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  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