Complex numbers and trigonometry

In 1748, while heading the mathematics section at the Berlin Academy, Leonhard Euler published in Latin a remarkable, epoch-making work under the title Introductio in Analysin infinitorum. This Introductio freed differential calculus from its grounding in the geometric figures that had until then given meaning and substance to its methods. To achieve this, Euler reversed the usual order of presentation and the hierarchy of disciplines: in the first volume, he developed the study of numerical and algebraic quantities through the study of functions before turning, in the second, to questions of geometry. This reversal enabled him to reveal the purely numerical relationships between trigonometric functions—until then defined geometrically—the exponential function, and complex numbers.

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Euler's formulas --------------------

After determining the series expansions of the standard functions of analysis, Euler gives the famous formulas that bear his name (x denotes an arbitrary real number):

$e^{ix}=\cos x+i\sin x,e^\{ix\}=\backslash cos\; x+i\backslash sin\; x,$ $\cos x={\displaystyle \frac{e^{ix}+e^{-ix}}{2}},\backslash cos\; x=\backslash dfrac\{e^\{ix\}+e^\{-ix\}\}\{2\},$