Fibonacci and Catalan Numbers: An Introduction

Chapter 11

More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science

Before we provide some instances where the constants α and β come into play, in Reference [22] we find mention of a rather interesting observation: the ratio of one's height to the height of one's navel is approximately 1.618—according to extensive empirical observations made by European scholars. Yes, the remarkable and mysterious number α continuously arises in all kinds of unexpected places!

Example 11.1: Let us start by recalling some trigonometric identities:

Now let θ = π/10. Then

So the angles 2θ and 3θ are complementary angles, and since the sine and cosine are cofunctions, it follows that

Therefore, from the above trigonometric identities, we find that

Upon dividing by cos θ (as cos θ ≠ 0), it then follows that

and

Consequently,

Since ...