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Trig w small angles
 On Being Approximate
Under construction

[sin/tan divergence]

[easy to remember approximations]

The angle multiplyer.I've said a 2° degree thumb gives you a x30 distance. It's really more like 28.7 (1/sin(2°)). The error is consistently about 5% high, which seems well worth the simplification.
 angle multiplyer error (closer) 1/4° 240 +5% (229) 0.5° 120 +5% (114) 1° 60 +5% (57.3) 2° 30 +5% (28.6) 6° 10 +5% (9.57) 10° 6 +4% (5.76) 20° 3 +3% (2.92)

Here is one simplification, admittedly not very simple.
We should work out one with just 3 or 4 numbers to remember, as this one's 5% error is excessively precise for most stuff.
 angle altitude factorsin(angle) ground distancecos(angle) 0° (it's on the ground) 0 1 10° (1 fist up) 1/6 1 20° (2 fists up) 1/3 1 30° (3 fists up) 1/2 5/6 40° (4 fists up) 2/3 2/3 50° (5 fists up) 2/3 2/3 60° (3 fists down) 5/6 1/2 70° (2 fists down) 1 1/3 80° (1 fist down) 1 1/6 90° (up) 1 0
which hovers around 5% error.

 A View from the Back of the Envelope Comments encouraged. - Mitchell N Charity