 Hi, I'm Zor. Welcome to a new Zor education. I would like to talk about something which personally I consider quite beautiful in mathematics. It usually happens when one particular part of mathematics is somehow is related to another, and all of a sudden you see things which you didn't really see looking at each of these two parts separately. Well, just for example, if you take a concept of a function, for instance, and then through coordinate systems, you are examining the graph of this function, you understand how the function behaves much better looking at its graph than looking at its formula. So that's where we connect together algebra and geometry. And some very interesting curves, geometrical curves like ellipse, for instance, or parabola are expressed in algebraic language in certain formulas, and you see this connection. So today, I'm going to talk about connection between trigonometry and the algebra of complex numbers. Well, first of all, I do recommend you to refresh your knowledge about complex numbers. There are lectures and exercises in this course in the algebraic section dedicated to complex numbers. So I presume that we know what complex numbers are, how they look like, etc. So now let's talk about connection between trigonometry and complex numbers, which again, personally I never expected to tell you the truth. And when I saw the first time of this beautiful formula, so let me just put the formula. E is a very special number in algebra and analysis and differential in tego mathematics. Ix is exponent and we don't really know what complex exponent actually is. And this is how it's expressed in trigonometry. Well, it's unexpected. It's called the Euler's formula. The name is, I think that's how it's spelled, Euler or Euler, depending on the pronunciation. It's a Swiss mathematician who went to work to Russia during the, I think it was the 18th century and basically spent his whole life in Russia. And he is one of the people who was in the beginning of the Russian mathematical schools. And so anyway, that's the formula which he suggested. It's called by his name. And I think it looks very unexpectedly, at least for me. Now I'm not going to talk about this formula, but I will talk about something which is very, very close to this formula. So again, complex numbers and trigonometry. But first of all, you know that the general way to express the complex number is real part plus imaginary part which has a real number multiplied by number i, which is invention of mathematicians. And the only invention has the restriction is this one. To be able to actually extract the square root of negative numbers, they have invented this number i, square, which is equal to minus 1. So these are numbers which we call complex numbers. And I was talking about geometrical interpretation of the complex number. Namely, if you have the system of coordinates, then point with coordinates a, b. So this is a, and this is b. Can represent this particular complex number. I did not talk much about how certain operations are reflected in the geometrical interpretation. Today I will. Now, if you consider not only this Cartesian system of coordinates, x, y, but also the polar system of coordinates. And if you remember in the polar system of coordinates, the point is defined by also two parameters, which are the angle, the polar angle, and the distance from the pole itself. So if this polar system of coordinate is coincides, if you wish, with the Cartesian coordinate system, the polar axis is coinciding with x-axis. Then this particular point can be defined in both systems, basically. a and b would be the coordinates in the Cartesian system. And angle phi and the lengths of this distance to the pole is another coordinate. So not only a, b, but also r and phi are legitimate coordinates at this point. OK. Now I will consider, at least temporarily, only numbers, complex numbers, where this length is equal to 1, which means that I'm considering the numbers which are on the unit circle. So r is equal to 1. And using the Pythagorean theorem, obviously, it means that a squared plus b squared, or square root of them, is equal to 1. Right? a squared plus b squared is r squared, so if r is equal to 1, that's what it means. It's called modulus. So modulus of this complex number is 1. It means it lies on the unit circle. OK, fine. But if that's true, then what is abscissa and coordinate? What are a and b in this particular case? Well, by definition of sine and cosine, a, which is abscissa, is equal to cosine of phi. Right? That's the definition. And b is equal to sine phi. So all of a sudden, we came to conclusion that our number can be represented as cosine phi plus i sine phi. Well, this is the first connection with trigonometry. But this is just the connection based on definition. We're just expressing one particular way of describing our complex number with another Cartesian system and the polar system of collisions. It's just a conversion from one system to another, no big deal. But here is a big deal. Let's consider we have two different complex numbers on the unit circle. I don't need this. So this would be my first number. And you know what I'll use the letter alpha instead of phi. It would be easier for me. So this is m dot alpha. Now let's consider we have two different complex numbers. Both are on the unit