Fibonacci Trigonometry 1: Basic Definitions

This is the first in a series of posts on Fibonacci trigonometry:

Part 1: Basic Definitions

Part 2: The Canonical Basis and Closed-Form Expressions

Part 3: The Basic Analogies

Part 4: Fibonacci Formulas Using sinh, etc. — Real Number Version

Part 5: Fibonacci Formulas Using sinh, etc. — Complex Number Version

Part 6: Trigonometric-Style Identities

Part 7: The Fibonacci Tangent and Cotangent Functions

Part 8: Recurrence Relations for Ftan and Fcot

Part 9: Analogue to de Moivre's Formula

Part 10: More on the de Moivre Analogue

Part 11: Fibonacci-Based Formulas for F(kn+m) and L(kn+m)

Part 12: Lucas-Based Formulas for F(kn+m) and L(kn+m)

Part 13: Summary of Formulas for F(kn+m) and L(kn+m)

Part 14: F(kn+m) and L(kn+m) mod F(n)

Part 15: F(kn+m) and L(kn+m) mod L(n)

Part 16: Summary of Modular Formulas for F(kn+m) and L(kn+m)

Part 17: Divisibility Properties for the Fibonacci Numbers

Part 18: Divisibility Properties for the Lucas Numbers

 

I think a connection between the Fibonacci numbers and trigonometry qualifies as a mathematical curiosity. It turns out that the Fibonacci numbers and Lucas numbers obey many identities analogous to the trigonometric and hyperbolic identities, with the Fibonacci numbers playing the role of sin or sinh, and the Lucas numbers playing the role of cos or cosh.

A generalized Fibonacci sequence is a complex-valued function f defined on the set of integers with the property that:

• f (n + 2) = f (n) + f (n + 1) , for every integer n.

A generalized Fibonacci sequence f is determined by the values of f(0) and f(1) (or indeed by any two values).

The Fibonacci sequence F is defined to be the generalized Fibonacci sequence with F(0) = 0 and F(1) = 1.

The Lucas sequence L is defined to be the generalized Fibonacci sequence with L(0) = 2 and L(1) = 1.

As we'll see, F behaves analogously to the sine function, and L to the cosine function. So let's also define:

• Ftan(n) = F(n) / L(n), the Fibonacci tangent function
• Fcot(n) = L(n) / F(n), the Fibonacci cotangent function
• Fsec(n) = 1 / L(n), the Fibonacci secant function
• Fcsc(n) = 1 / F(n), the Fibonacci cosecant function

We could write Fsin instead of F, and Fcos instead of L, but we prefer to keep the traditional names F and L for the Fibonacci and Lucas sequences.

F(n) is zero only at n = 0, and L(n) is never zero. It follows that Ftan(n) and Fsec(n) are defined for every integer n, and Fcot(n) and Fcsc(n) are defined for every integer n except 0.

Using mathematical induction, it's easy to prove the following facts:

• L(n) = F(n−1) + F(n+1) = 2F(n+1) − F(n)
• 5F(n) = L(n−1) + L(n+1) = 2L(n+1) − L(n)

Also, observing that the sequence of Lucas numbers modulo 5 consists simply of the numbers 2, 1, 3, 4 repeated forever in both directions, we note that:

• No Lucas number is a multiple of 5.