 Hi, I'm Zor. Welcome to News or Education. Continue talking about math plus and problems. Math plus is because whatever I'm talking right now is rarely addressed in schools. But at the same time, I believe all these problems are very important. They are actually forcing the student to think about certain things. And that's the purpose of the whole course actually. And the one which was before it. There is a prerequisite course called Math for Teens. So both presented on Unisor.com. I do suggest you to familiarize yourself with the Math for Teens course, the prerequisite. Just because it's very important to know all the basic stuff before you address something which is I call the Math plus and problems. Now, in particular today we will talk about something which is related to trigonometry. And I call this lecture trigonometry 01 as part of this course Math plus and problems. The problem is it's not really just trigonometry. Whatever I'm talking today about is basically a mixture of trigonometry, algebra and calculus. So, well, here it goes. Now, this lecture is basically about complex numbers and in particular the Euler's formula. Now, if you're not familiar with Euler's formula and whatever stands behind it, then it's too early for you to go through this lecture. You have to really go to lecture about Euler's formula, etc. I did present it in Math for Teens course. So you can go there. That's the one which right above the Math plus. So you have to go to trigonometry and find something which is called trigonometry and complex numbers. I think this is one of the last topics of that course. And in that topic there are a few lectures in particular about Euler's formula. So, the guy called Euler, he's a Swiss mathematician, but most of his life he spent in Russia. And I think it would not be exaggeration to say that he is basically the founder of the Russian School of Mathematics. One of the founders, but probably one of the most important founders. Let's put it this way. He published a few textbooks at his time. It was like 18th century, I believe. And what's interesting is that whatever he was kind of using as a material for his educational efforts at that time, still is present in almost the same format in the contemporary textbooks for geometry, for example. Not much changed actually. And quite frankly, if we will be as educated right now as his students were educated in 18th century, that would be quite an achievement to tell you the truth. In any case, so Euler and Euler's formula. Now, what is Euler's formula? It's quite famous formula. I mean, I probably would put this formula in its importance on the same level as Einstein's formula about relationship between mass and energy. E is equal to mc2, if you remember this. By the way, I do have the course called mass, not mass, physics for teens where I present many different things and then relativity for all where I do address this formula, Einstein's formula, and basically derive it in some way. So, back to Euler's. First of all, the complex number, the new number, the square of which is equal to minus one, usually it's called i, is an imaginary number and it was introduced into mathematics at a certain time. And the manipulation with this number is exactly, well, the same basically as manipulation with normal real numbers. For example, you can add them, you can subtract, you can multiply, you can divide, etc. Now, any complex number contains the real part and imaginary part. A and B are real numbers, i is this imaginary square root of minus one and this basically describes all complex numbers and we can manipulate these numbers as real numbers without any problems. So, one of the things which was introduced to complex numbers is basically arithmetic, plus, minus, multiply, divide. But then there are some other things like, for example, raising into some power. Well, obviously raising complex number into, let's say, integer number, that's simple, it's just multiplication. But what if you are using complex number in the exponent? That was not really defined because we don't really know what it is. So, Euler came with the formula e to the power ix is equal to cos x plus i times sin x. So, this is a real part, this is imaginary part. You can consider this as a definition, obviously, but any definition should be reasonable. So, this particular definition, if it's a definition, it's a very reasonable definition because all the formulas which we know about exponential functions define this way, it really holds. So, everything which we know about from the algebra, all the operations, like for example, we all know that something like a to the power x plus y is equal to a to the power of x times a to the power of y. We all know that from the real number theory. So, it stands even if x and y are complex numbers if you use this definition. So, everything seems to be fine. So, you get a lot of different logical foundation behind this formula. And it's very simple and it really opens for many different interesting things. And in particular, I will address two problems today where this particular formula plays a very important role and it's significantly simpler to derive what I want to derive using this than the way how we did it before. So, let's talk about trigonometry now. Now, the first problem is when we learned trigonometry, we had the formula sin of alpha plus beta is equal to sin alpha cosine beta plus cosine alpha sin beta. Now, in the course of trigonometry, this formula was derived basically using, you know, certain not very trivial manipulations. My first problem is derive this formula using the Euler's formula. So now, I would suggest you to pause the lecture and do it yourself. Now, I will continue in the following query. Let's start with E to the power i of alpha plus beta. What is this? Well, on one hand, this is cosine alpha plus beta plus i sin alpha plus beta. That's the definition. X is alpha plus beta. On another hand, according to the normal