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Fibonacci and Catalan Numbers: An Introduction by Ralph P. Grimaldi

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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:

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Now let θ = π/10. Then

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

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Therefore, from the above trigonometric identities, we find that

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

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and

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Consequently,

Since ...

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