 Hi, I'm Zor. Welcome to a new Zor education. previous lecture was about trigonometry and its roots in some practical problems and we have already defined certain functions of acute angles called sine, cosine, etc. Well, this lecture is more about abstract maths and I will define the trigonometric functions in a little bit more precise manner the way how it's basically defined right now for all kinds of angles, not only acute ones. So here is what I would like to present to you about this particular approach to defining trigonometric functions. Here rectangular coordinates and a unit circle which has the radius 1 and the center in the center of coordinates. Next, for any angle I mean any angle, acute, obtuse 90 degrees, 270 degrees, whatever we can find a point on this circle which is the result of movement of this particular ray which is coinciding with x-coordinates. To whatever angle we are talking about, let's say we are talking about this angle. So basically any angle can be constructed this way by moving this particular ray counterclockwise by the amount which is basically signifies the value of our angle. It can be in degrees, for instance, 90 degrees means we are moving this way or it can be in regions pi-ragions would be half circle, if you remember, right? Because the whole circle is 2 pi-ragions. Now, we can also move counter, not counterclockwise, but clockwise and that's considered to be a negative direction. So we can move by minus 45 degrees and that would be from here clockwise to this particular position. So what I would like to say is that for any numerical value of the angle, whether it's expressed in degrees or regions and there are only these two measurements we can always move our ray from original position to whatever this particular angle signifies and find the point which corresponds to this particular angle. Angle can be positive, angle can be negative not even necessarily it's supposed to be less than 360 degrees, it can be 400 degrees, which means we will move full circle that's 360 and then 4 more and that will be 400. So for any numerical value of the angle we always find the point. Great. Now this point A will have certain coordinates, let's call it A and B. Now, position of the point A as I was saying is basically reflecting the angle we are talking about. So let's say the angle we used to verify for this. Now, the definition and that's a very simple definition. This is A and this is B, right? Because these are coordinates. A means the projection onto the x-axis, B is this projection onto the y-axis. So and now I'm just defining the definition. There is no explanation here, it's a definition. Sine phi by definition equals to B. Cosine phi equals to A. So don't forget that the radius is 1. A and B are correspondingly abscissa and ordinate of the point which represents our angle. And these are definitions. Now it's quite different from whatever I have defined or at least explained about in the previous lecture. I was drawing the right triangle if you remember and I was saying that the sine of B is a ratio of B over C, right? And the cosine, so this is the sine is a ratio of A over C. So that's how I explained. Now, is it different? Well actually, well it's obviously different, but it's a natural expansion of this definition. And here is why. Let's consider our point is here which is, let's call it B and it has also A and B. This is A, this is B, this is the same as B. So this point B now represents an acute angle from 0 to 90 degrees, from 0 to pi over 2 radians. Now here, hypotenuse of this particular triangle let's call this Q. So we are considering triangle OPQ. Now the hypotenuse which is C is equal to 1 because if you remember we started from the unit circle which has a radius 1. Now, if this is a phi then B over C would be by my previous definition a sine, right? So sine is B over C. But C is equal to 1. So it's B. So it's B only. And that's exactly how I defined it, right? So my point here is that instead of drawing some triangle of some arbitrary size with this particular angle phi, I have basically said, look I don't need this kind of arbitrary decision. I don't know what kind of triangle. It's not good for a definition to say well let's take arbitrary triangle and measure the ratio. Now if I am fixed to a unit circle it's a precisely concrete radius 1 and I don't want to consider anything else. I could have probably said, okay let's draw a triangle with a hypotenuse equal to 1. Yes, I could have said that. But even that is not really enough to define the angle. Because then I have to really kind of build the triangle and I don't know really. I don't want to go into these details. This is much simpler. You have the circle of the unit circle. For any angle take this corresponding point take its ordinate and that would be the sine and take obsessive would be cosine. That's much simpler. And again for acute angles this corresponds to this exactly. Now with an advantage beyond this absence of arbitrariness well the advantage is that I can actually define these functions for any angle, not only acute angle. So I no longer restricted to acute angles because my definition doesn't involve right triangles at all. My definition involves a broader spectrum of angles and obviously this is more universal. And it encompasses in itself the definition which I used to have for triangles. So now we are talking about sine of let's say 90 degree which we couldn't really talk about if we were talking about triangles because there is no triangle with an acute angle of 90 degrees or 100 degrees or whatever or minus 20 degrees. So basically again this particular definition allows us to expand the concept of trigonometric functions to any angles. But let me continue further with other trigonometric functions and you will obviously understand why I defined it this way. Tangent is by definition sine over cosine. Now again if you remember tangent is equal to opposite catatouille ratio of the opposite catatouille to the adjacent one. So it's b over a. So sine which is b over a. So this is a definition of the tangent. And again it's universal because our sine and cosine are defined for basically any angle. Well obviously tangent does make sense only when cosine is not equal to zero which means cosine is our a. So the tangent is meaningful only for those cases where a is not equal to zero. So for angles which are not 90 degrees and not 270. And all multiple of these if you add 360. Now cotangent is by definition again cosine over sine. Which again corresponds to my triangular definition of a over b. What else is left? Second and cos second. So second by definition second f, second phi is equal to one over cosine of phi. And cos second phi is equal to one over sine of phi. So these are all definitions. There is no need to prove anything. But the roots of these definitions are in those triangular properties which we were talking about. These are true functions of angle only. So angle by itself given is sufficient to calculate basically all these different functions. And that's been done. Basically there used to be tables. Now it's all in the computers and calculators. So the value of the function sine or cosine or anything like that is already defined and calculated and it's built into our infrastructure. Now using these functions we can calculate many different things. For instance we can calculate the lengths of one calculus if you know the lengths of another calculus of a triangle and an angle. So for a triangle it's obvious. But at the same time, as I was saying many times, practicality is a great thing about mathematics. But what's probably even greater is that it's the perfect field for your creativity, for your mind work. And this universal definition of sines and cosines and other trigonometric functions allows to introduce a whole spectrum of different problems properties, theorems, etc. which we are going to explore in the next lectures about trigonometry. And it's just brain exercise to its fullest. So that's what I definitely suggest you to continue to approach with this particular attitude that forget about the practicality of these particular problems they are important but they have their own place. The purpose of this course is not to introduce you to practical recipes but to try to develop creativity and intelligence sufficient to solve all kinds of problems in the practical life regardless of the profession not even related to the mathematics. It's just to develop your brain and intelligence and mind. So trigonometry presents a perfect tool for this and that's what we are going to do in the next lectures. Thank you very much.