Earth Subtended Solid Angle from 2000 km Satellite

[stunner5000pt]

This isn't so much an astronomy question as one would think
A perfectly black satellite is in orbit around the Earth at a height of 2000 km
What solid angle does the Earth subtend when viewed from the satellite?

$$\cos \theta = \frac{6370km}{8370km}$$
WEll if we look at hte diagram the angle theta is 40.4 degrees or 0.705 radians

we want the solid angle which is defined as
$$\Omega = \int_{0}^{2 \pi} \int_{0}^{0.705} \sin \theta' d \theta' d \phi$$

is this fine?

 

[Meir Achuz]

Isn't that sin theta.? Remember Sohcahtoa.