rules of exponent, it's E to the power i alpha plus i beta, which means it's E to the power of alpha times E to the power i beta, which is E to the power of i alpha is cosine alpha plus i sin alpha times cosine beta plus i sin beta. Well, let's just multiply. Cosine times cosine. Sin times sin and i times i. i times i is i square and i square, as we know, is equal to minus one. So it would be minus sin alpha sin beta plus i times cosine. So it's i times sin alpha cosine beta. And i times this plus, I will put i out of the parenthesis, cosine alpha sin beta. Okay, so what do we have? We have the equality between two complex numbers. This complex number and this complex number. Now, if two complex numbers are equal, their real parts are equal and their imaginary parts are equal. Now, the real part of this is this and the real part of this is this. So we have the formula. Cosine of alpha plus beta is cosine alpha cosine beta minus sin alpha sub beta. Now, imaginary part of this is this and imaginary part of this is this. Therefore, sin of alpha plus beta is equal to sin alpha cosine beta cosine beta sin alpha. Exactly as I put it on the top. So we have derived it using, instead of really kind of tricky manipulations with angles, we have derived it extremely simple in extremely straightforward way. Just multiply a couple of things. It's really not a big deal. However, we obviously used all the theoretical material which we have learned before, primarily everything how to deal with exponents, the whole theory behind complex numbers, the manipulation with complex numbers, the rules when two complex numbers are equal to each other, that the real parts are imaginary parts should be equal, etc. So the progress in theory helped to solve this problem much simpler. And that's how it usually happens many, many times. If something is really kind of difficult to approach, some people really kind of struggling through some sophisticated maybe proofs or solutions or whatever, but then the general theory is moving slowly but surely in certain direction and helps to develop certain concepts making the whole problem much easier to solve or solution much simpler to present. So in this particular case, I'm using all the richness of the theory which we have developed before, complex numbers and plane algebraic manipulations with complex numbers to solve trigonometric problems. Now, my second problem is also about using Euler's formula for different purposes. I would like to know what is the derivative. Now, this is going to calculus now. What is derivative of sine and what is derivative of cosine? And I'll do it kind of similar. So we need to establish, by the way, how we derive formulas for derivatives of trigonometric functions. Well, if you remember derivative of sine is cosine and derivative of cosine of x is minus sine of x. Now, we derive these formulas in the course of calculus based on the definition of derivative. We give the increment to x like x plus delta x and then we had the ratio between increment of function divided by increment of arguments and took the limit to it. Well, yes, okay, that's fine. But if we kind of didn't do that but we do know some other things, for example, how to differentiate exponential function, it would be much easier. So again, I'm going to use the Euler's formula to derive these two from algebra and calculus of exponential functions, which is kind of easier. So exponential function, let's call it e to the power of x. If you remember, the derivative of this function is e to the x is itself. It's the only function which is equal to its own derivative. Now, if I will have something like a coefficient here, a times x, the derivative would be a times... But knowing this, I'm going to derive this formula much easier. And here's how. So let's take e to the power ix, which is cosine x plus i sine x. That's the Euler's formula and differentiate it. Now, on one hand, that would be derivative of cosine, so let's put it cosine of x plus i derivative of sine. So we know that just because derivative is additive function, so derivative of a sine is equal to sum of derivative. And if you have a multiplier, multiplier goes outside of the derivation. So these are all, again, theory, which we have learned about derivatives. And that's why I wrote it this way. On another hand, I'm using this formula with a is equal to i. So that would be i times e to the power of x, right? Derivative of e to the power of ix is i times e to the power of x. Same thing as here, where a is equal to i. Okay, what is this? Well, this is i times e to the power of x formula, Euler's formula, cosine x plus i sine x, which is equal to... Let me first put the real part. The real part would be i times i, which is i square, which is minus. Minus sine x plus i cosine x. So this is real part, this is imaginary part. This is real part, this is imaginary part. Which means what? Cosine derivative is equal to minus sine, and sine derivative is equal to cosine. That's it. Simple? Yes. But obviously, this simplicity is based on the whole theory of how derivative are supposed to behave, and how to have a derivative of exponential function, and obviously the whole theory of complex numbers, how they are multiplying, etc. So yes, again, the same thing. The progress of theory allows us to do something extremely simple. Okay, so these are just two usage of this famous Euler's formula, which basically combines together, as I was saying, algebra and trigonometry. And in some cases, actually, it helps you even in geometrical manipulations, but that would be kind of a story of another time, another lecture. So I do suggest you to read the notes for this lecture. To get to the notes, you have to go to the course on Unisor.com. You have to choose the course mass, plus, plus, and problems. Then in submenu, choose trigonometry. And that would be the first lecture in that trigonometric topic. Well, that's it for today. Thank you very much, and good luck.