Trigonometric functions.

Until the middle of the Seventeen century, the sines (or cosines, tangents etc.) were figures given in tables, lists that gave, for each value of the angle, the value of the sine, and, later on, its logarithm. Around 1650 a new point of view started to emerge - the functional one, or more accurately, the geometrical one, since the concept of function was not yet fully defined. The curve of the sines began to be studied, and with that, those of the cosines, of tangents and others. Thus the curve of equation $y=\mathop{\rm sen}\nolimits \, x$ started gaining "rights of citizenship" along with the better known ones, like the parabola (with equation $y=x^{2}$ ), or the hyperbole (

) or the circumference ( $x^{2}+y^{2}=1$ ). As curves, the same problems were set for them as for the others; to find their tangent in a point, or the area contained by them, or the volume of the solid obtained by rotating it around an axis, and so on. These were problems that could not always be resolved, but that bore witness to the imperfections of the methods used until then. The invention of infinitesimal calculus, by G. W. LEIBNIZ (1646-1716) and I. NEWTON (1642-1727), led to the solution of many of the ongoing problems. The trigonometric functions came to be studied from many points of view, their inverses were introduced, and the arctangent and its relation to the quadrature of the circle came to be recognised. Finally, unexpected relations were discovered, especially when, in the Eighteenth century, the complex numbers were systematically introduced. The most important ones were the so called Euler formulae :

$\begin{displaymath}\mathop{\rm sen}\nolimits \, \vartheta = \frac{e^{i\vartheta}... ...\cos \, \vartheta = \frac{e^{i\vartheta}+e^{-i\vartheta}}{2} \end{displaymath}$

that relate the sine and the cosine of an angle $\vartheta$ to the powers with imaginary exponent $i\vartheta$ and base the number

, base of natural logarithms. From these it is easy to obtain the relation

$\begin{displaymath}\cos \, \vartheta +i\mathop{\rm sen}\nolimits \, \vartheta = e^{i\vartheta} \end{displaymath}$

and therefore a complex number

, with module $\varrho$ and argument $\vartheta$ can be written in the form

$\begin{displaymath}z = \varrho e^{i\vartheta}. \end{displaymath}$

With emphasis given to the functional character of trigonometric quantities, now all grouped under the name of trigonometric functions, modern trigonometry had practically begun. This is characterised by doubling. On one hand, the trigonometric functions represent one part of the analysis, and are studied with little relation to the problems that brought them about. On the other hand, practical trigonometry continued, as in antiquity, to be the basis of for topography, astronomy and navigation. This aspect is going through a critical phase, since many of the surveys that required its use are now done automatically with complex machinery. The latitude of a ship is no longer calculated by measuring the height of the Pole Star but through radio signals sent out and received by satellites. Topography does not require slow geodetic work, but is performed through aerial survey and remote sensing, with the resulting data processed by computers. Similarly, the role of trigonometry has become more hidden. It hasn't disappeared, but has been transferred from daily operations to the planning and designing of the machines that took its place.

Index: brief history of trigonometry

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