 Hi, I'm Zor. Welcome to InUzor Education. I would like to continue solving trigonometric equations. As usually, I encourage you to do it yourself first before you listen to this lecture. So just go to Unizor.com. This is problems 2 in trigonometric equations topic. Again, as usually, I would like to mention that none of these problems, you will meet in your practical life, practical experience, whatever the work you do, except in the very, very few cases. So why do we do it? Well, as usually, I'm just staging that this is an excellent tool to develop your creativity, your intelligence, your ingenuity, and analytics. So that's the purpose. It's the development, mind development, not any kind of skill or whatever you can learn from here. All right, so let's solve these equations. Equation number one is very simple actually. Cosine to the force degree x minus sine to the force degree x equals to 8. Well, it shouldn't really scare you, this force degree, etc. Well, just because there are very simple transformations, which you can use to basically reduce this equation to something very, very simple, and here it is. First of all, cosine to the force degree is cosine square square, right? So that's what it is. Cosine square x square minus, same thing with the sinus, sine square, that's the same thing, right? Now, the difference between two squares, u square minus v square is equal to u minus v times u plus v. I'm sure you remember it and those who don't just multiply these, whatever u and v mixed will be reduced. This is plus and this is minus, so you will have only u square minus v square. So I will use this where u is cosine square and v is sine square. So it's a cosine square x minus sine square x times cosine square x plus cosine. So, oops, sine square. What is this? This is equal to one. Cosine square plus sine square at any angle is equal to one. So what's left is this. Now, cosine square x minus sine square x equals to p. Now, this is, again, I hope you recognize this formula. If you don't, go back to the previous lecture and you will see that this is exactly a cosine of 2x. Now, and this is something which we have solved many times before. It's 2x equals to arc cosine a plus 2 pi n, where n is any integer and this is plus or minus. From which, we conclude that x is plus or minus 1 half of arc cosine of a plus. So it's 2 and this goes, so I will reduce by 2 and it will be p n where n is any integer number. That's the answer. So that's very simple and there are only two things which you really should remember that one formula which I was using is the difference between two squares. And the second one is the cosine of a double angle, this one. Now, as I was saying, I usually don't remember the formulas. I prefer to derive them on the fly. But some formulas you probably have to remember. And these two, by the way, are among those which I personally do remember by heart, like cosine square minus sine squares, cosine of a double angle. And the difference between two squares is difference between them times some of them. Well, actually it's up to you what to remember, what to derive. I'm just sharing my personal attitude towards this. All right, next problem. So this is easy. Next would be slightly more complex. Cosine x times cosine 2x plus cosine 4x equals 1 eighths. Well, this is something which definitely requires certain ingenuity, certain artificial technique, certain trick if you wish. And if this particular technique comes to your mind, great, it's simple. If it doesn't, then it's probably very difficult to solve otherwise. So let me just share this particular trick with you. Let's multiply both sides by sine of x and also I will put this 8 on this side. So by 8 sine x. So on the right, we will have sine x. If I multiply by 8 sine x. On the left, I will have 8 sine x, cosine x, cosine 2x and cosine 4x. When I multiplied by sine x, what happens with solutions to this equation? I might actually add solutions which are not really the solutions of the original equation because I'm multiplying by something which contains an unknown x. So which extra solutions I'm adding? Well, those were sine of x is equal to 0, right? Because if sine of x is equal to 0, it's solution to this equation. Obviously, it's 0 here and 0 there. Now, is it solution to the original equation? Well, let's check it out. Now, if sine x is equal to 0, that means x is equal to pi n. Now, cosine of pi n, remember, cosine is abscissa. So pi n is 0, pi, 2 pi, 3 pi, et cetera. So it's these two points and abscissa of this point is 1. Abscissa of this point is minus 1. So this cosine is either 1 or minus 1. Cosine of 2x, if x is pn, is always this