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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$y-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>coordinates of the point <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$P_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> of the unit circle at the relative arc length t from the point (1, 0), are called <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\cos t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>cos</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sin t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>sin</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> of the unit circle. In this paper we show, through a new proof of the existence of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$P_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation>, 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.

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

期刊论文
出版模式：

开放获取
发表日期：

2026年1月5日

查看全文

Enrico Babilio,
Claudia Capone,
Alberto Fiorenza,
Filomena Galizia

摘要

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 $x-$ $x -$ and $y-$ $y -$ coordinates of the point $P_t$ $P_{t}$ of the unit circle at the relative arc length t from the point (1, 0), are called $\cos t$ $cos t$ and $\sin t$ $sin t$ , 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 $P_t$ $P_{t}$ of the unit circle. In this paper we show, through a new proof of the existence of $P_t$ $P_{t}$ , 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.