Western developments .

Trigonometry arrived in the west mainly through Arab sources. Developments during the Middle Ages were very slow, and there is no evidence of relevant contributions by European scholars until the fifteenth century. Once again, the needs of astronomy promoted trigonometrical studies. A higher precision of instruments requires tables always more specific in two directions, the sines are given with an increasingly bigger number of decimals and angles with always smaller intervals. Considering the interval between the arcs, GEORG PEURBACH (1423-1461) calculates a table of sines with intervals of $10^{\prime}$ , while JOHANN MÜLLER (1436-1476), who was known as the REGIOMONTANUS because he was from Königsberg (literally, "the mountain of the king"), composed a table with intervals of one prime.

Even exactitude increased significantly. This one was not given, as today, by the number of the decimal figures ,^1.3 but from the dimension of the radius of the goniometric circle. For example, taking the radius, also called toto sine , , the table reported the values of $R \mathop{\rm sen}\nolimits \, \alpha$ in integer numbers, which could go from 0 to 10000, corresponding to four decimal figures . In his tables, PEURBACH took the radius 600000, while REGIOMONTANUS first uses 6000000, and then 10000000, corresponding to seven decimals. Incidentally, this is the first time when the liberation occurs from the sexagesimal system for the sines (not for angles, that still lasts) and the base 10 is definitively adopted.

An impulse towards the development of trigonometry comes from typography, a discipline that, contrarily to astronomy, is totally based on rectilinear trigonometry. It is for the necessities of topographical survey that triangles and their solution are studied. These are problems that soon went beyond immediate applications and became the reason why mathematicians demonstrated their own capabilities and challenged their peers to the solution of increasingly elaborate and complex issues.

The first treatise of trigonometry written in the West, which remained the most important for a long time, is the De triangulis omnimodis by REGIOMONTANUS, written around 1464, but only printed in 1533. Many treatises followed, partly autonomous, partly as adjuncts to astronomy texts. Among these last ones we should mention the one that NICOLAUS COPERNICUS (1473-1543) was to include in his work De revolutionibus orbium caelestium, which was brought to light in an edition by G. G. J. RHETICUS (1514-1577) the year of his death. RHETICUS himself prepared a monumental series of tables of the six circular functions, with intervals of $10^{\prime \prime}$ and for a radius of , that were printed posthumously in 1596, with the title Opus palatinum de triangulis.

Among other things, in the work of RHETICUS for the first time the construction of tables appears starting from the formula

$\begin{displaymath}\mathop{\rm sen}\nolimits \, (n+1)\beta = 2 \mathop{\rm sen}\... ...\beta \cos \, \beta - \mathop{\rm sen}\nolimits \, (n-1)\beta .\end{displaymath}$

It would be difficult to trace back the numerous and often just perceptible contributions brought into trigonometry during the Sixteenth and Seventeenth century, and it is far to say that they did not wield decisive changes. It is worth mentioning the multiplication formulas of angles, which we owe to F. VIÈTE (1540-1603) and which were published for the first time by his disciple A. ANDERSON (1582-?):

$\begin{eqnarray*} 2 \cos \, x &=& u \\ 2 \cos \, 2x &=& u^{2}-2 \\ 2 \cos \... ...=& u^{5}-5u^{3}+5u \\ 2 \cos \, 6x &=& u^{6}-6u^{4}+9u^{2}-2 \end{eqnarray*}$

in which, because of the relation

$\begin{displaymath}\cos \, (n+1)x= 2 \cos \, nx \cos \, x - \cos \, (n-1)x ,\end{displaymath}$

every line is obtained multiplying by

the previous line and subtracting the line previous to that.

Index: brief history of trigonometry

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