Why Do Trigonometry Teachers Rely on Special Angles Rather Than Non-Special Ones?

Have you ever wondered why trigonometry teachers only focus on certain angles like 36° or π/5, instead of seemingly more practical angles like 37°? The reason lies in the simplicity and elegance of special angles, which can be derived using basic identities. In contrast, angles like 37° or π/5 (which is equivalent to 36°) require much more complex derivations, making them less appealing in a practical sense.

The Delight of Mathematical Beauty

Consider the cosine of π/5 (or 36°). The exact value, cos(π/5) (1 √5) / 4, is a truly beautiful and fascinating result. However, there's a stark contrast to how this is often treated in trigonometry classes, where teachers prioritize skills over the interconnected beauty of mathematics. Very few high school students appreciate the intricacies and the elegance of such results.

The Practicality of Trigonometric Formulas

When it comes to practical trigonometric formulas, we have a treasure trove of identities that make calculations easier. For example, consider the cosine of the sum of two angles formula:

cos(α - β) cosαcosβ sinαsinβ

By rearranging this, we can derive the double angle formula:

cos(2α) 2cos^2α - 1

From there, we can derive the half-angle formula:

cos(1/2θ) ±√(1/2 cosθ/2)

With these formulas, calculating the trig functions of half angles feels practical and straightforward. In contrast, calculating the trig functions of one fifth of an angle feels too complex to be practical, which makes it challenging to incorporate into the curriculum.

The Role of Geometric Derivations

The cosine of π/5 (36°) can be determined through geometric means. When we draw the diagonals of a regular pentagon, we create triangles with angles of 72°, 72°, and 36°. Using the properties of similar triangles, we can demonstrate that the ratio of the length of the diagonal to the side is the golden ratio, denoted as φ:

φ (1 √5) / 2 1.618033989...

Since the interior angle of a pentagon is 108°, we can calculate the sine of half the interior angle:

sin(54°) cos(36°) (1 √5) / 4

The Fibonacci Sequence and the Golden Ratio

The Fibonacci sequence is another fascinating link to the golden ratio, φ. The ratios of adjacent elements in the Fibonacci sequence converge to φ:

2/1 2 3/2 1.5 5/3 ≈ 1.666... 8/5 1.6 13/8 ≈ 1.625 21/13 ≈ 1.615... 34/21 ≈ 1.619... 55/34 ≈ 1.618... and so on...

This convergence is due to the fact that the matrix representation of the recursive relation for the Fibonacci sequence has a characteristic polynomial x^2 - x - 1, with φ as a root and an eigenvalue.

Penrose Tiles and Intricate Designs

For those interested in the extreme cases of angles that are multiples of 36° or π/5 radians, there is an interesting geometric construct known as Penrose tiles. These tiles show complex patterns that cannot be tiles with regular shapes, yet they perfectly fill a plane without gaps or overlaps. You can explore this fascinating topic further in the Penrose tiling section of the Wikipedia page. (Here is a link: Penrose tiling - Wikipedia).

In summary, while special angles like 36° are more straightforward to teach and learn, there is a vast world of mathematical beauty and practical applications that are often overlooked in favor of simpler, more practical angles. Exploring these topics can not only enrich the student's understanding but also reveal the intricate and elegant nature of mathematics.