So in today's session, we are going to start a new topic, which is inverse trigonometric functions ITF inverse trigonometric functions trigonometric functions. Okay. Now why I want to do inverse trigonometric functions because from functions this particular concept is logically coming out right so after doing functions and after understanding inverse of a function. It is very logical and obvious that we should start talking about inverse trigonometric functions. Of course, when we will be dealing with differentiation, we'll be also talking about how do we find derivatives of inverse trigonometric functions. Trust me, finding the derivative of inverse trigonometric functions is not that easy. Right. It is not a simple formula that you apply every time. Okay. So understanding the properties of ITF inverse trigonometric functions makes your differentiation process also quite easy to understand. Okay. So let us get started. First of all, the name of the topic has got, you know, two major words in it. One is inverse. Another is trigonometric function. So it's an amalgamation of your understanding of inverse of a function, which we already did under function chapter. And of course your trigonometric functions. Okay. So a big prequisite for this particular topic is you knowing your basic class 11 trigonometry very, very well. Okay. Just to revisit what is inverse of a function. Right. So if there is a function F, if there is a function F, let me draw it like a machine. Okay. In this function F, if you feed, let's say A. Okay. And this function gives you B as an output. After processing A, let's say this function throws out B as an output. Then how would the inverse of this function behave? The inverse of this function would be of course another function or another machine. Okay. Written as F with a superscript of minus one. As I already told you, this doesn't mean one by F. It is just a symbolic representation, which takes in your B, which takes in your B and gives you a back. Okay. That is how your inverse of a function behaves. Okay. Now, we have already seen in our function chapter that if f of x is invertible. Invertible means it has an inverse. Please note that for that f of x has to be a bijection or a bijective function. Okay. So what is bijection bijection means it is simultaneously both injection. Okay. And surjection. Right. All these words are quite familiar to you already know what's an injection. Injection is nothing but a one one function.  Right. So a function in order to be invertible must be bijective function or must be a bijection. That means it must be simultaneously an injection as well as a surjection. Correct. Okay. Now, let us try to talk about our trigonometric function. So what we are learning in this particular chapter is the inverse of trigonometric functions. Okay. So when we talk about trigonometric function, let us begin our discussion with sine x function. Okay. So for sine x function, let's call this as number one function. Let me take the exhaustive domain. So we all know that the exhaustive domain of sine x function is all real numbers. Okay. And co domain as all real numbers also. Okay. I mean it is up to the person who is defining it to define the co domain and I have defined it as all real numbers. So if I talk about sine x function in the present shape, that means when its domain is all real numbers and when it's co domain is also taken as all real numbers and we draw the graph of sine x function. What do we realize that in the present shape? So I'm just making a quick small diagram of sine x. So in the present shape, you realize that this function is neither one one nor one nor onto why why it is not one one because if I draw a horizontal line like this, right, what it is doing, it is cutting the graph at so many points. Correct. It is cutting at this point, this point, this point, this point, and so on and so forth. Right. So in the present shape, the function is the function is. Not one one. Right. And of course we know that the range of this function is from minus one to one. So it is not onto as well. Correct. So it is neither one one nor onto, which means it is not a bijection. It is not a bijective function or it's not a bijection. That means this also is tan tan. Right. So how are we studying this chapter? Because not only sine x, even cos x, tan x, c kicks, c kicks, cortex will show you a similar characteristic. Right. So they are not one one. They are not onto. So how are we studying the inverse of such functions? Isn't it? That means logically speaking, we should not be studying this chapter as a, you know, at all. Isn't it? So how are we studying this chapter? Anybody. So why are we studying this chapter? This chapter should not have existed. Isn't it? Can anybody tell me on the chat box or otherwise also why are we studying this chapter? Right. So in the present definition, that means with the present choice of domain and co domain, sine x may not be invertible. But what if I, what if I redefine this function like this? So what if I redefine it like this? So what I'm going to do is now I'm going to redefine this function sine x in such a way that I want to make it one one. Now, if you want to make a function one one, you should not allow the function to be cut by a horizontal line at more than one point. Right. In other words, you should not allow the function to actually take a turn anywhere. The moment it takes a turn, the function will be cut by a horizontal line at more than one point. Right. So, so look at this diagram and suggest me what should be the domain that I should choose for which the function is not allowed to take a turn back. Tell me anybody. Let me hear it out from you. Minus pi to pi if you take, see this is minus pi and this is pi. Okay. Minus pi to pi. If you see, there are a lot of turns that the function is taking. Correct. So in that particular domain, which you have mentioned, no, it will not be one one. Satyajit. Absolutely correct. So Satyajit mentions minus pi by two to pi by two. Okay. So he says that if I chop the function here like this and here. Okay. That means I only have this part of the function. I'm just, you know, making a bold white line there. Okay. If I have only this part of the function, I'm ensuring that the function is not taking a turn anywhere. Right. And of course, in that case, the function will be one one, but is that the only possible interval? You say, no, I can have pi by two to three pi by two also. Correct. So why not chop the function from here to here? Okay. So in this part, also the function will be one one only. Correct. So I can keep choosing such intervals like three pi by two to five pi by two, five pi by two to seven pi by two, seven pi by two to nine pi by two. I can even go to the left side. That is minus pi by two to minus three pi by two minus three pi by two to minus five pi by two and so on and so forth. Isn't it? Right. But let us say if everybody starts choosing their own sweet domain. Then what is the problem that will arise? Let us understand that. Okay. I will, I'll complete this in some time, but let us understand what will be the problem. If somebody starts choosing his or her own, you know, interval, where is the function is one one? Okay. So let's say there are three people. Okay. So let's say there are three people whom should I name? Let me pick any three of you. Let's say one is definitely going to be Ariashree. Okay. Other is let's say Bharat Makhija. Okay. And other one is let's say Nitya Gauri. Okay. So Ariashree chooses her interval where she wants it to be one one as minus pi by two to pi by two. Okay. And let's say Bharat Makhija chooses it from pi by two to three pi by two. Okay. And let's say Nitya Gauri chooses it from three pi by two to five pi by two. Okay. So these are three different mathematicians. Okay. Living in different parts of the, of the universe. And these people have chosen their own sweet domain for which the function f of x, which is sorry for with the function sign of x is invertible. Okay. Now let's say if, if Ariashree chooses this interval where the function sign x is invertible, of course, she has to also make the function onto. So for onto her code domain should be equal to the range, which is minus one to one. We all know. Okay. So this is the choice of Ariashree. Right. Now, if you also write the inverse of that function, of course, we all know that inverse works on the output of the function and gives you the input back. Right. So this is the output. This is the output. Right. Of the function. This is the input of the function. So inverse works on the output. That means for the inverse, the output of f is the input. Right. So this becomes comes here. Right. And this comes over here. Okay. Now, if Ariashree is asked a question, what is sign inverse half? Right. So she has defined a function sign x. Let's say I write the function name here, sign x. Okay. Whose domain is minus pi by two to pi by two and co domain is basically the range, which is minus one to one. And the inverse of that function will be sign inverse x. And for sign inverse, the domain is minus one to one and range is minus pi by two to pi by two. And she's asked this question, what is sign inverse half? How will Ariashree answer this question? Or what will be the answer that she will be getting from this? So tell me a value that she will be getting from sign inverse half. And that value should be between minus 90 to 90. What value she will get? Pi by six. Undoubtedly. Correct. So she will get pi by six. Okay. Now, let's come to Bharat Makhija. So Bharat has chosen his function sign x to be invertible in this particular domain. Okay. So this is his function sign x. And of course, when he is writing the inverse, inverse will be again minus one to one from pi by two to three pi by two. Okay. So this is his inverse of sign x function. Correct. Now, when Bharat tries to answer this question sign inverse half, what will he answer this as? What will he answer this as? Five five by six. Absolutely right. Absolutely right. So Bharat will answer this as five five by six. Why? Because he has to restrict his output between 90 degree to 270 degree. So the only angle which gives you sign of that angle as half between 90 to 270 happens to be 150 only. Right. So his answer will be only 150. Correct. Now coming to Nitya Gauri. Let's say Nitya Gauri. I mean, I'm just writing it the same thing quickly. For Nitya, she had chosen her domain for where sign x is invertible as three pi by two to five pi by two. Okay. So again, her function was sign x function. Okay. I'm just writing it down. And her inverse will again be between minus one to one and three pi by two to five pi by two. Okay. So this is the inverse of sign function. Now when she has to do this calculation sign inverse half, what would be the answer that she would be writing for such case? That means tell me an angle between 270 