 Hi, I'm Zor. Welcome to InDesert Education. I would like to analyze the properties of one particular function, which is not a simple, not a basic trigonometric function like sine or cosine, but a mix of them those. So the function which I'm talking about is this one. That's supposed to be sine. So it's a linear combination of sine and cosine. Well, let's start from the beginning. Sine looks like this, more or less. So this is a period from 0 to 2 pi. This is pi. And then it repeats itself. Now, the cosine on the same period looks slightly different, something like this. This is a cosine. The linear combination of them, just looking at the formula, you cannot really predict the behavior of this. One function is going down, another is going up. At this point, they are going both down. I mean, it's really not really obvious how the whole behavior of the function would pan out. Now, the function sine by itself or a cosine, they have this waving behavior, their periodical. And on one particular period, it's a wave kind of thing. And sine is different from the cosine by basically a shift. You can obviously make sure that this is true by considering this identity, right? So they are actually different by a horizontal shift by pi over 2, left or right, depending on which one you're talking about. But this one is not really so obvious that the behavior would really be repeated. By the way, the behavior of sine and the cosine, this behavior of waving and being a periodic function is called sinusoidal, from the sine, sinusoidal. So both sine and cosine have this sinusoidal behavior. Now, question is about this one. So although this is not obvious, but I will prove to you that the behavior of this function is also sinusoidal. And here is how. First of all, it's obvious that we are not considering case when a and b both are equal to 0 at the same time, because the function would be equal to 0, and that's the trivial case. So if you get about this, a and b are not simultaneously equal to 0. But if one of them is equal to 0, then it's obviously a sinusoidal function. Now, what I will do is the following. I will multiply and divide this function by this non-equal to 0, because they are not simultaneously equal to 0 factor. That's the same thing, right? I multiply by square root of a squared plus b squared, and I divide it, the function, by the same factor. Factor is not equal to 0, as we have just stated. So there is nothing different, this function, than this function. Good. Now, let's consider these two numbers, a over a square root of a squared plus b squared, and b over the same square root. Let's consider a unit circle. Since this I will call a, this I will call b. Now, it's obvious that a squared plus b squared is equal to 1, right? a squared would be a squared over a squared plus b squared plus b squared over the same thing. You will add together the numerators, and you will have exactly the same as the denominator, a squared plus b squared, and they're not equal to 0. So this is an important property of these two numbers. Now, let's consider a point which has coordinates, a comma b. So a is abscissa, d is ordinate. Now, again, it's obvious that since sum of abscissa square plus ordinate square is equal to 1, then this point is supposed to be on a unit circle somewhere. Right? So this would be a, and this would be b. And since a squared plus b squared is equal to 1, so the difference from the 0 to our point is 1, so it's on the unit circle. All right. Now, every point on the unit circle corresponds to certain angle phi, right? And by definition, cosine of phi is equal to abscissa, which is a, and sine of phi is equal to ordinate, which is b. So this is, by definition, a cosine, and this is, by definition, sine of this angle phi. So my point is that no matter what a and b are given to you, these two numbers, which I call capital A and capital B, would correspond to a point on the unit circle, and therefore they correspond to an angle phi. The cosine and the sine of phi are equal to exactly these two expressions. Regardless of the choice of a and b, as long as they're not together at the same time equal to 0, we can always talk about the angle phi, which is uniquely determined by these two numbers, a and b. Because from a and b, I uniquely determine the capital A and capital B, these expressions. And these determine the point on the unit circle, and the point determines the angle. So it's all uniquely determined. So the uniquely determined angle phi corresponds to a choice of lowercase a and lowercase b, and the cosine and sine are actually these formulas. So this is all from definition of the cosine and the sine. Now, using this, let me just rewrite this particular formula in the following claim. f-attacks equals square root of a squared plus b squared. And here I will put cosine phi cosine x plus sine phi sine x. Since this is the same as this, and sine is the same as b over square root. Now, and this is very important now. This is the last step. This is a cosine of x minus phi. If you remember the formula, if you don't, go to the corresponding lecture in this course. It's in the same topic of sum of two angles. So this is the formula for cosine and sine. And obviously, we still have to preserve this multiplier. So our function, f-attacks, is equal to this, where phi is some angle uniquely defined by a and b coefficients. So now it's obvious that this particular function is a type of which has a sinusoidal behavior, because it's still a cosine. So how can we draw the graph of this function? We take the function cosine, then we shift it to the right by phi, whatever the phi is, and we can determine the phi using, again, a and b. So we know the cosine of phi, we know the sine of phi, we know everything. So we shift it by phi, and then we stretch it vertically. So this is the horizontal shift, and this is the vertical stretch by square root of a squared plus b squared. So my point is that this relatively, well, it's hard to imagine the behavior of this function just looking at the formula. Let's put it this way. But still, which is very remarkable, it still has this sinusoidal behavior. And that's what's interesting. Quite frankly, I was a little bit surprised when I first learned about this type of thing. There are some other classes of functions which, if you mix them together, then the result, the mixture, belongs actually to the same class. So sine and cosine basically are in this type of class, and the mixture means you linearly combine them together. So that's it for this particular lecture. It's a relatively short one, and I just wanted to present to this very interesting property that sinusoidal functions are kind of closed within themselves. So if you combine two sinusoidal functions, you still have a sinusoidal function. Now, it's a different question of what happens if you have different arguments. You see here, you have x and x. But what if you have x and 2x, or x and 2x plus 3, or something like this? This is a different case. And that actually already leads us to a little bit more complicated functions, not just the plain sinusoidal kind of functions. But if the arguments are the same, then you still have a sinusoidal function. All right, thank you very much. And don't forget that Unizor is a website which does not only provide the lectures like these, but it's also filled with many problems which you can solve. And also for educational process, it has exams. It has ability for supervisor or parents to enroll students into this or that particular topic, check the result of the exam, et cetera. So please use the site. That's very, very beneficial for you. Thank you very much and good luck.