 Hi, I'm Zor. Welcome to Unisor Education. With this lecture, I would like to start talking about trigonometry. Usually, I'm emphasizing that mathematics is more abstract, if you wish, kind of a science or a knowledge or whatever. It's more about mind rather than the real things which we are dealing with. Granted mathematics really started its own existence from some practical problems. However, contemporary mathematics is an abstract kind of knowledge. Trigonometry is something which is definitely having its roots in some practical problems. Before I talk about real trigonometry which is a strictly mathematical kind of a subject, I will talk about some practical problem which people were facing some time ago, I guess, which led to the development of trigonometry as a separate branch in mathematics. It's not like a definition or theorem or anything like that which I will talk about today. It's about some practical situations and maybe a little bit of geometry to justify the existence of trigonometry as a separate branch. What kind of practical problem think about led to the development of trigonometry? Well, I'm sure there were many. I'm just kind of fantasizing what can be the problem which led to trigonometry. Here is, for instance, a simple example. If you have some kind of a mountain, it has certain height which nobody knows what exactly the height is. We are standing here on the ground and we would like to know the height of this mountain. How can that be done? Well, obviously, you can probably go up the mountain and use your GPS positioning system or something like that. Well, it was not available at the time. Plus, if you go up to the mountain, you still don't know how to measure your height. So, here is the way how people actually tried to solve this problem. People knew how to measure the distance. So, let's consider this distance is given, let's say a, from this particular person standing, watching, looking at the mountain. Also, people knew how to measure angles. So, if you remember, the angles can be measured by degrees. If you have an angle, it's divided into 360 degrees. So, this is 90 degrees and this is 180 degrees and this is 270 degrees and 360 degrees. So, they can measure the degrees of this particular angle. Another measurement unit is region. Region is an angle which has an arc equal in lengths to the radius. So, this is the angle of one region. So, no matter what the measurement is, whether it's a region or it's a degree, we can measure the angle. So, let's consider that we have this angle. Let's use letter 5 for this. So, with all this particular distance to the mountain and below the angle we view the top of the mountain. Is this enough to find out the height? Well, in theory, if you remember the course of geometry, these two characteristics are sufficient to construct a right triangle. Which means that yes, in theory, if you can construct the right triangle by using one catheters and an angle, it means this is somehow defined. But let's just talk about practicality of this. Practicality is that you cannot really construct it. It's not a piece of paper or anything like that. So, still you have a problem of measuring this particular height. But here is what somebody really understood the geometry of that thing offered. Let's have a rod or a stick or something much shorter than the length of the mountain. Now, if I will position it in such a way on the same line that I view these three points on the same line and I know the length of this particular shorter rod and I know this distance, let's call it C. What can I say about these two right triangles? Obviously, they are similar. Now, similar triangles have proportional sides. So, basically what I can say is that C over X is equal to C over A. Now, B I know, A I know, and C I know and that's how we can get to X. So, X is equal to C times A times B over C. Right? A times B over C. Now, this is how we can actually find the height of the mountain. A can be measured. B, we know, this is the rod which we actually have, we also can measure. And C is the distance on which if you would install this rod B, it will be viewed at the same angle. Okay, this is the beginning of trigonometry. And here's what I mean. Let's calculate B divided by C for all possible angles phi. Well, it's just 360 degrees or somehow in decimals you can have this ratio for one region for two regions or for 0.5 regions etc. and basically have a table of value of B over C B over C for any angle you have. Now, let's consider that B over C for any angle you have is tabulated somehow. So, there is a table which has the correspondence. Now, how can we construct this table? Well, basically we can have one particular B, one particular rod, and install it on one place, measure the angle, measure the C, and calculate. Put it a little further, angle will be smaller, B would be the same, C would be a little bigger. So, this B over C ratio would be a little smaller. So, basically moving B back and forth we can tabulate the values of phi, the values of this ratio for every phi. Now, when this is done, when this is a table which contains all the possible ratios B over C for every angle phi, we can say, you know what, we can measure the height of any mountain. How can we measure? Well, very simple. We install our viewing device at some place, we measure the angle phi, we measure the distance A, then we go to our tables for this phi we find this particular ratio, and now all we are saying is that X is equal to A times this