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[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 factor sin(angle)	ground distance cos(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.