 Hi, I'm Zor. Welcome to a new Zor education. I would like to spend some time talking about inverse functions for trigonometric functions which we have already learned about. We did study sine, cosine, etc. different trigonometric functions. Well, they establish the value for an angle, usually measured in regions. So, these are functions. Now, the question is about the reverse transformation. If you know, let's say, the value of a sine or a tangent, can we determine the angle which has this particular sine or tangent or something? Well, that's basically the purpose of having inverse functions. Because sometimes we do know, even for some practical results, sometimes you do know the trigonometric function of an angle and you just have to determine what the angle is. Well, unfortunately it's not simple with trigonometric functions. And the most important problem is that they are periodic, which means the value of a function at some particular angle is exactly the same as the value of this function with another angle. The simplest way is just to add 2 pi to an angle and all the trigonometric functions repeat their values after 2 pi. So, we will address this particular issue. Now, what are these inverse trigonometric functions? They all have a very specific name. If you're talking about trigonometric function like sine, then there is an inverse function called arc sine. Cosine, arc cosine, tangent, arc tangent, etc. So, we always use the word of the name of the function itself and the prefix arc in front of it to designate the inverse function. Why? Well, the reason is actually quite simple because most of the trigonometric functions and properties are explained or defined using the unit circle. And the unit circle, especially if you are measuring angle in radians, the measure of the angle in radians is exactly the same as the lengths of this arc on the unit circle. Because this is one and this is one and this is one region. So, the radius is one, the arc is one and the angle is measured as one. So, in this particular case, angle is basically equivalent to an arc and that's why we are adding a prefix arc to designate it. We want to find an angle or the corresponding arc of the unit circle which has this value of a sine or a cosine, etc. So, that's the etymology of the names. Now, as I was saying, the problem with trigonometric functions in defining the inverse is periodicity. Now, it's not such a simple problem and we are supposed to address it quite well actually in this lecture. What's important is to realize what exactly is a function and inverse function and related concepts of domain and codomain and range. So, even considering that I have addressed all these issues in the corresponding more general lectures about functions, I will just repeat certain things here about functions in general and their inverse ability just as an introduction to inverse trigonometric functions. So, what is a function? Well, let's recall that you have to have one set of elements called domain. You have another set of elements which is called codomain and you have certain rules which put into the correspondence every element of a domain to some element in the codomain. What's important is that if you know an element of the domain, using the rule you can always find the corresponding element in the codomain. Now, are these images of these elements filling up completely the codomain? The answer is no. Codomain is a theoretically possible values for these images, but actual images can be concentrated in a narrower subset of a codomain called range. Just as an example, if you have a function let's say y equals to x square which is usually defined as the function from any real number to any real number. So, this is all real numbers and this is all real numbers, but the range is actually only non-negative real numbers. It's a subset. So, these are important concepts, domain, codomain and range. What also is important, look at this picture. I put the image of this point and this point into one point, one element in the codomain or in the range. Is it possible for the function? The answer is yes. An example is exactly the same. y is equal to x square. If you have 2 square you will get 4. If you have minus 2 square you get 4, right? So, this is minus 2 and this is 4. So, images are the same although the prototypes are different. That's fine. There's no problem with that. This is still valid functions. Now, let's talk about inverse function. Inverse function is the function which will allow us to find a prototype if you know the image. In this particular case when 2 and minus 2 are mapped or are transformed into the same element 4, it's impossible to find the prototype. It can be either 2 or minus 2 so we cannot really find it. It means that in this particular case there is no inverse function and function y is equal to x square. As we defined it, as the function defined for all real numbers, this function does not have an inverse function. Graphically it can also be viewed in such a way. If you have a graph of y is equal to x square, now what does it mean that you have to find a prototype? It means if you take any particular y and you would like to find what is the prototype, well, you put the horizontal line through this point until it intersects the graph and then you drop the perpendicular. So obviously by definition of the graph this point is the point which has coordinates x and y which satisfy this particular equation. So x which is this, square which is this would be equal to this. But now you see we have a different intersection point. So we have two intersection points. So this is x1 and this is x2. Both x1 and x2 result in the same y. So if there is a horizontal line which intersects our graph in two points, this is a sign of the fact that the function does not have an inverse. So that's the graph. We have more than one point of the argument, more than one argument which maps into the same image. Is there a way to overcome this difficulty? Well, obviously there is always a way, we just have to find it. And here is what usually is suggested in this case. You should remember from general discussion about the functions that monatonic functions are always established one-to-one correspondence between the domain and the range. Y, well that's basically by definition. If x1 is greater than x2, we have a monotonically increasing function. Y1 is equal to f of x1 greater than y2 which is f of x2. If you have this function monotonic and you have one argument is greater than another, then the result of the function is exactly the same. And I'm talking about strictly monotonic function, not equal, no greater or equal sign in either case. Now the same thing actually would be in case of monotonically decreasing function. The only thing is this particular would be less than sign, but in any case if these are different, these are