Trigonometry, The Bridge That Unites all Mathematics

Although it is difficult for us to recognize it, mathematics makes our lives easier, since the beginning of time. And today too

Surely most readers have heard or read the famous phrase “If I have managed to see further, it is because I have stood on the shoulders of giants”, attributed to Isaac Newton, in which he acknowledges the work carried out by all those who preceded him in the search for answers to the questions he was investigating. And it certainly is, since a large part of what surrounds us, comforts, knowledge, and others, have been possible thanks to the effort and intelligence of our .ancestors. Nowadays, any young person can feel much safer and more comfortable than someone thirty years ago anywhere in the world (well, with a few exceptions, but that is another matter), since they have in their pocket the ability to do and Theoretically driving whatever you want, from knowing how to get anywhere, knowing where you can eat the menu you want and at the price you consider reasonable, buying anything, managing the next trip in the means of transport you want,

Materially, everything is in your hand. Everything? Well, assuming coverage, working Wi-Fi, battery memory, etc. But that gives it, wrongly from my point of view, of course (I think you always have to have a plan B, just in case). It is the sufficiency that current science and technology give us.

However, if we were to consider the absurd game of only using what we know its foundation for (I am not saying that we can build it, or know the details that make it possible, but only have a vague basic idea of ​​how it works), we would probably be much more lost than that young man thirty years ago. Or maybe not. Let’s do a test.

When we attend school, we learn a large number of things from different subjects (at that time we always question their usefulness, it is inevitable, but as we grow we end up understanding why; not all of them, but many). Most of them we learn by heart, and over time, if we don’t use them, we end up forgetting them. But rarely have we asked ourselves the reason why they are like this and not otherwise, and even more, the efforts that other people had to make to reach that conclusion. Like despising that effort, which, you will agree with me, is nothing short of criticizable.

For example, going to the mathematics that is the basis of this section, we memorize some formulas to calculate the area of ​​a series of common objects, which surround us from all sides. To deduce some, we don’t have to think much: the area of ​​a rectangle, a square, a triangle, even a rhombus or a trapezoid, because in these last two cases we must have been taught (and bad, if it wasn’t ) that can be broken down into the sum of other more elementary areas.

But what about the area of ​​a circle? Or with the length of a circumference? Would we be able, right now, to justify how they are obtained? Where does that mysterious constant denoted with a Greek letter come from? No, it is not worth saying, we do this or that integral and that’s it. We have not reached such sophistication. The way in which those Greek citizens of centuries before Christ obtained the formula that we all know today, was magnificently explained to us by our colleagues Urtzi and Miriam in a previous review, ‘Archimedes and the measure of the circle’, with video included. There they made it clear to us what Archimedes’ exhaustion method was like, and how he calculated different approximations by increasing the number of sides of the inscribed and circumscribed polygons to the circle (up to 96 sides to approximate π !!).

One may think that, once what is sought has been proven, the matter is settled. But mathematics offers us multiple points of view, nothing is ever completely closed. In this case, we are going to see that Archimedes, like Newton, was also able to see further, thanks to his predecessors.

Because the method of exhaustion that we all relate to the great Syracusan was actually described earlier by Eudox of Cnidus around the fourth century BC. C., although unfortunately we only know this through references to other authors, since nothing written by him is preserved. In Book XII of his famous Elements, Euclid also refers to this procedure, and it is on him that Archimedes later bases himself, but with different methods. Euclid is not so applied, it is more theoretical. Let’s take a look at his arguments (of course with modern notation and arguments, as we do with all ancient texts).

Consider a circle of unit radius OA (see attached drawing). Euclid suspects that when we divide arc AB in half, the area of ​​triangle ACB (determined by a chord of the circle, AB, and the tangents to the circle at the ends of that chord) is more than halved. Let’s see how he reasons.

According to the expression for the area of ​​a triangle, we have that

According to the definitions of elementary trigonometry, we have that

hence

So, going to the initial expression, we have that

In the image, we also see that the triangles AOH and ACH are similar since their sides are perpendicular. Observing then the angle α in the triangle AOH, we verify that

hence

Those of us who are not used to it should not drive us crazy, we have not done anything more than express the area of ​​a triangle by means of very elementary trigonometric functions. This from the point of view of a mathematician (even from the point of view of a high school student), is not ‘ná de ná’.

Proving that the area is reduced by more than half is equivalent to see if the inequality is true

Or what is the same, after simplifying, that

We know that

so we would have to show that

Or what is the same, that

But this is obviously true, since if we call x = cos α, the inequality

And that expression is obviously positive since it is the sum of positive quantities. Archimedes will later use this idea to show that the area of ​​a circle coincides with that of a right triangle whose legs are, respectively, the radius of the circle and the other the length of the circumference, as appears in the review by Urtzi and Miriam. Archimedes is not shy about talking about the length of an arc of a curve, which forces him to introduce new concepts such as concavity and convexity. Euclid does not, and the paradox is that an argument similar to the one above could have been followed to show that the perimeters of regular inscribed and circumscribed polygons with 2^n sides both tend to 2π (the length of the circumference). It would have sufficed if they had used equality

deduced from the previous drawing, and the trigonometric inequality

In Archimedes’ defense, the widespread use of trigonometry did not become common until the second century AD. C., a tool that, as we can see, greatly simplifies many very complicated arguments using only elementary geometry. Trigonometry, which offers us relationships between the sides and angles of triangles (and by extension, of any plane figure; but etymologically it is what trigonometry means, the measure of triangles), has allowed unexpected calculations of magnitudes that not even are within reach (such as the measurement of distances in astronomy, or incredible perspectives in architecture, to give two simple examples). One more of the many great ideas that we constantly use (mobiles, in this case, going back to the initial idea) that our ancestors left us.