Trigonometry and Fibonacci numbers

Abstract

This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

Integer sequences that behave as Fibonacci-Lucas pairs

“…S. Di DOMENICO There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1]. S. Di DOMENICO There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1].…”

“…We begin by writing the recurrence (1) in what appears to be a very contrived way: There are many ways of going about this [1,2,3,4,5] but we give an alternative method.…”

“…The answers to the Nemo page from November on why mathematics is good for you were: 1 Congratulations to Henry Ricardo for identifying all these quotations. In this issue, we tum our attention back to Sir Isaac Newton, who also figured in this column in November 2005.…”

Números de Fibonacci e números de Lucas

“…Diversos matemáticos viram nessas fórmulas analogia entre sen a e F n e entre cos a e L n , isto é, a menos dos coeficientes, as fórmulas possuem os mesmos ingredientes. Listas mais extensas de identidades trigonométricas e suas análogas de Fibonacci e de Lucas podem ser vistas em [46,53]. Por exemplo, observe a analogia entre as fórmulas para soma de arcos…”

“…Uma aplicação mais elaborada tirada de [46], é a obtenção da soma de uma série envolvendo números de Lucas através de uma série telescópica. Consequentemente, fazendo θ = l e usando as relações (3.71), obtém-se Para concluir o uso de números complexos nesta seção, serão mostradas maneiras de se estender a definição dos números de Fibonacci para índices complexos e também apenas para índices reais.…”