An airplane over the pacific sights an atoll at 8 degree angle of depression. If the plane is 520hm above water, how many kilometers is it from a point 520 m directly above the center of the atoll?

Assume the Earth is flat.
Draw a figure with flat horizontal Earth surface along the bottom. Put three points on the figure:
P (airplane)
A (atoll on the ground)
B (point 520 m above the atoll)

Draw the (right) triangle PAB. The angle at P is 8 degrees, aimed below horizontal to point A

It should be obvious that
BA = 520
PB = 520/tan 8 = 3700 m