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# 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. (0-2**π**) 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.

VoilĂ , la very practical equation:
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**.