particular ratio B over C, which we can found for this particular angle phi. So, this is all possible because the triangles are similar. So, for every fixed angle phi, this ratio B over C is exactly the same. This over this or this over this, this ratio is always the same. So, this ratio can be considered as a function of the angle. So, B over C, the ratio of the calculus which is opposite to the angle to the calculus which is adjacent to the angle is constant for every angle phi regardless of what exactly the size of triangles are. This can be bigger, this can be smaller, but the ratio is exactly the same. This is the beginning of the trigonometry. So, whenever people realize that they can calculate this ratio once and for all, let's say for every degree from 0 to 90 or something like this. So, it's just 90 calculations. Okay, fine. They did it once, they tabulated. And after that, they can measure the height of any mountain, any tree, any building or anything else using just a simple calculation. They have to know the distance from the observer to that particular object and the angle at which the top of the object is viewed from this particular thing. And the angle is also measured by somehow some device, obviously. And then you just multiply whatever the table says for this particular ratio, for this particular angle, by the distance to this object. So, this is the beginning of trigonometry, as I said. The foundation is similarity of all the right triangles which have the same angle, acute angle phi. And now people can actually expand it a little bit. Not only this particular ratio between the opposite calculus and the adjacent calculus is constant for a constant angle, acute angle phi, regardless of how big the triangle is. But all other ratios of different sides of this right triangle are exactly constant if the angle phi is fixed. Now, and here is something which they actually have defined as terminologies concerned. If you have the right triangle with these categories, and this is an angle phi, let's say. So, people came up with the idea that any ratio between these sides, if angle phi is fixed, is actually also fixed regardless of how big the triangle is. As long as this angle is phi, then the ratio of A over B, A over C, B over A, B over C, C over A, and C over B. All these ratios are constant for a constant phi, regardless of how big the triangle is. It can be this big or again to that big. As long as this angle is phi, then the ratio between these sides all are depending only on phi. So, people came up with terminology. This thing, A over B, which is opposite cartridges to adjacent, is called tangent. A to C is called sine. B to A is called cotangent. B to C is called cosine. C to A is called cos second. And C to B is called second, if I'm not mistaken. So, all these ratios are actually functions of the angle phi, of an acute angle phi. So, whenever we are talking about, let's say, cosine of the angle phi, what does it mean? Well, in this case, it means build a right triangle with one acute angle phi and take the ratio of adjacent cartridges towards hypotenuse. That's what it means. And regardless of what kind of triangle you build, as long as the angle is phi, this ratio will be the same. And that's why it's legitimate to call this ratio function of the angle. So, these functions are introduced, where introduced, in the beginning for acute angles in the right triangles. And again, there are some practical roots which we were talking about. Now, what's wrong with this definition? Well, it's okay for acute angles. No doubts about that. The problem is that, well, the contemporary mathematics again is much more abstract. We are not only talking about acute angles, we are talking about any other angles. What if it's angle 90 degrees? There is no acute angle of 90 degrees. There is no right triangle which has two 90 degrees angles. What if it's obtuse angle? I mean, then we are not talking about right triangles at all. And we would like to expand the functions and there were certain practical applications probably as well, but we are not talking about this right now. What's important is that this application is not really sufficient from mathematical standpoint. Since we have all different kinds of angles, we really would like to define these functions for all the values of the angles. Well, let me just give you an example. When people invented square root, they knew how to make square root of positive numbers, but they were basically completely helpless if we are talking about negative numbers. They didn't know what this is a square root of minus 1. So they kind of expanded the repertoire of their numbers and added complex numbers just to be able to apply the same function which is square root to any numbers which we know. And they knew about positive and negative numbers. So it looks like they were kind of restricted by having square root only to positive. So they expanded the numbers, expanded this set of different objects which they are dealing with. Same thing here. We have a function defined here for acute angles only. And this is actually not sufficient. Mathematicians were always not satisfied with these restrictions. So they always tried to expand the definitions in as much as possible. So the next lecture I will talk about the real definitions of trigonometry in contemporary mathematics. And that would expand the definition of all these functions to all kinds of angles. But that will be subject to the next lecture. Thank you very much.