circle. So this is alpha and this is b. Now, what I would like to do, I would like to multiply them. So I know that one number on the unit circle is expressed this way. Another number is expressed that way. And now I'm looking for representation of their product. So this is called z1 and this is z2. Now z1 times z2 equals. Well, let's just multiply these two things together. Now, you remember i times i, which is i squared, gives minus 1. So members which do not contain the imaginary part, members without i are cosine times cosine and sine times sine, but with a minus sign because it's i times i would be minus 1. So the real part would be cosine alpha times cosine theta minus sine alpha sine theta plus i plus. OK, what's the imaginary part? With i, I will have cosine alpha sine theta and sine alpha cosine theta, right? Sine alpha cosine theta plus cosine alpha sine theta. That's the result of multiplication. Well, what is this? Well, you should really recognize this formula. It's a cosine of alpha plus beta. And this formula is sine of alpha plus beta. So this is, again, one of the very, very beautiful part of this theory. When you multiply two numbers with a modulus of 1 and the angles alpha and beta, you get another number, which is also of modulus 1. And the angles are some of these angles. So multiplication by another number is basically a rotation by a corresponding angle. This is a really very unexpected and very strange, at least personally. That's how I feel about this. And that's what makes these mathematics very, very interesting and beautiful, if you wish. So what does it mean for us? Well, let's just think about the consequences of this interesting conclusion. So multiplication of the complex numbers actually results in addition of the angles which are associated, the polar angles which are associated with these numbers. Does it remind you anything? Well, let me express this rule slightly differently. And you will recognize what I'm talking about. Let's consider a function. Now, this is a function of real argument. x is a real argument. It's an angle, right? Angle measured in radians. And the value of this function is a complex number. So it's a function which has domain real numbers and codomain complex numbers. And what's an interesting property this function has? F of 0 is equal to what? This is 0, so cosine of 0 is 1. And sine is 0, right? What I have also proved right now, F of alpha times F of beta equals F of alpha plus beta, right? If I multiply these two, I get also cosine of alpha plus beta plus sine of alpha plus beta. I just proved it, right? So in this particular notation, does it remind you anything? Well, remember the function, exponential function? Well, a to the 0 is equal to 1, right? And a to alpha times a to beta is equal to a. And the exponents are heading together, right? So if you don't remember this, go ahead and re-examine the exponential functions. They do exist in the course. So you see the resemblance. The only difference is this is a function of real argument with real value. And this is a function with real argument with complex value. Well, yes, I understand there is some difference, but there is a huge similarity in the properties of this function, right? And that's what actually prompted Loyler to his formula I was talking about at the very beginning. All right, so be it as it may, I was just trying to show how interesting trigonometric functions can be related to algebra, which we have already learned, the algebra of exponents, OK? What else I can actually extract from this interesting thing that if I multiply two numbers like this, their polar angles are added together. Here is, for instance, very interesting thing. Let's again consider the unit circle. And we have agreed that if I have some kind of a number, which is complex number, which is represented by a point on this unit circle, and then I have another number, so this is angle alpha, I have another number, then their product would be something like this, which is alpha plus beta. So when I multiply, I add the angles, OK? So if I multiply by itself, so if I will do a square of a complex number, I will do cosine alpha plus I sine alpha squared. So what does that mean? Well, it means that the polar angle alpha, which was associated with this number, with this complex number, becomes two alpha, right? Alpha plus alpha for the square, OK? And alternatively, if I want to take a square root of something, instead of doubling the angle, I have to halve it, right? So if I want to do something like square root of cosine two alpha plus I sine two alpha, that would be cosine alpha plus I sine alpha. So this is two alpha, and this is one alpha, right? So that's quite interesting. What does it mean in this particular case? I would like actually to use it for one interesting purpose. Let's say I start with angle this one, angle pi. Now what's the coordinate of this point? Minus 1 comma 0, right? So we are talking about ax plus bi equals minus 1 plus 0i, which is minus 1, right? So this particular point represents this particular complex number. It has, in polar coordinates, r is equal to 1, obviously, because that's the unit circle. And pi is equal to pi. Pi is equal to pi. Now what if I want to have the square root of it? Well, as I was just explaining, I have to divide the angle in half. So half