to 450 for this sign gives you half. Right. Her answer is going to become 390 degrees. Right. So 390 degrees is 13 pi by six. Correct. Yes or no. Now one, this one, this create a confusion in the field of mathematics that one mathematician is claiming sign inverse half to be pi by six. Other mathematician is claiming it to be five pi by six and one more mathematician is going claiming it to be 13 pi by six and da da da da. Right. So the function like sign inverse half is giving you multiple outputs that itself is not a characteristic of a function. Function gives you a unique mapping for a particular pre-image for a particular pre-image, which is half. I can have only one image. I cannot have five by six, five pi by six, 13 pi by six and so on and so forth depending upon the whims and fancies of the mathematicians. So this is going to create a non-standard system in the field of mathematics. This is going to create a havoc or a big, big chaos or a big, big confusion in the field of mathematics. Right. So what was decided that, I mean, see individually, these mathematicians were correct in choosing the domain in the respective way that they wanted it to be. So Arayashree was correct in choosing from minus pi by two to pi by two. Nithya Gauri was correct in choosing from pi by two to three pi by two. And Nithya Gauri was correct in choosing from three pi by two to five pi by two. But if everybody starts choosing their own sweet domain for which the function is invertible, it is going to create confusion in the field of mathematics. Right. So what was decided that one of these intervals will be called as the principal value branch. Right. And that branch was actually chosen out to be the one Arayashree chose minus pi by two to pi by two. Okay. So this particular branch, this particular branch was what was called the principal value branch. Okay. Principal value branch. Okay. And why do we call principal value branch? It is to standardize the process of, you know, how the inverse of these functions will behave. And this is the branch which even your calculators follow. Even your softwares follow. So if you have a calculator with you, no matter whether it is in your phone or whether it is on the, on your computer or whether it is on your actual calculator, if you do sign inverse half, it will only give you pi by six. Right. So this standardizes the output that I would be getting from an operation like sign inverse half. So everybody cannot start writing their own sweet, own sweet result. Okay. Now, please note that in your J, especially in your J main J advance kind of an exam, the examiner may basically redefine the inverse of a function by choosing a different domain. Don't expect always that minus pi by two to pi by two would be the interval for which he will be making sign x invertible. He may choose pi by two to three pi by two and then give you a question based on the same. So if the question set decides to give you a different domain for which sign x is invertible, you have to work according to that requirement or according to his, you know, definition. Right. And questions like this have been asked in the J main and J advance exam. Right. So all your formula, all your properties that you have learned or you will be learning treating minus pi by two to pi by two as the principal value branch for sign x for which it is invertible. They will all change. Getting my point. Okay. So please understand unless until stated otherwise, you can take the domain for which sign x is considered to be invertible as minus pi by two to pi by two. But if the question setter mentions a different domain to you, you need to better follow what he has mentioned. Are you getting my point? So if let's say, if let's say I'm a question setter and I say, Hey, my sign x is invertible. And my domain where I have made it invertible is pi by two to three pi by two. Okay. So that is a domain on which I'm working. Now tell me what is sign inverse half. Then your answer should be five pi by six only. In that case, you cannot say pi by six. Understood what I'm trying to say. Okay. So unless until stated otherwise, you can, you have to take minus pi by two to pi by two as the principal value branch. Fine. So now when the sign x, oh, sorry, I'll choose my yellow pen only. Now in this particular definition, if you see the graph of sign x function, I'm just going to draw it separately over here. In this particular definition, the graph of sign x function will look like this. Okay. So it will start from minus pi by two and it will end at pi by two. So it is definitely one one because if you draw any horizontal line, if you draw any horizontal line is going to cut the graph only at one point. And is it on two? So one one. Yes, it is one one tick. Is it on two? Yes, because I have made the core domain as the range. I made the core domain as the range. So minus one to one. Okay. So I made it as the, as the core domain. So it is on two as well. So it is both one one and onto, which means it is a bijection function or it is a bijective function. And now it is invertible. Okay. And now it is invertible. Okay. Is it fine? Any question, any concerns? Similar process will now be followed for other trigonometric function as well, which I'll be covering one by one. Okay. So let me again start with sign x only. I have already done half the work, but there's no place on this page. So I'll be taking it on the next slide. So is it fine? What is the concept of principal value branch? Why was the principal value branch concept made? It is very similar to your IUPAC system. Right. If IUPAC system, it was not there in chemistry for naming of, you know, chemical compounds or organic compounds. There would be a big confusion, isn't it? Everybody will start naming a compound in their own sweet way. Okay. Is it fine? Any questions, any concerns here? So why are we studying this chapter? Finally is because even though in their exhaustive domain, the word is exhaustive, even in the exhaustive domain, sign, cost, and secos, they were not invertible.  And then we can study their inverses. Is it fine? Is the chapter exists in our syllabus? Is it fine? Any questions? Any questions, any concerns? Okay. If no questions, no concerns, we will start our discussion again with Sinex function, but I will not take much time. So let me call this as one again. So for Sinex function, we choose the principal value branch as minus pi by 2 to pi by 2. And we choose the co-domain to be the range. Okay. In the present format, the Sinex function graph will look like this. Okay. Clearly it is a one-one function. And of course, since I have chosen the range as its co-domain, it is onto as well. So inverse, I'll be writing to its site. So inverse, we do the opposite. As I told you, the range will become the domain of the inverse always. And the domain of F will become its range, which is minus pi by 2 to pi by 2. Right? So what does it mean? It means the answer that you should be writing for Sine inverse should always be in minus pi by 2 to pi by 2 interval. So you can't write any answer that you feel like. So if somebody is saying, what is Sine inverse? Half. You can only write pi by 6. No 5 pi by 6. No 13 pi by 6, et cetera. If somebody says, what is Sine inverse? 1 by root 2. You can only say pi by 4. Only say pi by 4. Nothing like pi minus pi by 4. Nothing like 2 pi plus pi by 4. All those results will not be accepted. They will be marked wrong. Okay. Now, how would the graph look like? So you already have seen this in the function chapter that a function and its inverse and vice versa. Right? Inverse and the function. By the way, both are inverses of each other. So a function and its inverse. How are they graphically related to each other? How are they graphically related to each other? A function and its inverse. How are they graphically connected? Mirror images about which line from it? About Y equal to X line. Correct. So in order to make the graph of Sine inverse X, you all need to tell me how would be the mirror image of this line, this graph about Y equal to X line. So let's say this is Y equal to X line. Okay. I'm just making it in green. Okay. So this is your Y equal to X line. Let me write the name also in green. Okay. Now. Imagining the mirror image of this particular graph about Y equal to X line is very easy. Right? As you can see, this part will come on this side. This bulge will come on the other side. Right? Similarly, this bulge will come on this side and this part will come on this side. So overall, if I just reproduce the graph, it is going to become like this. Okay. Please note that the Pi by two, which was on the X axis will now go on the Y axis. The Pi by two, which was minus Pi by two, which was on the negative X axis will now go on the negative Y axis. The one here will come here. This is one. Okay. And the minus one will come over here. That means this point is one comma Pi by two. This point is minus one comma minus Pi by two. Okay. So this is how the graph will actually look like. Okay. Now, just to tell you that even Geo G will show you the same graph. Okay. So if I show you the graph, let me just show you the graph. Yeah. So the command for sign inverse is arc sign. Okay. Arc sign. Where is my equal to one? Yeah. Arc sign. Okay. Oh, I think I wrote double R. Yeah. So you can see on your screen, this is the graph off. This is the graph off. Sign inverse X that it will show you. Okay. So even Geo G will follow the same definition. And hence it is called the principal value branch principle means something which is followed by many of the people. Right. Important branch. All right. Now, few things which are noteworthy about this particular inverse function. Let us note that down because that is very, very important for our future working with this particular function. Okay. So few noteworthy points point to be noted. Number one. What kind of function is signed inverse X? Even odd or do you think neither? Even odd or neither? What do you think? You tell me. Correct. It's definitely an odd function. Yes. So it's an odd function. Why? If you see the graph is symmetrical about the origin. Right. If you remember your function chapter where I discussed with you even odd function. I categorically said that if a function is odd, its graph will be symmetrical about origin. Right. So whatever you have drawn in the first quadrant, the same thing is there in the third quadrant as well. And whatever you have drawn in the second quadrant, the same thing is there in the fourth quadrant. However, there is nothing in the second and the fourth, by the way. Okay. So because of this, it will follow the property f of minus X is equal to, in fact, you know, I have to write it in terms of sign inverse. So let me write it like that only. So because of this, it will follow the property that sign inverse of minus X will be negative of sign inverse X. Okay. However, I will