The question may seem unrelated to sailing, or may seem trivial when viewed through modern eyes, but the question is of critical importance for navigation. The sine function is pretty much in any problem that involves a map. For instance, for a chartplotter to display a course over ground (COG) arrow, the sine function is involved.

Today, those values can be computed pretty fast, thanks to calculus, and are buried somewhere in the circuitry of a chartplotter. Thus, no pilot has to worry about trigonometry. Prior to the invention of computers, calculators, or even sliding rules, having what is known as a « sine table » was however of critical importance.

x (degrees) | sin(x) |

1 | 0.017 |

2 | 0.034 |

… | |

90 | 1.000 |

How did we find those values?

I am currently reading « Heavenly mathematics » to better understand the haversine formula and sight reduction tables. Both comes handy when doing celestial navigation. The understanding of the origins of formulas is overkill for practical purposes, but this is how some of us are having fun. Amongst other things, the first chapters of the book go over the history of trigonometry and the construction of the sine table.

The first known values (namely for 0°, 30°, 36°, 60° and 90°) were derived from geometric figures. From those, all multiples of three degrees can be found through the trigonometric identities:

\begin{align} \sin(\alpha+\beta) &= \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)),\\ 1 &= \cos^2(\alpha) + \sin^2(\alpha). \end{align}

Those were known by the Greeks as early as 190 AD. So exact values for sin(3), sin(9), … all the way up to sin(360) by multiples of three were found early in the history of trigonometry. However, these multiples leave gaps. Knowing the sine of 1° would be of critical importance, as one could then reconstruct, by simple iteration, all other sine values by the use of equation 1.

Amazingly, such value cannot be found without solving a cubic equation. This means that practical navigation could not rely on a complete *exact* sine table before the the 16th century! The solution to the cubic was only discovered then. For more than a thousand years, navigators worked with approximations of a sine table!

The exact solution for sine of 1° comes from the generalisation of equation 1 to three angles:

\begin{align*} \sin(\alpha + \beta + \delta) = \sin(\alpha)[\cos(\beta)\cos(\delta)+\sin(\beta)\sin(\delta)]+\dots \\ \dots + \cos(\alpha)[\sin(\beta)\cos(\delta)+\sin(\delta)\cos(\beta)] \end{align*}

Plugging 1° for every angle gives, with a bit of reorganization:

\sin(3) - 3\sin(1)+2\sin^3(1)=0

The sine of 3° is a known value (roughly 0.052), which means this equation is a (depressed) cubic. Prior to knowing the solution, various approximation techniques were used, such as the squeeze technique or what would be called today Newton’s method. If one is willing to accept that the power of three term is fairly small, then \sin(3)/3 – basic interpolation – is also a good approximation.

I doubt any reader will use this information anytime soon, but you may sleep a little bit wiser. It may also comfort you in knowing that for a few centuries, sailors did navigate without knowing trigonometric values.