I want to plot the region of a sphere that satisfies the equation

$$
\tan\theta\geq-\frac{rz}{rx\cos\phi + ry\sin\phi}
$$

where $rx$, $ry$, $rz$ are known constants. The spherical coordinates $\theta$ and $\phi$ are linked together, making the desired region a smooth cutout of the total sphere. 

The basic parametric plot code for a sphere is: 

```
ParametricPlot3D[{Cos[ϕ]*Sin[θ], Sin[ϕ]*Sin[θ], Cos[θ]}, {ϕ, 0, 2 π}, {θ, 0, π}]

```

I essentially want to apply the above inequality to the limits of this plot, but cannot figure out how to successfully do so. I could just create a table of points in this region and use `ListPlot3D` or something, but I would prefer a smooth surface.