talk officially about this even odd property in one of our properties of itf's, which will come a little later on in this chapter, but this is a straight away inference that we can draw from the graph. Okay. Second thing I would like you to tell me what kind of a function is this? Is it increasing? Is it decreasing? Or is it neither? What do you think? Is it increasing function? That means if you increase the input, output is also going to increase. Or is it a decreasing function? That means if you increase the input, output is going to fall. Or is it neither? That means sometimes it increases, sometimes it falls. Neither. Okay. Now, somebody has said neither. Okay. Now, let me tell you something which is very, very common sensical here. If a function is invertible, right? It must be one one, right? And it must be monotonic. Remember, I told us when I was talking about types of functions. So even sign inverse is an invertible function because it's inverse is sign X. So if sign inverse X has to be a monotonic function, it will either be increasing or it will be decreasing. It can never show neither of the two property. Right? So when I gave you the option neither, please note that none of the inverse signometric function. In fact, no function which is invertible can be a case of neither. Because if it is a case of neither, that means it's inverse never existed. Right? If it was never existed, that means that function itself never existed because it came from the inverse of something. Isn't it? Okay. So never ever, never ever categorize any invertible function as neither of, neither increasing nor decreasing. It can never be. It will either be increasing or it will be decreasing. It cannot show both of them. Okay. So yes, most of you have answered this correctly. It's an increasing function for sure. Okay. Graphs. Why is graph confusing you? See, if I put a value here, minus half. Okay. It's value is here. Correct. If I put a half, it's values here. So if I increase the input, my output also has increased. So why does confusing you? Okay. Isn't it? So come from the left side. If you increase the input, is the graph going upwards? Yes. It is going upwards. So it's an increasing function, plain and simple. Okay. So it's an increasing function. Okay. That means, that means if sign inverse X1 is greater than sign inverse X2, then what can you conclude about X1 X2? Then you'll say, then definitely X1 should be more than X2. Okay. See, don't take these simple, you know, things slightly because a question is based out of it. Okay. A question has been based out of this in the competitive exams. Okay. Next. Next. Yeah. What can you comment about its continuity? What can you comment about its continuity? Of course, from the graph, you can say it is continuous in its domain. Right. This function is continuous in the domain of the function, which is minus one to one. Okay. But it is, is it differentiable also in its domain? No. It is differentiable only in the open interval minus one to one. Right. It is more differentiable at one and minus one. Because if I draw a tangent at one or a draw a tangent at minus one, these two tangents, they will have infinite slope. Okay. So when it has infinite slope, we don't consider the function to be differentiable at such points. Right. So yes, if somebody asks you, is the function continuous in the domain, which is minus one to one, the answer to that is yes. But is it differentiable in its domain? The answer to that is no, because at the end points of, of the domain, the function has infinite slope tangents. So it is not differentiable. Okay. Now, this is an extra information, which I'm going to give you. In fact, I'm going to discuss about you, discuss about this in the differentiation chapter. Derivative of sine inverse X is one by under root of one minus X squared, but your X should be in the open interval minus one to one. Okay. So if somebody asks you, what is the derivative of sine inverse X at two, you will say is boss at to the function doesn't exist. So there is no question of it being differentiable. Okay.  Okay. Any questions? So now let us do a similar exercise quickly with cost function as well. Note this down, note everything down because I'm going to switch my slide. You know, today my keyboard is behaving very weirdly. All right. So now I'm going to switch my slide to the cost X function. So there are six trigonometric functions. So we'll be talking about six inverse trigonometric functions corresponding to them. So number two, we'll talk about costs. Now, when I talk about costs X, I will leave the domain and co domain for you to tell me such that my costs X function is invertible. Right. So on your chat box quickly tell me what should be this question mark and what should be this double question mark. Right. So all of you make the graph of costs X on your respective notebooks or make it in your mind and ask yourself. Right. What interval should I cut or what should interval or what domain should I choose for which my function is one one. Now guys listen to this very, very carefully. Tell me an interval where the function is one one and it covers the entire range of the function and preferably it should not be discontinuous. Now you would be thinking that costs X is anyway is not discontinuous anyway. Right. But yes, the other two criteria you should be satisfying. See, you can choose a part where the function is one one, but it may not cover the entire range of the function range of costs X is minus one to one. So tell me that domain or tell me that interval of X for which costs X covers the entire range, which is minus one to one. And of course it should be one one and preferably it should not have a discontinuity. So what are the answers that come in your mind? Okay, zero to pi is one answer that comes in your mind for the domain. Pi to two pi can also come minus pi to zero can also come two pi to three pi can also come. But yes, what we choose as the principal value branch is zero to pi. So your calculators, your softwares, any tool that you use which has got these calculation abilities, they only work into the principal value branch. So if you do cost inverse of any number which is between minus one to one, your answer is going to come between zero to pi only, right? It's never going to exceed pi. It's never going to be below zero, okay? And in order to make it onto, we already know that I should keep my core domain as the range, which is minus one to one. And yes, under this particular interval, my costs X function is invertible. How would the graph look like? The graph will look like this, okay? So this is zero. This is pi. This is one. This is minus one. Okay. Now, inverse, how would I define it? Exactly the Ulta. It's domain will be minus one to one. Range will be zero to pi. And symbolically, we write cost inverse X as this, okay? Please, I hope there is nobody who is treating it as one by cost X. No, it is not one by cost X. I think those days are already over trigonometry days. This is not C kicks, right? C kicks is reciprocal of cost. This is inverse of cost. Both are different things, okay? Don't treat reciprocals as inverses, right? Now, how would the graph look like? I would request everybody to make the graph of cost inverse X. Yes, done.  Trust me, many people are not able to do these imaginations, okay? And that includes me as well, okay? So I'm very, you know, candidate admitting that when I was a student, I had a tough time imagining the reflection of any graph about Y equal to X line. See, imagining about X axis reflection or imagining about Y axis reflection, that is very easy that anybody can do. But when it comes to a mirror which is inclined and we want to imagine the reflection about an inclined mirror, that is not everybody's cup of tea, right? So those who are able to imagine well and good, you are blessed with that art of doing it. But those who are not like me, what do I suggest is something which I had discussed in the bridge course as well. So to reflect a function graph about Y equal to X line, follow these two step mechanisms. Step number one, first reflect f of X about X axis, okay? So first reflect the graph, whichever graph you are given to reflect about Y equal to X line. First reflect that graph about X axis. And then second is rotate the graph, rotate the graph, obtain in step one, obtain in step one, 90 degrees, anti-clockwise, okay? So what I'm going to do is I'm going to follow this two step mechanism in order to make the required graph. So what I'm going to do is I'm going to make some miniature graphs over here. So this guy first I will reflect it about the X axis which goes like this, okay? And then what I do, I rotate this 90 degree anti-clockwise. So you can do it on a piece of paper by taking the help of a piece of paper, but I can also do it mentally as well. So when I rotate it, it's going to look like this, isn't it? So this becomes your reflection of your graph cos X about Y equal to X line. So I'm going to reproduce it over here. So this is how the graph is going to look like. It's not that completely flat over here. It's something like this. Yeah. Okay. I'll show the graph on GOG graph also. So this is one. This is minus one, okay? This is zero. This is pi and this cuts here at pi by two, okay? So when I was studying this particular chapter in order to remember the graph, I used to think of ful, okay? When we learn the Hindi alphabet ful, the foe of the ful is how this graph actually looks like, okay? So let me show this to you on GOG graph as well so that you can relate to it. Let me mute this guy. Yeah. Y equal to arc cos. So there is arc cos. Now arc cos is different, okay? Arc cos, arc shine, they're different, okay? That is hyperbolic inverse function. So we don't have to deal with them, okay? So arc cos is not to be chosen. Arc cos has to be chosen. Check your input. What happened? It is not showing me any output for, oh, equal to equal to, sorry. So this is how it looks like. Is it fine? Any questions? Any questions? Okay. So right now my graph is not in the pi form, but this top end is at pi by four. This guy is at pi by four. Okay. Oh, sorry. Pi, I'm saying pi by four. This is pi by two, okay? This is zero, of course, okay? And this is state at minus one, okay? So these are dead ends for this graph, okay? Don't make arrows. Some people have a habit of making arrows, okay? You're not making the graph extend all the way up, okay? It has a dead stop at those points, right? Is it okay? Any questions? Any concerns here? So let me raise this. Okay. Now let's talk about few noteworthy points about this function. Okay? Few points to be noted. Number one, is this function even odd or neither? So let's talk about it's even odd or neither characteristic. Is it even? Is it odd? Or do you think it's neither? What do you think? Remember, if it is odd, it will be symmetrical about origin. The graph will