As far as the trigonometry is concerned, think of it this way:
With any lens, the light rays from the edges have much farther to travel than the ones from near the centre, so by the time the light from the edges reaches the film it is tired, so the edges are darker .
Speaking slightly more seriously, it is just the nature of the optical realities.
In general, the result is both natural and aesthetically pleasing, except when the shot is of something flat which we expect to be illuminated evenly - something like a wall.
The desire to illuminate the scene evenly when the angle of view is quite wide probably comes from viewing to many photographs on TVs/monitors - they look quite natural on prints.
“Photography is a complex and fluid medium, and its many factors are not applied in simple sequence. Rather, the process may be likened to the art of the juggler in keeping many balls in the air at one time!”
Ansel Adams, from the introduction to The Negative - The New Ansel Adams Photography Series / Book 2