Trigonometry

I went to 3 different trashy soviet schools (because of bullying) and none of these taught mathematics properly. All they did was showing you a crappy soviet textbook full of formulas to memorize. I still remember my struggling with trigonometry because no one bothered to explain to us what it is all about.

This is, probably, the most fascinating story on Earth - how humans generalize and abstract out recurring patterns they observe. I am not in a position to tell it, but at least I could say a few words.

Mind creates abstract concepts and from these concepts the whole conceptual frameworks. What we call branches of mathematics (almost everything beyond arithmetic and Natural Numbers) are development of this or that particular conceptual framework.

Back to trigonometry. The tradition says that ancient Greeks, those exempt from a forced slave labor and concerns about where the food will come from, have studied shapes (drawn with a stick on the soil).

One of such fundamental shapes (besides a straight line and a right triangle) is a circle, which they used to draw using a stick and a piece of rope. They even developed the special instrument to draw circles precisely.

So what is a circle? Well, it is what we see shining in the sky above our heads. To be precise - it is a generalization and then an a well-defined abstraction of what we see in the sky.

For these ancient Greeks it was a all points at the same particular distance from the center (at the length of a rope). They have no concept of a Set back then.

So, this is an abstraction, which involves a distance from the center. Abstraction of What?

Well, there is a universal notion (which we, intelligent observers, can observe and generalize from) of a distance. In any human language this notion has been captured by “how far” or “how close” something is from the other.

This notion is universal because, it turns out, something what we call a “force” (a pull, so to speak) diminishes with (proportional to) a square of a distance. So, our abstracted notion of a distance is indeed based on “something real”.

This is why we see some spheres in the sky and all the other spheres - this shape emerges from a process which involves distances from the center. Our generalizations and abstractions of a circle (as a projection of a sphere) turns out to be “perfectly captured” (just right).

Our minds can “naturally” measure lengths in terms of straight sticks (because these naturally occur in our shared environment) by superimposing it as a “unit” of length and then counting how many units it takes.

In our language-based shared culture we have developed a standard methodology for measurements - to superimpose an imaginary “scale” or a “grid” upon everything we observe. The tradition says that it was Descartes who developed this methodology in a systematic way, for what we nowadays call a “coordinate system”.

So, to study an abstract (generalized, ideal) circle, one imagines and superimposes a “coordinate system” upon it, in a way that the origin co-insides with the center of a circle. One also assumes (imagines) that the radius of a circle is an one single unit (of the scale). This is what we call a unit circle.

We also need a concept of an angle. This is another generalized abstraction, and has its own superimposed “scale” of measurement of a “degree” of an angle (you can think of a bent or curved “scale ruler”).

In the case of modal arithmetic, by the way, the scale ruler (the Number Line) turned into a spiral and being read up from the top. Anyway.

The universal notion we have captured, generalized and then abstracted out with the concept of an angle is how it differs from a perpendicular or a flat line. There too are readily occur in our shared environment - certain kinds of trees versus a flat surface of the soul.

Tradition holds that a perpendicular corresponds to the 90 degrees, and that such angle (versus the horizontal surface) is called the right angle.

Again, they were onto something real, because what we call orthogonality is a special property - it is sort of the “longest distance” from the surface. Yes, a distance again.

Greeks considered it as a special line, which is what a perpendicular is a name of, and it is unique (there is only one such line through any point).

Now, when we have all these concepts and have superimposed our coordinate system (Greeks do not call it this way, of course) we can observe emergent abstract patterns.

The observed that each point on a unit circle (which has exactly the same distance from the center) has its own unique (different from all other) projection onto each axis.

The difference is due to the angle between the radius - a line from this point (to the center) and the axis (one and the another).

This is what we call a relation or an unique correspondence between two measured observations.

This is a mapping from an angle (in its units) to a distance from the axis or the length of a perpendicular (in its units).

Tradition chooses to call the functions (particular mappings) from an angle to the length of a projection to the axis – sine for the X-axis and co-sine for the Y-axis.

The formalism of a “right triangle” (based on the Pythagorean theorem) is used to calculate the actual lengths.

This is how to begin to teach what is behind these cryptic strings of Sin and Cos.

Everything has to be explained from the first principles, so all the formulas and underlying relations can be sort of re-discovered instead of merely memorized (which is what bad soviet teaching was all about).

Knowing the principles frees one from memerizing all the details. This is how mathematics should be taught - to always zoom in and out from the observed recurring pattern, to a generalization, to an abstraction and its properties and “laws”.