be symmetrical about origin, okay? If it is even, the graph will be symmetrical about y-axis. But now let me ask you a very logical question here. A common sensical question. Can a function which is invertible be an even function? If a function is even means it is symmetrical about y-axis. Means if you make any horizontal line, what will it do to the function? It will cut it at more than one point. So can an even function, sorry, can an invertible function ever be even? Ever be even? No, right? So no inverse trigonometric function. In fact, this is applied to any function which is invertible. It can never be even, my dear. So never ever by mistake also say even, okay? So either it will be odd or it will be neither, okay? So in this case, it is neither even nor odd because it is neither symmetrical about, so it is not symmetrical about origin. So if it is not symmetrical about origin, it is not odd. And even it is out of question. Even it is out of question, right? So it is neither, right? So it is neither even nor odd, okay? Now don't get confused here. Many people think sir, cos x was an even function, right? But note down, even cos x is not an even function in the present definition. So if you see 0 to pi cos x, it is not even. Because if it was even, it would have been symmetrical about y-axis. So this is the only part of the cos x which we are using. Right? So it is not even anymore, okay? Because we have curtailed the domain in such a way that this function is no longer even function, correct? So because this guy has to behave as an inverse of cos inverse x, right? Correct? So cos x is what? Cos x is the inverse of cos inverse, isn't it? Right? And vice versa. So even this guy cannot be even, right? So in the present form, cos x is not even function. And cos inverse x is definitely also not even, okay? Alright, so please note that no invertible functions can be even. It can either be odd or neither. Alright, coming to the next point. What kind of a characteristic do you see in this graph of cos inverse x with respect to increasing, decreasing? So is it increasing or is it decreasing? Neither is out of question. I already told you why neither is out of question. Yeah, it's a decreasing function, correct? So it's a decreasing function clearly. Okay, remember this? So that means that if somebody says cos inverse of something, let's say cos inverse of let's say x1 is more than cos inverse of x2. Okay, how is x1, x2 related then? Then you'll say x1 must be less than x2. Okay, so please make a note of this. Second is with respect to continuity as you can see from the graph and I don't want to waste your and my time asking the question because it is very evident from the graph. It is continuous for all x. It's a continuous, what happened to my marker? Yeah, it's continuous for all x belonging to minus one to one. But it is differentiable, but is differentiable for all x belonging to minus one to one open interval. It is not differentiable at the endpoints because as you all know, it will show you infinite tangents at these positions. Okay, infinite slope tangents will come at these positions. Infinite slope tangents will come at these positions. Okay, all right. Now just an extra information here, which anyways we'll discuss. Anyways, we'll discuss this in our differentiation chapter. The derivative of cos inverse x is negative of under root 1 minus x2. Okay, and you can only find the derivative in the open interval minus one to one. Okay, so as you can see this term will definitely be a negative quantity. Why? Because this is positive, this is positive, but this guy is making everything negative. So if you see from the graph also logically, if you make a tangent at any point, it will always have a negative slope. Okay, if you make any tangent at any point on the graph other than minus one and one, it will always show you a negatively sloped tangent. If you find any question, any concern with respect to the bio data of cos inverse x, so it's an either even or odd, it's a decreasing function, continuous in minus one to one closed, differentiable in minus one to one open, and derivative is given by negative of the derivative of sin inverse. Remember sin inverse derivative was one by under root one minus x2. Here an additional negative sign has come and there is a reason for it. We will discuss that reason in our subsequent properties. Find any questions? Let's move forward. Sorry, one hour we are discussing two functions only. Okay, but before I move on, I would like to give you a few questions. Okay, so let's take a few questions based on whatever we have done so far. My first question to you would be find the domain of sin inverse log of x by three to the base of three. Find the domain of this function sin inverse log of x by three to the base three. Okay, very good. Very good. Very good. Very good. So I think other than Michelle, everybody is giving the same answer. Only Michelle answer is different. Okay, let's discuss it. See, sin inverse of anything. Okay, whatever is this thing? Let me put a question mark. Okay, that particular thing should actually belong to minus one to one because sin inverse can only process such inputs which lie in this interval minus one to one. If you feed anything beyond this, sin inverse will say boss, what is this? I don't know how to process it. Okay, so this question mark in our question was log of x by three to the base three. So this guy should lie between minus one to one. Okay. Now, since you're dealing with a base, which is more than one, we can easily use this property. I think all of you are familiar with this property. Correct. So please recall, I'll write it down here. Please recall that if you have log, let's say, you know, any function x, maybe, or whatever, and there's a base which is more than one. Okay, and you're saying this is greater than B, or maybe, you know, whatever, lesser than B or something. Then you can directly say x is greater than a to the power B provided your a should be more than one. So in our case, our base is three, which is definitely more than one. So I can use this kind of a property here, which clearly means x by three, x by three is between one third and three. Just multiply three throughout, x should lie between one to nine. So your domain is one to nine. This is your answer. Is it fine? Any questions, any concerns? Clear. Everybody. Okay. Let me ask you one more question. If cos inverse x plus cos inverse y plus cos inverse z is three pi, then find the value of, find the value of x to the power 2017, y to the power 2016, z to the power 2015. Very good Vishal. Satyajit, awesome. So you guys have done this chapter in school. Despite that wonderful performance. Good. So people are giving right answers one after the other. Nice. Okay. Yes. So anybody else who wants to reply other than Satyam Tanvi Vishal Satyajit. Okay. Okay. So please understand here that you are dealing with three such cases or three, three such, you can say expressions, which are all based out of cost inverse, right? Now cost inverse, whatever input you give, the answer for this will always be between zero to pi. Okay. That's the reason why I gave you that, you know, definition of each of these functions. So will we always take the principal value branch? Yes. Unless until stated otherwise, you will always take the principal value branch, which for cost inverse is zero to pi. Getting my point. Right. Similarly, this guy output will also be between zero to pi. This guy output will also be between zero to pi. Now you want all of them to add to three pi. That can only happen when you have picked pi, pi, pi for each one of them. So this situation, this situation is only possible. Only possible when each one of these become pi, pi each. This is also pi. This is also pi. And this is also pi. Why? Because if any one of them is lesser than pi, there is no way that any other one will become more than pi to compensate for that, you know, value, right? So each one of them have to be at their max, that is pi, pi each in order to get a value of three pi. Okay. In other words, if you want your cost inverse x to be pi, x has to be equal to minus one. So y also has to be equal to minus one. So z also has to be equal to minus one. In that case, your answer will become this to the power 2017, this to the power 2016.  This becomes your answer. Is it fine? Any questions? Any concerns? Now this particular type can be asked in different, different shapes and sizes. Sometimes the teachers may give you a sine inverse x plus sine inverse y plus sine inverse z is equal to three pi by two. Right. So at that position, pi by two, pi by two, pi by two is what they will each take. Okay. Or they can give you minus three pi by two like that. Or in this case, they can give you minus three pi. So different versions of, you know, questions can be formulated of the same type. Is it fine? Any questions? Any concerns? Okay. And this, we quickly move on to one more question. In fact, this question is a very trivial one. You will find it very simple. I'm going to ask you a few sine inverse values and are you going to answer that? What is sine inverse negative half? Quickly. Quickly. Quickly on the chat box. Sine inverse negative half. The faster you reply, the faster, the more questions we can take. Yes. Minus pi by six. Absolutely correct. Okay. What is sine inverse negative root three by two? What is sine inverse negative root three by two? Minus pi by three. Very good. What is cost inverse minus half? What is cost inverse minus half? Radiance. Radiance. Please give your answer in terms of radiance. Okay. Yes. Two pi by three. Okay. Yes. Don't make those, you know, silly mistakes like minus pi by three and all. Even cost minus pi by three is not minus half. Okay. So even that would be wrong. See why I'm asking you these questions is because you should be appreciating the principal value branch concept that I discussed with you. What is cost inverse negative root three by two? Five pi by six. Excellent. Okay. What is cost inverse negative root two? Yes. Does that exist? Doesn't exist. Okay. Why? Because this guy is beyond minus one to one. Okay. So all these kinds of questions don't get tricked. This doesn't belong to minus one to one. Okay. So I would request everybody to do a quick exercise after today's class as a homework. Okay. Today when I'm done with all the functions, make four columns here. Okay. And then you write your inverse function, write its domain, write its range, and also make its graph. Okay. Do it for all the functions that you are going to take up. Okay. Sign inverse, cost inverse, et cetera. So please fill it, cost inverse, fill it like that. Okay. Preferably you can turn your notebook to landscape mode and then start doing it. Okay. So this should be remembered because you will not get a time to you know, sketch this over and over again in the examination domain. Trust