point, so it's always 1. And 4x is 4pn, it's also 1. So it's minus or plus 1 times 1 times 1. It can never be equal to 1h, right? So any solution of this type is no good. So if at the end, after we solve this particular equation, we will have certain solutions, we must check. And those which are one of these, we should really exclude. They are extraneous. They don't belong to the original solution. So let's just remember this, that x should not be equal to pi n, where n is any integer number. After we have identified these extraneous solutions, let's solve this guy. And this is very easy. Look, out of this 8, I will have 2 sin x cos x cos 2x cos 4x equals to sin x. Now, why did they do this? Why I multiplied 4 by 2 to get 8 because of this? You recognize this formula? Again, that's something which I do remember by heart. It's sin of 2x. Sin of 2x is 2 sin x cos x. Because it's x plus x, it's sin x cos x plus cos x sin x, which is 2 sin x cos x. All right, great. So one simplification we made. We reduced two members, sin and cos, to 1. Now, what's left? 4 sin 2x cos 2x cos 4x equals to sin x. Now, let's replace 4 with 2 by 2. Now, do you recognize this? It's exactly the same thing. It's a sin of a double angle. Double angle in this case is 4x. So this is sin of 4x. This is 2. This is cos sin 4x equals to sin x. You see how everything is rose in? Do you recognize this? It's also a sin of double angle. In this case, double angle is 8. So it's sin of 8x is equal to sin x. Well, that's much simpler, right? So let's wipe out this thing, this thing. And that's the only thing which we have. And we actually did solve something very similar. But I will do it again. So it's sin 8x minus sin x is equal to 0. And the way to do it is to convert it into a product. And if product is equal to 0, means every component is equal to 0, either or. All right, so how to convert this into a product? It's different between two signs, right? Well, the way to do it is if I'm not mistaken, I have to take the middle point between them, which is 8 and 1. It's 9 halves, right? So sin, instead of 8x, I will put 9x over 2. And the difference between them is 7. So it's plus 7x over 2. So minus sin 9x 2 minus 7x over 2. So again, this is middle point, which is half of their sum, 8 plus 1, 9 over 2. And this is the half of the difference. This is 1, this is 8. So we take the middle point, middle point plus or middle point minus. So this is 9 half and this is 7 half and 7 half. So 8 is 9 half minus, plus 7 half and 1 is 9 half minus 7. Now, why do they do it? Because if you will open this up, it's sin for sin. And this will be also sin cos. So the sin cos will be reduced. Okay, let me just do it in details. Sin 9x over 2 cos 7x over 2 plus cos 9x over 2 sin 7x over 2. Now minus sin 9x over 2 cos 7x over 2. Now minus, but this is minus, so it's plus cos 9x over 2 sin 7x over 2 equals to 0. And this thing is reduced. So I have only this, which is the same as this. So it's 2 cos 9x over 2 sin 7x over 2 equals to 0. Which means either cos 9x over 2 equals to 0 or sin 7x over 2 equals to 0. So let's check what solutions to this and this. In this case it's easy, 7x over 2 equals to pi n. Which means x is equal to 2 pi n over 7. However, however, we should really exclude this thing, right? So this formula is true for all integer n, except those which are multiple of 7. If n is multiple of 7, then this will be reduced and I will have 2 pi something, which means one of these, and these are not the solution. This formula is good, but only if n is not multiple of 7. Now similarly here, solution to this is 9x over 2 equals to pi over 2 plus pi n. So 9x is equal to pi plus 2 pi n is equal to pi times 2n plus 1. So x is equal to pi 2n plus 1 divided by 9. Again, it's only for those cases where 2n plus 1 is not multiple of 9. Because if 2n plus 1 is multiple of 9, then we have a problem. By the way, in the answers to this, I'm specifying conditions slightly different and saying that something like if n minus 4 is divisible by 9, then it's not the solution, which means exactly the same thing as this. And it's a good exercise for you actually to check that this is exactly the same as whatever the website says the solution is. Well, basically that's it. What's this lesson which you actually might think about from this particular problem? You see, we have multiplied by sine of x and the whole thing on the left just rolled in from x to 2x to 2x to 4x and from 4x to 8x. Well, it's a gas, it's a trick, whatever you want to call it. But again, the more problems like this you will solve, the more it will be inculcated in your mind that there are certain tricks which