 Hi, I'm Zor. Welcome to Inuzor Education. Today we will talk about function arc secant, which is an inverse function to a secant, which in turn secant is defined as you remember one over cosine, so arc secant is inverse to secant. And as usually, I will talk first about secant and its graph, and then I will inverse it. And by the way arc secant, this is the fifth lecture in the mini series about different inverse trigonometric functions. We have already covered arc sine, arc cosine, arc tangent, and arc cotangent. They're all on Inuzor.com, so this is number five and the one which will remain after that will be arc cosecant. Okay, so let's talk about secant. Now as I was saying, this is one over cosine, so let's first start with the graph for a cosine, which looks like this. This is zero, this is y, x, pi over 2, minus pi over 2, pi, this is 3 pi over 2, and this is minus pi and minus 3 pi over 2. So, we start with a cosine. Now let's do one over cosine. Well, obviously in the cases when the cosine is equal to zero, one over cosine would go to infinity, so we will have asymptototus here. So, this is an asymptote, and this is an asymptote, and this is, and this is, okay. Now, in between the asymptotes, now that's very easy. At point zero cosine is equal to one, so one over cosine is equal to one as well, so we start from this point. Now, going left and right from zero, cosine is getting smaller and going down to zero, which means one over cosine will get bigger and will go to infinity. And in both cases, it's plus infinity because the cosine remains positive. Now, very similar situation is here. At pi, cosine is equal to minus one, so one over cosine is also equal to minus one. But then, left and right from this point, cosine goes to zero, which means one over cosine goes to infinity, but in this case it's a negative infinity because the cosine is negative, so it goes like this. And similarly here. So, this is the graph of secant. So, let me wipe out cosine. We don't need it anymore. So, what's the inverse of the secant? Well, in this particular case, there is no inverse function for a very simple reason. What is an inverse function? I'm just repeating myself for every trigonometric function. What is inverse to secant in this particular case? Inverse to any function. It means that if you have this function, y equals to secant of x. So, direct function, the original function for every argument, which happens to be an angle in regions in this case, you can find the value of a secant very simply. What's the angle? Let's say it's this one. You draw the perpendicular until it crosses the graph, go across to the y, and that's the value of the function. Now, in case of inverse function, we have to do the inverse procedure. Given the value of the function, find the argument, the angle basically. Well, the problem is that some values of the function have absolutely no argument which corresponds to them. Let's say this one, which is one half. Well, there is no value of the argument, there is no angle secant of which is equal to one half. It's either one and above or minus one and below. There is nothing like one half or zero or minus one half. So, that's number one complication. Number two complication is that if you have a value of something like this, then you have two different angles, cosine secant of this angle is equal to this value and secant of this angle is equal to this value. So, we have more than one. So, in some cases, we have no arguments. In some cases, we have two arguments or more. Actually, if you will continue the graph, you will have infinite number of arguments. All of them have the same value of the secant. So, we cannot define a function in inverse to a secant if we deal with this particular definition of a secant. What usually we do in these cases? Well, very simple. We restrict the domain and we restrict the range, we restrict everything in order to be able to do this inverse procedure. Now, first of all, how can we restrict domain? So, we can have no more than one value for our argument if given the value of the function. Well, there are probably many ways of doing this, but one of the ways which I suggest is to use this interval from 0 to pi. Now, in this interval, the function is monotonically increasing here, monotonically increasing here, but it's always unique, which means if you have a value of the function, it has only one intersection with the graph. So, these branches, this and this, the combination of these branches, or if you wish, secant with a reduced domain from 0 to pi actually can be inverse. Now, the inverse function is the function which gives you the value of the angle in regions if secant is given. In this case, we can define this function, but there is one more complication. For every function, we have to know two things, the domain and the range. So, in this case, we know we have reduced actually domain from 0 to pi, actually I have to exclude the ends because that's where, oh no, the ends, I don't have to exclude, there are no asymptotes here. Okay, so ends are included. Now, y, however, is a little bit more difficult. It's two pieces, it's one and greater or minus one and smaller, which can be actually described as absolute value of y greater or equal to one, which basically means the same thing, either y is greater than one or less than minus one. So, this is domain and this is the range. So, whenever we invert the function, whenever we define an inverse function to this, what usually happens? Whatever used to be a domain becomes a range of the inverse function, whatever used to be the range becomes domain. So, our new function, which we call our accident of x. Now, x belongs to basically this area, which means it's from minus infinity to minus one, including the one union with one comma plus infinity, or in other words, absolute value of x greater or equal to one, that's the same thing. And y would be from zero to pi. So, that's how we define our inverse function. Now, let's talk about the graph. You know that the graphs of the main function and inverse function are symmetrical relative to the bisector of the main angle, right? So, first of all, let me wipe out these pieces where the function is not defined anymore, our original function. I mean, we don't need it. That's our original secant. Now, let's think about what happens if I want to reflect it relative to the bisector of this angle. This used to be the domain. Now, it will be the range, right? So, from zero to pi, I will have the range of my function. Now, somewhere in the middle, pi over two, I have this asymptote. It will still be an asymptote for the inverse function, but instead of vertical, it will go horizontal, right? And my graph would be like this. Now, this is minus one. This is one. So, the combination of these, as you see, x is absolutely greater than one. That's the domain. The range is still from minus one to one. The only difference is, you see, the domain here is not just a consecutive number of points from zero to pi. I have to exclude one midpoint where we have, so x not equal to pi equal to, right? Which means here, in my range, I also have not all the values from zero to pi, but not equal to pi over two. So, there is a restriction on the domain here. At this point, function secant is not defined. So, at this point, there is no, so it's not in the range of the function cosecant. So, this is the graph of secant. This is the graph of the cosecant. Range is from zero to pi, except pi over two. The main is to the left of minus one and to the right of the plus one, which is absolutely greater or equal to one. Well, basically, that's it. The function is not odd or even, so there is nothing to talk about in this case. What's important is we have reduced the domain to prevent multiple values of the argument for a single value of the function. And that actually allowed us to inverse the function with a couple of restrictions that the main is not equal to pi over two, and the range is outside of the absolute value of one here. And that was transferred to the inverse function when we have the main outside of the one and the range was in zero to pi without the pi over two point. Well, that's it for function arc secant. To tell you the truth, I don't remember the problems related to arc secants. For whatever reason, out of these inverse trigonometric functions arc sine and arc cosine rarer but still used arc tangent, arc cotangent, arc secant, and arc cosecant for whatever historical reason are not used much in the problems. So in any case, the definition is there and that's what's important. This function is no better, no worse than anything else. And it's interesting basically that we have such a complicated play with domains and the range in this case. Thanks very much. Please examine again the notes for this lecture on Unisor.com. And as I was saying, this is lecture number five. We will still have the arc cosecant as the last lecture in this mini series. Thanks very much and good luck.