me. If you know the graph, everything about that particular function is gone to you. Okay. Picture is worth thousand words. And that's how we just started our class 11th. Isn't it? Let's not talk about tan function. So again, I will leave some blanks over here. Fill in the blanks. So tell me a suitable domain. And a suitable code domain. Such that the function tan X is one one. It covers the entire range. And preferably it should not have a discontinuity in that intro. Tell me, tell me. And for that's chosen domain and code domain, let us sketch the graph as well. Right. We shall. Everybody knows the graph of tan X. So in your mind or on your notebook, you make the graph of tan X and then see how can you choose that interval as a domain for which the function is one one takes care of the entire range and preferably doesn't have a discontinuity. Right. So many of you would have chosen minus pi by two to pi by two. Some of you would have chosen. Pi by two to three pi by two open. Somebody would have chosen three pi by two to five pi by two open. You can go backwards also, but normally when we talk about the principal value branch, this is chosen to be your principal value branch. So this is your principal value branch PV. Okay. And of course your domain, your range should be all real numbers. In fact, your core domain should be your range, which is all real numbers. So under this particular definition, tan X graph looks like this. So this is pi by two. This is minus pi by two. By the way, next week onwards, we will be visiting your premises. So offline classes will commence. Yeah. So inverse. You have to write the opposite. It will be from real to minus pi by two to pi by two. Fine. Okay. Now, all of you please sketch the graph of tan inverse X. Okay. Everybody. I would request you sketch the graph of tan inverse X. Either you can imagine the reflection of this graph about Y equal to X line. Right. If you are good with your imagination, but if you're not, you can follow that two step mechanism, which I discussed. What was it? Reflected first about X axis and rotate that 90 degree anticlockwise direction. So once you're done, just write it down on the chat box. So have you made the graph on your respective notebooks? Just say I'd done if you are done with it. Done. Are you done? Okay. Yeah. So this is how the graph will look like. Okay. This will be pi by two. Okay. So please note that it will never touch Y equal to pi by two or Y equal to minus pi by two. It will be asymptotic to it. And please make arrows at the end because this graph is going to go on and on and on forever. Let me show that to you on GOG browser. So Y equal to this is all the graph looks like. Okay. And let me tell you, it is never going to touch this line. Y equal to pi by two. Okay. So Y equal to pi by two will be one of the horizontal asymptotes. Same goes with Y equal to minus pi by two as well. Is it fine? Okay. So let's go back to our blackboard and let's write down few things which are noteworthy for us with respect to this graph. Number one, what can you comment about even odd nature of this particular graph? Okay. Because you know, even it cannot be, so it can either be odd or neither. So what do you think? Right then we, it's going to be an odd function. Excellent. Okay. And because of this, this property will come into picture tan inverse minus X is minus tan inverse X. Next. What kind of a function is this? Is it increasing? Is it decreasing? Or is it neither? Neither again is out of question. Is it increasing or is it decreasing? Please note that it's, it stays below Y equal to pi by two, but it doesn't mean it has stopped increasing, right? It keeps on going. Okay. Vishal, it's an increasing function. It's an increasing function. Okay. That means if somebody says tan inverse of X one is more than tan inverse of X two, then what can we predict about X one X two X one should be greater than X two. Okay. Next. The function tan inverse X is continuous for all X. Okay. That means it is everywhere continuous. And same goes with differentiable. Also, it is differentiable for all X. Right. X time formation here, derivative of tan inverse X. In fact, many of you would be already knowing by this time, the derivative of tan inverse X is one upon one plus X square, all these derivations I'll be doing when I do the differentiation chapter with you. Okay. And of course it is worth noting down that it has got two horizontal asymptotes, two horizontal asymptotes. What are they? Y equal to pi by two and Y equal to minus pi by two. Okay. So you can say limit as X sends to infinity for tan inverse X is pi by two and limit as X sends to minus infinity for tan inverse X. What happened? Tan inverse X is minus pi by two. Okay. Is it fine? Any questions, any concerns related to the kundali of tan inverse X?  Is it fine? Any questions? If not, if not, then can we move on to the fourth one? Okay. So this is the glimpse of the slide. Now we'll move on to the fourth one. Everybody copied? Okay. Let's move on to fourth one. Fourth one, we will be discussing about Corsic function. Okay. Now, all of you recall the graph of Corsic function or draw it if you want to and tell me a suitable domain that I should be choosing for which Corsic X function is one one, it covers or justifies the entire range and preferably, preferably the word is preferably means that means we can even sacrifice that particular clause. Preferably it should be continuous. Preferably clause can be sacrificed also. Okay. Mind you. Okay. So I'll just draw the graph of Corsic X. I'm sure many of you would remember it, but many of you may have, I mean, it should ideally not step out of the mind. It is as vital as remembering your, you know, basics of mathematics. Okay. I'm just making. So this is pi, this point is pi by two, this point is zero. This point is minus pi by two. This point is minus pi and so on and so forth. The branches. Yeah. This is two pi. This point is three pi by two, et cetera. Okay. Now, how should you chop off this function? In order to ensure that it is one one. In order to ensure that the entire range of the function is taken care of by the way, the range of the function is one to infinity and minus infinity to minus one. How should you chop? Now, Vishal is saying minus pi to pi minus pi to pi. It has taken four, two turns by the way. In fact, four, four turnings are there. So Vishal, that cannot be the answer. Correct. Would you like to change it or anybody for that matter? How should you choose the interval of X for which the function Corsic is one one. Correct. Second, it covers the entire range. And third, preferably, preferably means this particular information or this particular criteria can be dished also can be sacrificed also. Right. Minus pi by two to pi by two. But that is not exactly correct. Minus pi by two to pi by two. Excluding zero. Now, here, please note that if I choose a branch where it is continuous, okay, I cannot ensure its one one nature. I cannot ensure that the entire range is taken care of. Correct. So I have to sacrifice on the continuity part. Right. That is the reason why I chose this interval. So let me remove the part which I don't want. Okay. So basically I chose minus pi by two to pi by two. Okay. Excluding zero. Excluding zero. To be my interval for which this function is one one and onto. Okay. One one and covering the entire age. Yeah. Yes. That continuity has to be sacrificed in this case. Right. That is why I said preferably. That means if not, then it is fine. So this is the domain for which the function is one one. And how should you choose the core domain? Of course core domain should be the core domain should be the entire range of the function, which is minus infinity to minus one union one to infinity. This particular thing is also mentioned some in some books as are minus minus one to one. Okay. Both are same things. Okay. Is it fine? Okay. So under this definition, Cossack X will be considered to be invertible. Okay. So how would I write the inverse? So as you know, the range becomes the domain for inverse and domain of F become its range. Okay. So all these things which I am writing, they should become a part and parcel of your memory. Okay. So this is how Cossack inverse X function is written. Cossack inverse X. Okay. Now, please everybody sketch the graph of this function. Look at this graph. Try to imagine its reflection about Y equal to X line. Okay. Or do the two step mechanism which I discussed with you and let me know with a done. Are you done? Then we'll discuss it. Done. Okay. So how many of you are able to imagine about Y equal to X line? In fact, I would, I wouldn't have, right? Maybe some of you would be able to imagine. And how many of you, in fact, many of you have not responded yet. So I'm waiting for people to respond. Okay. We shall also done. Now, I'll tell you a third way to make such graphs. Okay. I will neither imagine about Y equal to X nor I would reflect and turn at 90 degrees. Okay. So what I normally do, I use a common sensical approach. Okay. See, this tells me that the domain is to the right side of one and to the left side of minus one. Isn't it? So my graph will exist to the right of one and to the left of minus one. Yes or no. Okay. This tells me that my graph will exist between minus pi by two to pi by two. And of course it will become asymptotic on the X axis. Correct. Now, having this restrictions in my mind, I will ask myself when I put one in Cossack inverse, what should I get the answer as Cossack inverse one? What should be my answer? Pi by two. Correct. So keep your pen over here. Okay. And now see you have to go to the right. Correct. You can't go to the left because you can't be between minus one to one. You can only go to the right side and you have to become asymptotic to X axis. So the only way forward is you go like this. Like this. Correct. No. Simple. Okay. Now we know that when I put minus one in Cossack inverse, I should get the answer as minus pi by two. So keep your pen over here. Now remember again, you have to go to the left. Correct. And you have to be asymptotic to the X axis. So the only way forward is your graph can go like this. Simple as that. Simple. Okay. So I don't have to really imagine the reflection.  