How to Define Sine and Cosine as Functions over Reals Rigorously and with Minimal Prerequisites

Abstract Sine and cosine as real functions on the real axis can be defined in several ways. However, the standard way used in undergraduate courses in Calculus is the unit circle definition: shortly, for a given real number t , the $$x-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> and $$y-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> coordinates of the point $$P_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> of the unit circle at the relative arc length t from the point (1, 0), are called $$\cos t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>cos</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> and $$\sin t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>sin</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> , respectively. The heart of the matter is that the notion of arc length is either postponed after the exposition of integral calculus, or, when it is given through the notion of polygonal path, such notion seems never used to prove the existence of the point $$P_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> of the unit circle. In this paper we show, through a new proof of the existence of $$P_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> , how the definition of sine and cosine can be formalized using only a minimal knowledge of classical Euclidean Geometry and properties of real numbers, avoiding to use the notions of area, limit, derivative, series, integral and complex number.