From: Frank Reed
Date: 2016 Aug 29, 19:10 -0700
David Pike, you wrote:
"So Range = 115/tan0.15 x 5280 = 8.3 statute miles."
Yes. I got 20 miles with a somewhat different estimate of the thickness of that broken horizon. It's a pretty good trick though. Familiarity with the uncertainty of the sea horizon provides us with an ability to estimate distance that most people would never suspect is possible.
Regarding the calculation, I would just note here that you never need to invoke a tangent when you're dealing with small angles unless you are absolutely sure you have that specific geometry. You could just as easily use a sine. We rarely have any assurance that either geometry is correct. Better than either of these for practical calculation, remember that angles are ratios, and for small angles you can just remember the angular ratio rule:
(minutes of arc)/3438 = size/distance.
This is a versatile, practical trick that saves us from having to carry around trig tables (or slide rules, or calculators) just to solve simple angular size problems. It's 99% accurate for angles less than 10°, and it's 99.9% accurate for angles smaller than 3°. And it is based on the fundamental definition of an angle as a ratio of "distance out" to "distance across" the line of sight, where the latter is measured on a circular arc. A unit angle, where the ratio is exactly one, is usually called one radian though it needs no units on it. We convert these pure angles to conventional angular measures by comparing a full circle to the number of conventional angular measures in a circle. In the case of minutes of arc, that's 60·360/(2·pi) which is 3438. That's a number worth memorizing.
Here's a little astronomical trivia you can toss around to impress the astronomically-impressable: the number of AUs or astronomical units in one parsec is 60·3438. Why is that? Well, I'll leave that as a little entertainment for the reader... Oh, and of course while you may have to explain what an astronomical unit is to your impressable audience, the parsec has entered pop culture thanks to a famous little technical error (fixed retroactively by fans) in the original Star Wars movie nearly 40 years ago