta$ and $y = \rho \sin \phi \sin \theta,$ we get
\[\tan \theta = \frac{y}{x} = \frac{-18}{10} = \boxed{-\frac{9}{5}}.\]