This is from our common sense. I can make the graph. Is it fine? Any questions? Any concerns with this? Those who made the graph, did you get the same figure or you got something else? Same. Okay. We'll quickly discuss some noteworthy points about Cossack inverse. Number one. Even odd or neither. Even to any basis out of question. Is it odd or is it neither? Tell, tell, tell quickly. Odd functions. Yes. It's an odd function because the graph is symmetrical about the origin. So Cossack inverse of negative X is negative of Cossack inverse X. Please remember this. Okay. Second, is it increasing? Decreasing or neither? Again, neither is out of question. So is it increasing or is it decreasing? It's decreasing. Yes. Decreasing function. Now here, few people have this concern that cell. Let's take a very special case at minus one. Your function is giving you minus pi by two. Right. But when you put a one, you're getting pi by two. Right. So that means if I increase my input from minus one to one, my output is increasing. Isn't it? So how come you're saying the function is decreasing? Right. This question will appear in your mind. Many of you would be getting this question in your mind. Okay. Now the answer to this is in your understanding of increasing decreasing function, which has not been officially given to you, but let me give it to you in very, you know, plain and simple word. See, when you talk about, when you talk about a function increasing at a point X equal to C, then what happens? The value of the function at C should be more than what it had at C minus H. And in turn should be lesser than what it had at C plus H. If H is a very, very infinitesimal quantity, this is a definition when we say a function is increasing. Okay. Similarly, when we say a function is decreasing at a given point C, then the value of the function at C should be less than what it has at C minus H. Correct. And in turn, it should be lesser than what it has at C plus H. Okay. Where H is a infinitesimal quantity, right? But now when, if a function abruptly stops at some point, let's say, if I talk about the function characteristic in an interval, let's say A to B, and it is abruptly stopping at B. Okay. And let's say abruptly starting from A, then how will you say what is the nature of the function at B? So what do we do? We just check what happens to the function value. Let's say I don't put the symbol. So what happens to the function value at B and B minus H when H is very, very small? If this is more, then we categorize it as increasing. Okay. And if this is more, then we categorize it as decreasing. Right. So if you want to study the nature of a function or you can say the monotonic nature of a function at a point where the function abruptly stops, then these two criteria will not work. No, because there's nothing right to that point. So we base our conclusion, what is happening to the left of that point. So same thing is what is going to happen at the point minus one. At minus one, the function has abruptly stopped because, you know, minus one plus is not in the domain. Isn't it? So we see just before minus one, that means at minus one minus and at minus one, which is more. This guy is more. So this function shows a decreasing characteristic at minus one. Getting my point. Similarly, if you want to study at one, so check what happens at one and what happens at one plus. Now this guy is more. Correct. So at one, the value is more than what it has at a higher point that is one plus. That means the function is decreasing at one. So even at these two points minus one and one, they will claim the function to be decreasing. We are not going to assess the function that oh, at minus one, it was minus pi by two and at one, it became pi by two. So there is a rise in the value. No, there is no rise in the value there. Okay. It is still tagged as it is still labeled as decreasing function even at minus one and one. Is it clear any questions, any concerns, we will talk more about it in the application of derivatives chapter, which will come little later down the year for you. All right. Any questions? Any questions, any concerns? Okay. Next one. Is it continuous in the domain? Yes, it is continuous in the domain of the function, which is minus infinity to minus one, union one to infinity. Is it differentiable in the domain? Now the answer to this is no, it is differentiable in open interval minus infinity to minus one, union one to infinity. Why? Because at the end points, at the end points, one and minus one, the slopes of the tangent they become infinity. Okay. So at this point, at this point, if you sketch a tangent, their slopes will be infinity. Okay. So we cannot include one and minus one to be the points of differentiability. What happened to my spelling of differentiable? Yeah. And let me just demarcate this because this is the previous discussion that we had. Is it fine? Any questions? Okay. Again, an extra information I'm going to give you here. The derivative of cosec inverse X is given by negative one by mod X under root X square minus one. Okay. Now there are some books which mention this as minus one by X under root X square minus one. Okay. Now I'll tell you the problem with this notation. Okay. So if you're claiming your derivative of cosec inverse X to be this, then what is going to happen? Now let's try to analyze a point on the left of minus one. Let's take this point. Okay. So let's say this is my X value. Okay. What is this X negative or positive? If I take an X to the left of minus one, what will be the sign of that X? Negative. Correct. Okay. Now see if I sketch a tangent at this point. Okay. The graph seems to suggest that this slope will be negative slope. Isn't it? Like this negative slope. But if I use this information, see this is anyways negative. This is also negative. And this is positive. Correct. So overall this answer will become positive, right? Which is not the right answer because ideally if I take any negative X also I should get a negatively slope tangent. Isn't it? Yes or no? No. So many books which write this as the derivative of Caussic inverse X, please note that. I mean, of course they have made some assumptions and all. Ideally you should be writing mod of X over there because here if you see irrespective of whatever X you take, it will always give you a negative answer, right? Because this will always be positive. And with a negative sign, you'll always get a negative answer. Getting my point. So these are small, small things which you know many people neglect or many people ignore. Okay. So keep this in mind. Any questions, any concerns? Okay. So bio data of Caussic inverse X is ready. So let's now move on to seek inverse. I think this is fifth on our list. Yeah. So again, same exercise will do quickly. Tell me the domain that you would choose for to seek. So that number one, it's on one one function. Number two, it takes care of the entire range. And number three, preferably that's a preferred thing that I want, but you may choose to ditch it also preferably should be continuous. Very good. Very good Satyam. Yeah. So zero to pie excluding pie by two. This is what is considered to be the principal value branch. All right. And in order to make it onto, we should choose its core domain to be the range, which happens to be same as what we had for Cossic, which is this. Right. And under this definition, the graph is going to appear like this. So spy by two. Is it fine? So zero to pie. Okay. As you can see, this is the domain zero to pie excluding pie by two. And its range is one to infinity. And minus one to minus infinity. Clear. Okay. Now let's do the trickier part of the game. Let's write its inverse definition. So this part is easy, stating its domain and core domain. You just have to flip the two of the function. And symbolically, symbolically we write seek to be like this. Yes. Time for the graph. How would the graph look like? Again, either use your imagination about reflection about y equal to X line, or you choose that two step mechanism, or you choose your common sense. I would prefer the last one using the common sense. Draw it on your notebook and let me know if you're done. Then everybody. Okay. Good. We'll discuss it again. So look at the domain. So I have to be, I have to ensure that I have to be on the right of one and left of minus one. Correct. And I have to be between zero to pie. Let's choose pie over here. Okay. And pie by two is this line. Okay. Now apply common sense. See, let's, let's ask ourselves if I put one in seek inverse, what answer will I get? What is seek inverse one? Zero. Correct. So at one, I will be here. Now remember, I have to go to the right and I have to become asymptotic to pie by two line. That means I cannot cross it. So the only way forward is me going in this direction like this. Simple as that. Similarly, put your value as minus one. You get a pie. Again, I cannot go to the right. I have to only go to the left of minus one because I have to go from minus one to minus infinity. And again, I have to become asymptotic to pie by two line. So the only way forward is this. There you go.  Say yes, if you're getting this graph. Awesome. Okay. Now let's look into some points which are noteworthy about seek inverse function again. Odd or neither. Odd or neither. Right. Neither even nor odd function. Okay. So by mistake also don't write seek inverse minus x is seek inverse x. No, it has a different formula. We'll discuss about that later on. Okay. Yeah. Increasing function or decreasing function. Increasing function. Or decreasing function. Correct. Increasing function. Okay. What about continuity? Continuous in the domain? Yes. It's continuous in the domain of the function. Right. Is it differentiable in the domain? No. It is differentiable. But in the open interval minus infinity to minus one union one to infinity. Right. Please note again at one and minus one. If you draw a tangent that tangent will have infinite slope. Okay. This will also have infinite slope. Okay. And again, an extra information. The derivative of. Seek inverse x is given by. This. It's exactly the negative of what we saw for co-seek inverse. Again, that mod is very important. Because if you don't put a mod, you will end up getting an answer which says negative. So far values of x lesser than minus one, but that is not the case. Okay. Is it fine? Any questions? Any questions? Any concerns? All right. So now moving on to the last in this list. The sixth one. We'll talk about cortex function. So tell me. Again. A preferred. Interval of x for which cortex behaves as a one one function. It justifies the entire range of the function. And preferably it should not be discontinuous. Okay. Now for the benefit of everybody, I'll make the graph of cortex in case you want to refer to the graph. Okay. So it goes on and on in both directions. I'm not going to make everything. Yeah. Very good. Now here, please understand. See, many people. May may also choose minus five by two. Let's say close at minus five by two. Two pi by two excluding zero. Right. Because even in this interval, it is one one. It is taking care of the entire range. But the problem here is that it has a discontinuity at zero. Okay. So all this while when I was saying preferably not discontinuous preferably not discontinuous. Here you could have both the options. You could either choose zero to pi, which most of you have said on your answer. Right. In zero to pi also it is covered. It is one one. It is covering the entire range. And it is not discontinuous. Even