different as well. So different points here would make me to different points there. And that's actually enough to establish the one-to-one correspondence. And if we can establish one-to-one correspondence, that means we can inverse the function. The only thing we have to define the function properly and say that first of all, we will consider the function not on the entire domain, not defined for everything which the original domain was. We can actually reduce the main and define our function only here, but the purpose is to choose this subdomain, if you wish, in such a way that the function would be monotonic, well, one-to-one correspondence would establish with the range and it will still cover the whole range. So we are changing the definition of our function. Instead of function y is equal to x squared, which we define for all real numbers, let's define a new function which looks exactly the same. It has the same transformation rule from x to x squared, but only for x greater or equal to zero. So only for non-negative numbers. This is not the same function y is equal to x squared as we used to have it before. And it has a different graph. Instead of graph having the whole parabola, it has only one part of the parabola because it's not defined here. So it's basically a completely different function. We changed the function, but well, completely different, but not that different. So we are still retaining the rule of transformation. And that's also important as before the range of the original function was all the non-negative numbers. And the range of these functions is also non-negative numbers. So we are still covering the whole range. And now what we are saying is let's define a brand new function which has the same transformation rule, reduce the domain to the subdomain where the function establishes one-to-one correspondence with the range and having actually the range as a co-domain so we don't have the difference between range and co-domain. So this new function, and in this case, for instance, it's a function defined on all non-negative arguments taking values in the all non-negative real numbers. This is a reduced definition, but it's not that dramatically reduced because still the transformation is the same and the whole range is covered. If we do this, then this function is reversible. And the inverse function is actually this one. Now this x belongs to a range and this y, the range of the original function and that's why it belongs to a domain of the original function. So this is non-negative and this is non-negative and everything is still fine. And this function is called the principal square root when only the positive, non-negative value of the square root actually is taken into consideration. So the square root of 4, and I'm talking about the principal square root, is 2. Not minus 2. There is no such thing as minus 2 in this definition of the function and its inverse because minus 2 doesn't really belong to a domain of the original or range of the inverse function. So we are changing the definition of the original function. We are reducing it to a different sub-domain and different co-domain which is actually equal to the range. That's number one. Number two, inverse function is supposed to be defined, its domain is supposed to be the range and its range is supposed to be the original domain of our function. So that's how they are inverse to each other. Now if we start from here and apply the original function we get there and then we apply the inverse function we go back here into the same point and that's why we are saying that if you combine these two functions one after another the original function first and then the inverse function we will get identity. In this case identity in this particular area in the domain of the original function. Well you can start from here and first use inverse function which goes this way and then direct function will go this way and we still get an identity. We will still return into the same element. So that's what's very important, very important to change the definition of our function in such a way that on a reduced domain and reduced co-domain down to the range we will have one-to-one correspondence and to establish one-to-one correspondence it's easier to use such a reduction where our function becomes monotonic because monotonic function always assures the one-to-one correspondence. This is the rule which is applicable to trigonometric functions as well. And here is how. I'll just do a very simple example which will show that that's how we can do it. If you take for instance the function sin which is like this. So obviously if you take this value of y and draw a horizontal line you will have more than one actual infinite number of intersection and each of these values of argument have a sin equal to this particular y which means we don't have a one-to-one correspondence. But let's do it this way. I reduced my domain to an area from this to this and in this area function is monotonically increasing which means it establishes exactly the one-to-one correspondence with this domain, reduced domain but the range is exactly the same. So I'm not changing the range, I'm just reducing the domain. I'm not changing the formula, the rules of how to calculate the sin or anything like this. Everything is exactly as before. I just reduced the domain. And in this particular domain I can find an inverse function. I can always find from a single y I can always find a single x. So from a single value of a sin for instance I can find an angle and that's only one angle which means the inverse function is really a function. So this is an approach and that's basically an introduction into my explanation about what's an inverse function for each of the trigonometric function. So for each of those trigonometric functions I have to determine where exactly this function is monotonic and its values cover an entire range. If I will be able to establish it in this particular reduced domain I will be able to define an inverse function and that would be subjects of subsequent lectures about inverse functions. That's it for today. And as usually don't forget to go to unizord.com. Notes for this lecture actually includes all these explanations which I was trying to convey now. Maybe you should read it again just to be a little bit more fluent in this material. And again my main point was that without some drastic change we can still change a little bit the definition of the trigonometric functions by reducing the domain where it's defined, reducing the angles where it's defined and for this reduced domain we can define an inverse function. Original trigonometric functions with angles defined across any real number there are no inverse functions as there is no inverse function for a regional y is equal to x where x is any real number. Okay, thanks very much and good luck.