 Hi, I'm Zor. Welcome to Unizor Education. This is the third lecture in the mini-series, which defines different inverse trigonometric functions. In this case, we're talking about arc tangent. This is the function which is inverse to tangent, obviously. So before talking about arc tangent, let's talk about tangent. And what's very important about it's graph, the main range, these type of things. So tangent is the function which you can define as sine over cosine of a national. Now, let's build the graph. Well, if you remember, it's great. If not, I'll just try to kind of derive the graph from the graphs of sine and cosine. So if this is pi over 2, this is pi, this is 3pi over 2, this is minus pi over 2, this is minus pi, et cetera. So sine goes from 0 to 1 and then to 0 and then to minus 1, et cetera. Minus 1, 1, et cetera. That's how sine goes. Now cosine is symmetrical. It goes like this. So when we divide one by another, sine by cosine, obviously it points pi over 2 and minus pi over 2, and 3pi over 2 and minus 3pi over 2, where the cosine is equal to 0. We will have asymptotes. Now here, if you divide this graph by this graph, you will have 0 here because numerator sine is equal to 0 and the cosine denominator equals to 1. And then the graph grows and finally it grows to plus infinity because the denominator goes to 0 and sine is 1. So it goes something like this. Now similar in this case, the situation is very much alike except it's negative because the sine is negative and the cosine is positive. So when you divide one by another, you will have negative results. But also in this point, you have an asymptototent that goes to minus infinity. So this is our tangent. Now after that, it goes by a period of time. All right, so knowing what the tangent is, now let's talk about what can we say about inverse function, arc tangent. Well, obviously this function, as I have represented it in red, does not have an inverse function because for any value, there are multiple arguments. In each of those arguments, function takes the same value. So you cannot determine by the value of the function tangent by having y defined. You cannot define x. And we will do the same thing with the tangent as we did with sine and cosine in exactly the same situation, which means we will reduce the domain to the area where the function is monotonic, which in this case, traditionally, is from minus pi over 2 to pi over 2. The function is monotonically increasing. And it takes all the values from the range, from minus infinity to the plus infinity. So we wipe out everything around this as if it does not exist. And we are left with the function, which will be called new tangent or whatever, which is defined only for an interval from minus pi over 2 to pi over 2. I don't remember if I mentioned it before, but whenever I'm using parentheses, it means the boundaries are not included. Whenever I'm using square brackets, that means the boundaries minus pi over 2 and pi over 2 are included. In this case, we cannot include boundaries because that's where we have asymptotes. So the function is not defined at minus pi over 2 and pi over 2. It's defined in between. That's why I have round parentheses. So this is my domain. My range and my co-domain are all real numbers. So this is the domain, and this is the range. Now we can define an inverse function or a tangent of x. Well, the sense of this function, the meaning of that function is, well, inverse. If you have some kind of a value of the tangent, we can find the angle in radians tangent of which is equal to x. So that's what the meaning of the inverse function. Now, obviously, the x can be any value which this function takes from the range, which is minus infinity plus infinity. And the range of the function, so this is the domain, and the range of the function would be the domain of the tangent function, which is minus pi over 2 pi over 2. Now let's do the graph. Now, we know that there is a symmetry between the graphs of the function and its inverse function. Symmetry means relative to the angle bicep. I think I need a new picture. This doesn't look good. The picture would look like this. Minus pi over 2, pi over 2, this is x, this is y, and the graph looks like this. That's my tangent. Now, if I will reflect this relative to the angle bicep, that's how it would be. My range would be from minus infinity to plus infinity, sorry, my domain for this function. My range would be from minus pi over 2 to pi over 2, which is pi over 2, and this is minus pi over 2. So whatever asymptotes used to be, they will be turning into my, basically, maximum and minimum. And the function would look like something like this, symmetrical relative to the angle bicep. So the range is from minus pi over 2 to pi over 2. The main is all real numbers, and that's the graph of the function arc tangent of x. It's quite an interesting function. What's interesting about it, it can be basically modified. And right now, it's from minus pi over 2 to pi over 2. But we can actually adjust it with proper multipliers and shifting, et cetera, to any range. And I was actually thinking that function like that, and this is a practical implementation of something, function like that can actually be used in some practical situation, namely, we know that we have something which is called progressive text, which means that the percentage of the taxation is greater with base amount greater. Now, what it means is that you might have some minimum taxation, and you can have a maximum taxation, and the percentage grows from the minimum to the maximum well in something like this type of a fashion. So we can actually establish this instead of minus pi over 2 to pi over 2, we can establish this graph from a to b basically by aging something and multiplying something by this graph. And also, we can regulate the stickness of this curve by multiplying x by certain factor. So it can be used for the purpose of defining a very simple formula. OK, this is the formula for taxation. Whatever you earn, just multiply by this percentage expressed as arc tangent with certain multiplier here and multiply here and maybe horizontal shift, and you'll be fine. So that's one of the practical implementation of taxation using function like this. So it gradually moves towards certain asymptote on both sides. All right, that's it for arc tangent. I do recommend you to go to unisor.com to this lecture, to notes to this lecture. Just read it again. I think it would be very helpful. Other than that, good luck.