in minus pi by two to pi by two, let's say close at any one of them. You can close it at pi by two also and keep it open at minus pi by two also. In that interval also it is one one in that interval also discovering the entire range of the function. But that has an issue that it has got a discontinuity at zero. So out of the two, I will prefer zero to pi because that has no discontinuity in that interval. Are you getting my point? So this is not preferred. This is not preferred. What is preferred is I'm just raising the arm, which I don't need. What is preferred is this song. Okay. So the principal value branch that is chosen is zero to pi open. Please do not put closed. Don't mix it with the Cossack's interval for which it was invertible. The difference between the Cossack's interval for which it was invertible and Cossack's interval for which it is invertible is the brackets at zero and pi. So they're the brackets where square brackets. Here the brackets are open brackets. Okay. And of course your code domain should be the range, which is all real numbers. Okay. So it's inverse will be from R to zero to pi. And symbolically we write it as cot superscript X. And how would the graph look like? I would request everybody to give me the graph quickly. Or at least do it on the notebook and say it done if you're done. Done. Very good. Yeah. The graph is going to look like this. Okay. So please note. It is going to be from zero to pi and both of them are asymptotic. It is not going to achieve it. This is pi by two. Okay. Is it fine? So as you can see its domain is all real numbers. So it is shown by arrows at the end because it is going all the way to cover the entire number line of X axis. Ranges between zero to pi cannot touch zero. Hence it is asymptotic to y equal to zero cannot touch pi. And hence it is asymptotic to y equal to pi as well. Okay. All right. So let's talk about few noteworthy points here. Number one. Odd or neither. Odd or neither. What do you think? Correct. Neither. Correct. Right. So it's neither even nor odd. Okay. Now this the confusion here. Many people think sir tan inverse. So you made odd. Why are you doing some some this step treatment with caught inverse? See again, try to understand the graph that you had for tan for which it was invertible that itself was odd. Okay. But tan inverse also came out to be odd. And the cortex graph that we're considering for it to be invertible, this is not in odd function. Right. Because for it to be odd, it has to be symmetrical about origin, which is which it is not. Right. So don't expect those kind of you can say correlation that you actually had in your normal trigonometric functions or your normal standard sin cost and functions. Okay. So don't expect those kind of analogy in your inverse trigonometric function. Like, other day somebody was saying sir, isn't not tan inverse x sin inverse x by cos inverse x. No, there is no such relation. Okay. Don't extrapolate. Don't extend your basic trigonometric identities to your inverse trigonometry. Inverse trigonometry identities are quite different. Okay. Because here you don't deal with ratios. Yeah. Here you are dealing with angles. There are some values which lie in their respective principle value branch. Right. So don't expect the same kind of a treatment that you gave to sign cost and to be given to their respective inverses. Okay. Yeah. So it is neither even nor odd. Okay. Second, increasing, decreasing. What do you think? I mean, it is very evident from the graph. Decreasing. Correct. It's a decreasing function. Okay. And as you can see from the graph, it is continuous. For all x. It is differentiable also for all x. What about the derivative of caught inverse? One by minus one by one plus x squared. Okay. It's again, the negative of what we had, what we had as a derivative of tan inverse. I'll tell you the reason why they are negative of each other in one of the subsequent properties. It's fine. So after one hour, 42 minutes of class, we are able to cover six basic inverse technometric functions along with their graph and few noteworthy points about these functions. Okay. Is it fine? Any questions? Any questions? Any concerns? All right. Now, time has come that we take up few properties of these inverse technometric functions. Okay. But before I move on. A quick round off question answer session. Okay. Again, simple ones. One line is, okay. What is tan inverse of, okay. Simple ones. One line is, okay. What is tan inverse of minus root three. Come on. Fast on chat box. And inverse of minus root three. Minus five by three. Very good. What is caught inverse of minus root three? What is caught inverse of minus root three? Caught inverse of minus root three. Okay. Satyam. So Vishal, you think caught five by six is minus root three. Or now you change your answer. Good. Yes. Just two people responding. Okay. Very good. Very good. So excellent. I think most of you have got this concept pretty correct. See again, many people I mean in a hurry, we'll say minus five by six. Okay. Please note that minus five by six is not in the value branch for caught inverse. Caught inverse output. Unless until mentioned. Otherwise has to be between open interval zero to five. Correct. So the only angle between open interval zero to five, which gives you caught of that particular angle to be negative root three is five, five by six. Don't say any other answer. Any one. Okay. Tell me what is Cossick inverse of half past. Yeah, this was a googly question. Because this is not in the domain. What is domain domain is minus infinity to minus one close union one to infinity. So Cossick doesn't know how to process half. Yeah. Tell me what is seek inverse two. Sorry, minus two. Minus always creates a lot of confusion. Yes. Seek inverse minus two. Correct. Two pi by three. But many of you taking a lot of time to answer this. I don't know why. Okay. Tell me. Cossick inverse of negative root two. Tell me caught inverse of minus one. Okay. Let's do one by one. So first one Cossick inverse of negative root two. Minus pi by four. Correct. Yeah. Caught inverse minus one caught inverse minus one. Three pi by four. Correct. Okay. This is this is where you need to know me very fast. Right. So please do that exercise which I had asked you a couple of sites back. That is very, very important for your understanding. Okay. Initially it will be. See now you have to think in terms of angle. Right. In class 11, you used to think in terms of value for that angle. So from here, from value, you have to think of the angle and that angle should be in the principal value branch and you cannot go. You cannot say any other, any angle you want to. Getting her fun. So that is the thing that you need to practice. Okay. Okay. So we will not be moving on to certain set of properties of ITFs. They come in sets. Of course, because there are six inverse stick functions. So they will come in set. So there will be, I mean, I don't exactly remember the count, but they will be around 10 to 12 such set of properties. Okay. And each of them will take some time for us. Okay. And what is there in this chapter? This chapter has only got a couple of parts. And in fact, understanding inverse trigonometric functions, then understanding properties of inverse trigonometric functions. And then the application of properties, which comes in the form of solving equations or finding the sum of series.  So it is mostly knowing the function and knowing the property. And of course there are applications based on the same. So we are now going to start with the very first of our properties of ITF. Okay. This property is basically coming from your inverse function property, inverse of a function property, which I'm sure many of you remember. In our inverse of a function, we did a property like this. F of F inverse X is X. That means it is an identity function. So this guy is an identity function. Okay. So let's say F is a function from A to B. It's an invertible function. Okay. So let's say this is an invertible function. Then F of F inverse X is going to be an identity function, but we call this as IB. Why? Why IB? Because the X that you get, the X that you get comes from the set B, which is the domain of F of X. So this, it comes from this set. Okay. That is why it is called IB. So based on this property, we have the very first property of ITF. And those properties are as follows. Sine of sine inverse X is X. And again, X has to be in the domain of sine inverse. What is the domain of sine inverse? Minus one to one. Okay. Similarly, cos of cos inverse X has to be X. And this X has to be in the domain of cos inverse, which is again minus one to one. Tan of tan inverse will be X. This X could be any real number. Cosec of cosec inverse. Okay. This X has to be in the interval minus infinity to one, union one to infinity. C of sec inverse X. That is also X. And cot of cot inverse. So just on the basis of the first, you know, property that we had discussed in our inverse of a function, this property is coming up. Okay. So this is the simplest of all the properties that you will get in this particular chapter. Note this down. So now this is, this is very easy to understand. This is very easy to remember as well. But the problem comes. Sorry. But the problem comes when you have a mixture. That means, let's say if I ask you, what is sign of cos inverse X? Or what is cos of sign inverse X? Or what is sign of tan inverse X? Or what is tan of cosec inverse X? There the people start, you know, making mistakes. Okay. So we'll take, we'll talk about that also in our next slide. And I'll tell you a mechanism how to deal with those situations. So first make a note of these six basic properties and then we'll move on to that one. And maybe after that, we'll take a break as well because it's almost two hours of the class gone. So Monday school is reopening. No. No doubt about that. No pushing off dates and all. Okay. Now why I asked is because high alert, red alert was declared because of this, I think the monsoon things. Yeah. Okay. Was it raining at your site today? I mean, it's all gloomy or cloudy, but not raining yet here. I mean, it rained in the morning, but not now. Yeah. It's all all overcast. Yeah. All right. So now let's talk about the extension of this property when there is a different trigonometric ratios working. For example, if I say, what is sign of cost in Merseys, how will I find this out? How will I find this out? Okay. Now the process of finding this is the same as used to find out sign of any angle given to you. Remember class 11. If I gave you sign of some angle. Okay. Let's say sign of some theta. What do you use to do? It helps you to use to have a sign. Correct. And it used to have a magnitude. Are you able to recall this sign we used to get for our quadrant system. We used to see in which quadrant theta lies, whether it is in the first quadrant, second