of pi is pi divided by 2, which is this point, right? Now this point, this is pi divided by 2. It has coordinates 0, 1, right? Now 0, 1 is 0 plus 1 times I, that's I. So according to my rules, square root of minus 1 is equal to I, right? Which is actually absolutely correct thing. That's how we defined I, that I square is equal to minus 1. So it completely corresponds to whatever the theory we started from, the theory of complex numbers. Now here is a slightly different last trivial. I mean this is trivial because this is basically a definition. But this is a little bit less trivial. Let's start with I. Now in one of the lectures about complex numbers, I was presenting you with a problem. Express square root of I in the regular A plus Bi form of the complex numbers. Now what did they do to find out A and B? Well, I squared both sides. I got I is equal to A plus Bi square, which is A square minus B square, right? Bi square would be B square and I square, and I square is minus 1, so it's minus plus. So this is the real part. 2AB5, this is the imaginary part. Now I wanted to be equal to I, which means this supposed to be equal to 0 and this supposed to be equal to 1, right? So I had a system of two equations with two variables. Then I solved these equations and I found that A is equal to 1 over square root of 2 or square root of 2 over 2 and B equals 1 over square root of 2. Actually, with the minus sign it's also true because I can both have minus 1 over square root of 2 and minus, and the square will be exactly the same. But let's concentrate on this solution. So it's not a very complicated solution, but it's a solution, right? Now let's go back to my trigonometric thing. Let me start with I, which is this point, which has r equals to 1 and phi is equal to pi over 2. And let's extract the square root of it using the rule I was just talking about by dividing the angle in half, right? So dividing angle in half means that the new angle would be here. It has pi over 4. And what's the cohesion that's at this point? Well, cosine of pi over 4 and sine of pi over 4, right? So square root of I is equal to cosine pi over 4 plus I sine pi over 4, which is equal to square root of 2 over 2 plus I square root of 2 over 2, the same solution. So purely trigonometrically, I can come up with exactly the same solution I've got from solving the algebraic system of two equations with two variables. Isn't that wonderful? I love it. So basically, that's the general introduction to this very interesting and intricate relationship between trigonometry and the algebra of complex numbers. I did not derive completely the Euler formula because I didn't really explain what the number E is. I didn't talk about this, but I didn't really define it properly yet. But I did actually hinted that the properties of the complex numbers expressed in the trigonometric form are very much like the properties of the exponential functions. And the only difference is that exponential functions and algebra functions from real numbers to real numbers. And these are from real numbers to complex numbers. But that's a normal expansion. I mean, that's how we always expand our field of knowledge. We realize that we can do something this way, and then we invent something else. We define new numbers. We define new operations, et cetera. Now, I've not really touched what to do with numbers which are not on the unit circle. But this is really a very easy extension. Because if the number is not in the unit circle, let me just make this presentation quite quickly. I can always represent this number as scaled up, or if it's closer to 0 point, or scaled down the unit radius, the radius which has the length of 1. So I can always say that this particular AB, A plus BI, is equal to some kind of a modulus, which is this distance, which is greater than 1, or less than 1, or whatever, times this particular number, which is, again, cosine phi plus i sine phi. So when I multiply two numbers like this, my modulus plural is moduli. My moduli are multiplied by each other. And this part, when I multiply cosine alpha plus i sine alpha by the cosine beta plus i sine beta, would actually add the angle. So if I have something like m1 cosine alpha 1 plus i sine alpha 1, and I multiply it by m2 cosine alpha 2 plus i sine alpha 2, I will get, as a result, m1 multiplied by m2 multiplied by cosine of alpha 1 plus alpha 2 plus i sine alpha 1 plus alpha 2. I mean, that's obvious, right? Because this is just the multiplication m1 times m2. This is a new modulus. And this part, the normalized part of the complex number, contains some of two angles, some of two polar angles. Well, that completes this particular lecture. I hope it was entertaining and fun, not only interesting from the informational standpoint. And again, the more you know, and this is not just for mathematics, it's like everywhere, the more you know, the more you realize the connections between certain things. And it really kind of opening your mind to significant growth. I mean, that's the very important part of learning experience. First, you start with learning this, this, this, and this. And the more you know, the more you actually integrate all this knowledge and combine together, it really represents a beautiful picture. On this now, I would like to end this lecture. Thank you very much, and good luck.