 Hi, I'm Zor. Welcome to a new Zor education. The subject of today's lecture is the function arc cosine. We will examine its properties. I recommend you to go through introduction to inverse trigonometric functions. This is number two lecture in series of relatively short lectures about properties of inverse trigonometric functions. So today it's about arc sine. And whenever I talk about functions, I do prefer to talk about the graphs of these functions because graphs really help to understand how the function behaves. Now, talking about inverse trigonometric functions, as I have explained in the introduction, the traditional trigonometric function, any trigonometric function, has no inverse function. And the reason is very simple. So if we are talking about arc cosine, so let's start from the graph of the cosine. Now, the graph of the cosine is this and then an infinite number of waves, right? So this is zero, this is pi over two, this is pi, three pi over two. This is minus pi over two, minus pi, minus three pi over two, etc. X, Y. The domain where the function is defined is all the real numbers. The real number, in this case, represents an angle in regions. And the range is from minus one to two to plus one. Now, the function does not have an inverse for a very, very simple reason. What is an inverse function? We have to find an argument if you know the value of the function. Well, if this is the value of the function, then we have to draw this line and see where it crosses our graph and some other point. So this point is, this argument has this value of the function of the cosine. And this argument also has exactly the same value of the cosine. So if you have the value of the cosine, you cannot uniquely determine the value of the argument. In which case you just say that there are no inverse functions in this case. If you cannot determine an argument by the value of the function, function is not inverseable. Well, but we want to inverse it, right? And as in the case with arc sine, for instance, function, we just have to reduce our cosine function, define the new cosine function, if you wish, reduce the domain where this function is defined. How can we reduce it? Well, right now the function is defined on an entire x-axis. Any real numbers is good as an argument. Now, what's the reasonable way to reduce it? Well, we will reduce it to the interval where number one on this interval function is monotonous. If it's monotonous, it's invertible, or inversible, whatever you say. And the second point is that in this interval, we would like function to take everything within its range. So all the values which cosine in theory can take, it does take this reduced variant. So where exactly I can choose an area on the x-axis, where the function cosine is monotonous, and takes all the values from its range. Well, its range is from minus one to one to one, and as some kind of one of the variants, what can be suggested is from zero to pi, and the function has the graph this on this interval. Now, let's think about it's monotonous, it's monotonously decreasing from zero to pi, function is monotonous, and it takes all the values, minus one and one. Can we choose some other interval that say from minus pi to zero? Yes, we can. Then the function would be monotonous, it would be monotonously increasing in this case, from minus one to one, so it's good as well. But traditionally, we choose from zero to pi. Why? I don't know, because it's positive. There are some reasons, but it doesn't really matter. Any interval is good, where the function is monotonous, and takes all the values from its range, additionally from zero to pi is taken as this particular interval. So everything outside of this interval does not exist anymore. A new function cosine is not defined outside of this interval, and on this interval function is reversible, is invertible, or invertible, whatever. Because it's monotonous, and it's convenient that the function takes all the values from the range. So whatever the value of y is in this particular equation, we can always find the proper value of x if we are not using a traditional original cosine function, but the new cosine function, which is reduced as far as its domain is concerned. So now we can talk about the function y is equal to arc cosine, arc cosine of x, and have its domain exactly the same as the range of the original function. So I can always put any value into the x from the range of this function, and as far as its range, it fills up completely zero pi interval. So the function y is equal to cosine of x, the new function cosine, maps domain from zero to pi to a range from minus one to minus z to plus one. The function arc cosine x maps uniquely, obviously, domain of minus one to one, which is a range of this function, into the range zero to pi, which is the domain of this function. Now, you know that the function, which is inverse to the original function, has a graph symmetrical relative to the angle bisector to this one. Well, let's basically show it in this particular case. You can need this place. So this is y is equal to cosine of x. So now, domain of this function minus one to one. So it's something like this, minus one to one. Now, its range is zero to pi. So the y takes the value from zero to pi. So it's in this rectangle. By the way, the rectangle from the original function is this one, right? So as you see, we have just turned this rectangle by 90 degrees to get the graph symmetrical relative... I'm sorry, I didn't really turn it by 90 degrees. I turned it over around the bisector. That's more precise. I turned it over around the bisector. So let's just see how it goes. Now, our original function cosine at zero equals to one, which means this one at x is equal to one should give me zero, right? So that's the point. Next, at pi over two, original function cosine has the value of zero. So the value of zero of this function is pi over two. So at zero is equal to pi over two somewhere here. And finally, for pi, my original function is equal to minus one, which means for minus one, this function should give the pi, which is this. And the graph would be like this. So these two graphs, this one and the red one, are symmetrical relatively to the n-group bisector. And the symmetry was explained, actually, where I was talking about inverse functions. That's very simple. Well, that's it. These are properties of the cosine. We have established that this is... the arc cosine is the function which is defined on the interval from minus one to one, and it's decreasing from pi to zero on this interval. That's as much as can be said, and it uniquely maps this domain into the range, and the range covers basically everything, which original function reduced to whatever the interval of monotonousness was defined. Okay, that's it. I recommend you to go through notes for this lecture at Unisor.com just to be able to, you know, refresh and understand everything, whatever it is. And that's it for this lecture. This is the second lecture out of six for each trigonometric function. I will explain what's the inverse function. Thank you very much, and good luck.