Mathematics: Sextant angles

Vertical sextant angle

The triangle OBL (see fig. below) can be described in terms of H, α and Distance:

Distance = H/tan(α)
The angle in rad. (0-2π) and both height and distance in metres.

From rad. to degrees: α = A * π/180, 'A' being the same angle in degrees.
To describe angle A in minutes total, then A*60 = a, thus α = (a/60) * (π/180). So, α = a/3438, 'a' being the angle in arc minutes.
FACTUM: tan(x) = x, if angle x is small.

Resulting in (with π = 3.14): Distance (m)= H * 3438/a

Furthermore, distance in nm. = distance in meters/1852.

Voilà, la very practical equation:

Distance = 1.856 * H/a

It contains just two approximations, both of neglitible influence. First, we left out the tan function and second we used 3.14 for π.
Please realize that a smaller angle improves the approximation of the tan. Yet, as an opposing effect the instrument error of a smaller sextant angle increases.

All in all, the factor 1.856 is not a typo, and just by chance near to the nautical mile: 1.852 kilometres. If you are still reading, you are very brave person and might perhaps agree that it originates from: (60 * 180)/(π * 1852).

So far we considered a perfect triangle (OBL) and forgot that life isn't always perfect. Height h is usually quite small, but distance SB sometimes is not. This leads to an extra premise, which is seldom mentioned by other navigation textbooks:
Angle OLS should be bigger than 15°.

Further reading:
Online navigation courses.
Flotilla sailing holidays.
RYA & ASA sailing schools in Greece.
Yacht charters guide.

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