Last night I had insomnia. I laid in bed for a few moments, before thinking about the validity of a certain mathematical ‘truth’ which I’d always been told. I was always told that dividing by a fraction was multiplying by its reciprocal, or (in symbolic terms):

\[ \frac{a}{\frac{b}{c}} = \frac{ac}{b} \]

Anyways, this is trivial to verify, but led me to consider another property. Namely:

\[ (ab)^n = a^n b^n \]

This is also trivial to prove, but led to more and more properties to investigate. Eventually, I ended up with a (certainly not unique) proof of the Pythagorean Theorem. After the fact, I pondered the so-called “special” triangles I learned about in high school geometry class, namely the 45-45-90 and 30-60-90 triangles. A lot of study was put into the properties of these triangles, as they had a well-defined ratio of sides. However, the Pythagorean Theorem states that any right triangle inherently has a defined ratio between its sides. In fact, for any x-(90-x)-90 triangle, the ratio is

\[ 1:\frac{1}{\tan{x}}:\sqrt{1+\left(\frac{1}{\tan{x}}\right)^2} \]

where x is the angle of the opposite side. Why are the 45-45-90/30-60-90 triangles special enough to devote standardized tests and quizzes to? I can only assume that since the ratio is trivial for angles that have simple trigonometric identities, that this was simply “easier to teach.”