you can actually use, like multiply by something, divide by something, add something, convert sum into product, product into sum. All these techniques are your tools in your toolbox. And another lesson actually is that if you multiply by an expression which contains the variable, we multiply by sine x in both cases, from both sides, you might add certain new solutions which are not really the solutions of the original equation and that's why you have to be very careful. And that's why I excluded these cases at the very end. That's it. Next. Sine x equals sine 2x times cosine 3x. Okay. Again, there is something which we have to do to reduce it to more or less solvable shape and form. Well, my first guess would be to present sine of 2x as 2 sine x cosine x and then reduce by sine x. Is it a good guess? Sure. Why not? So, however, if we just reduce by sine x without saying anything, we might lose solutions in this case. In the previous problem, we multiply by sine x and we edit solutions which are not really solutions of the original equation. Here, obviously, if sine x is equal to 0, which means x is equal to pi, and these are solutions to the original equation because this is 0 and this is 0. However, that's not the case if we will reduce. So, we might lose these solutions. So, what we have to say is, okay, we already got a certain number of solutions. These are solutions. Not all the solutions, but solutions, nevertheless, to the original equation. Now, we can reduce it by sine x and we will have 1 is equal to 2 cosine x cosine 3x. Okay. So, what's next? If you don't know what to do and you have the product, try to convert it into a sum. If you have a sum, try to convert it into a product. That's all I can say. Now, what's that? Well, this is the, let's see, cosine of 2x plus x plus cosine 2x minus x. So, again, between 1 and 3, 2 is a middle point. And so, the x is 2x minus x and this is 2x plus x. Now, if you will open the parentheses here, it's cosine 2x. No, I need 3x. Sorry. It's 3x. So, it's cosine 3x times cosine x minus sine sine. This is cosine times cosine plus sine sine. So, sines will be reduced and 2 cosines will give me exactly this. So, the equation which I have is this one. So, 2 cosine x times cosine 3x is this sum. Now, is it better? Well, let's present it in this way. Here is another observation. This is double this, right? 4x is double 2x. You remember the formula for double angle, cosine 2 phi is equal to cosine square minus sine square, which is equal to, if I will replace sine with 1 minus cosine, I will have 2 cosine square phi minus 1, right? So, I can always replace cosine of 2 phi with 2 cosine square phi minus 1. And that's what I'm doing. That's exactly in this case where phi is actually 2x. So, it will be 2 cosine square 2x minus 1. That's what cosine of 4x is. Plus cosine 2x equals to 1. Now, what is this? How to solve this? Hey, this is a quadratic equation for cosine of 2x. So, let's just put y is equal to cosine of 2x. And now the equation is 2y square plus y minus 2 equals to 0, right? That's a quadratic equation. If I will get y, then from y I will get x, correct? Y, okay, formula for quadratic equation. This is 4. This is minus 1 plus minus square root of 1 plus 16, which is equal to minus 1 plus minus square root of 7, square root of 17. This is a cosine of 2x. Now, think about this. This is slightly more than 4, right? So, if I have minus here and minus here, I will have greater than 4 in the numerator and 4 in the denominator. So, my ratio will be less than minus 1. So, by absolute value, it will be greater than 1. Cosine cannot be greater than 1. So, minus actually doesn't work for us. Only plus. So, cosine of 2x is equal to square root of 17 minus 1 over 4. That's what we have here. The solution to this is 2x is equal to plus minus arc cosine of this monster plus 2 pi n. And the final solution is divided by 2 plus pi n. That's the solution. Plus this, which we already got from the very beginning. These are two sets of solutions. Well, that's it for today. I do recommend you to go through these problems again just by yourself and just try to solve and check the solutions. The answers are on the webpage. See if you can get the same thing. Just to, you know, to inculcate into your brain that there are certain techniques, tricks, whatever you want to call them, which can be used to solve. Every time you solve another problem, you train your mind to find your way in the unknown situation. Because that's what the skill which you can develop and which you will have invariable during your, whatever the professional life you have. That's it. Thank you very much and good luck.