quadrant, third quadrant, fourth quadrant or whether it is a magnet. And magnitude we used to get from reference triangle. Are you able to recall your class 11 days? This is what we did for finding sign of any angle. Isn't it? It used to have a sign. And it used to have a magnitude. Correct. And together it gave us that sign of that particular angle that we needed. Isn't it? Same procedure is going to be followed here as well. Okay. So what I'm going to do here, I'm going to call this as an angle only. See, at the end of the day, inverse trigonometric functions return an angle to using it. Sign inverse, cost inverse, sign inverse, seek inverse, go seek inverse, cot inverse. What are they? At the end of the day are some angles. Of course, in their respective principal value branch. So I'm going to call this as an angle theta. And you already know that this has to be in the principal value branch. Correct. So the theta is restricted between 0 to pi. It cannot go beyond it. Of course, unless until mentioned otherwise. Okay. So if the question center initially gives you the domain and range as per his whims and fancies, then you have to restrict yourself to that. But if nothing is mentioned, you consider it to be the principal value branch, what we have already discussed in the previous slides. Okay. Now, somebody is asking you sign of theta here, isn't it? Because this guy is theta at the end of the day. Right. So it will definitely have a sign. Correct. Okay. And it will have a magnitude. Correct. Now, what do you think will be the sign? If my angle is between 0 to pi and you're finding sign of theta, what will that sign be? You will say that positive only because 0 to pi is either in the first quadrant or in the second quadrant where sign is known to be positive. Okay. And what about the magnitude? So for magnitude, what do we do? We make a reference triangle that we all know. Right. So what I'm going to do, I'm going to call this as my theta. Okay. I'm going to call this as my theta. Okay. So cos inverse x, if it is theta, that means cos theta is x. Now many people say, sir, how? Okay. I mean, of course, initially you will get these questions, but that is simple. You just take a cos on both the sides. Okay. Cos of cos inverse x is x that we already seen in the, in the previous slide. Isn't it? That's how cos theta becomes x. Okay. Now write this x as x by 1 and treat this as your perpendicular and treat this as your base. Okay. I'm sorry. I'm sorry. I treat this as your base and treat this as your hypotenuse. Okay. Now here, when I'm making the reference triangle, I'll probably write a mod x over here because I'm so sorry. Mod x over here in the base and 1 over here. Why? Because my x could be negative also. No. So how could I make a triangle with a negative side? So I'll make a mod x there. Correct. Now in the very same triangle, if I say what is sin theta, correct? How will you find out? You will find out by taking the ratio perpendicular by hypotenuse. So what will be perpendicular here under root of 1 minus mod x square, which is as good as x square. Correct. So and if you take a perpendicular by hypotenuse, what you will end up getting? The magnitude this by 1. Yes or no? In short, in short, what have I figured out? I figured out sin of cos inverse x and that answer is under root 1 minus x square. Is it fine? Any questions? Any concerns? Any questions? Any concerns? Okay. Let me ask you one more. Maybe that is something that you can try out. Tell me. Tell me what is sin of tan inverse x? Please give me a response on the chat box. Okay. Very good. Okay. Time to discuss. Should we discuss it out? Okay. Let's discuss it out. Now, all of you, please pay attention here. This is something which will, you know, enlighten you, you know, and many books write down the result, you know, very easily and a lot of doubts and voids in the understanding remain in the minds of the students. Okay. So let's listen to this very, very carefully. Okay. Now, let us start with assuming tan inverse x to be theta. Okay. Now, if I just go by the face value of the principal value branch, this theta should lie between minus pi by 2 to pi by 2 open. Correct? Right? In other words, theta could be in the fourth quadrant and the one first quadrant, but in the fourth quadrant, sin is negative. In the first quadrant, sin is positive. Yes or no? Yes or no? Okay. Now, what I'm going to do, I'm going to break this situation into two pieces or into two cases. Okay. One of the case where I will take my theta to be between minus pi by 2 to 0. That means I'm restricting it to be in the fourth quadrant. Thereby, I'm restricting my x to be negative. Yes or no? Only when my x is negative, tan inverse x will be from minus pi by 2 to 0. Right? Yes. Yes. Yes or no? Okay. Now, in this situation, in this situation, if I start finding sin of theta, then what will happen? Sin of theta, first of all, the sin will be negative because you are in the fourth quadrant. Right? This guy is in the fourth quadrant. Right? And what about the magnitude? So for magnitude, you are going to do something like this. You're going to take this reference triangle. You're going to call this as theta. So let's say if I'm saying tan inverse x is theta, that means tan theta is x. Right? So your opposite is this. Your base is this. Correct? In other words, your hypotenuse becomes this. Correct? So if I ask you from this particular diagram, tell me what is sin theta? Magnitude, you will say opposite by hypotenuse. Correct? In other words, what we figured out is that sin of tan inverse x is negative mod x by under root of 1 plus x square. If my x is negative, right or no? Yes or no? So left part, left arm is clear to everybody. So what will be your answer of sin of tan inverse x when x is negative? That is clear to everybody. But I say yes, no. Say maybe whatever. Yes. Say something, say something, say something. Yes. So Smithy has said yes. Okay. Now let's look into the right arm. Okay. In the right arm, I'm going to take that situation where my theta is between 0 to pi by 2. Okay. 0 to pi by 2. In fact, 0 could be included at both the places. It doesn't make a difference. So I'll include air also. No issues. Okay. So if your tan inverse x is giving you an answer which is between 0 to pi by 2, x should have been positive. Yes or no? Sir, we don't have to take two cases, right? Because both tan and sin are negative and positive 2. How does it, how are you, why are you getting sin and tan to each other? Tan inverse, something that's some angle. I'm talking about sin of that particular angle. Now that angle could be in fourth quadrant. It could be in the first quadrant also. So how is it related to sin inverse and tan inverse both being in the first and the fourth quadrant? There is no such relation. One is a trigonometric ratio and other is an angle. Okay. And you're finding that trigonometric ratio for that angle. Okay. Yeah. So now, now what we are going to do here is see it. Now in this case, if I want sin of theta, it is going to have a positive sign because now you are in the first quadrant. Okay. Magnitude wise, nothing will change. Magnitude wise, so why do I go? Okay. So in other words, can I say, in other words, can I say sin of tan inverse x is giving you mod x by under root of 1 plus x square when x is greater than equal to 0. Okay. Now let us take the two results into our consideration. So sin of tan inverse x gave you this result minus mod x by under root 1 plus x square when x is negative and it gave you mod x by under root 1 plus x square when x is greater than equal to 0. Now just try to take that mod thingy out of the picture. Just rephrase it. Just rephrase this whole stuff. When you rephrase this whole stuff, what do you see is that negative of mod x where x is negative will become negative of negative x. Yes or no? Correct. When x is less than 0 and when x is positive, it will be as good as x by under root 1 plus x square. Isn't this as good as saying that sin of tan inverse x will be just x by under root 1 plus x square no matter whatever is your x. Most of you gave this answer but without realizing that the end result came from so much of analysis. And in school, your teacher will directly tell you this. Do this. Take a reference triangle. See, reference triangle never gives you the sign. It only gives you the magnitude. This is from our class 11 days. I'm not saying something new here. So when you are finding sign of something, you have to take into consideration that sign can be both positive negative in the first quadrant and fourth quadrant. But miraculously, the result comes out to be this. And most of you got it by chance. So here, the conclusion that I would like you to draw is as a matter of shortcut. You don't have to do all this process. You just have to make a reference triangle and get your result as if the sites where you're in terms of x. For example, if I have to I will just say let tan inverse x be let tan inverse x be theta. That means x is tan theta. So I'll call this as theta. I'll call this as x. I'll call this as 1. I'll call this as under root of 1 plus x squared. And what I have to find, I have to find sign of tan inverse. That means I have to find sign of theta. I will just take opposite and divide it by the hypotenuse. Miraculously, this works. Okay. So many people said, then this entire sign and magnitude business. Don't we do it? No, not required. If you want to waste your time, you can do it. But in a practical sense, the sign is automatically hidden within x. So everything will be taken care by this expression itself. From your side, you don't have to put an additional negative or positive sign anywhere for that matter. Yes. We'll take more questions. Let's try to take more questions. Let's try to take this question. Tell me. Tell me tan of sign inverse x. What should be the answer? Tan of sign inverse x. Write down on the chat box. Very good. Again, what we'll do here. You'll say, okay, let this be theta. Okay. Then sign theta is x. Okay. So I'll make a reference triangle. Yeah, I'll make a reference triangle and I will treat the opposite to be x. Hypotenuse to be one. So this guy becomes under root 1 minus x squared. Correct. And this is my theta. Now the question is asking me tan of theta. So from this diagram, what is tan theta tan theta is x upon under root 1 minus x squared over. Now I don't have to put a negative sign positive sign and all those stuff because the sign is hidden within this x. Let us say let us say this theta was in the fourth quadrant. Correct. Your tan of that angle should have been negative. But in that case, even x will be negative. So this guy will automatically become negative. Okay. Of course denominator is always positive and it will take care of the sign. You don't have to worry about the sign at all. So whatever I did in the previous slide was to give you a better insight. But whatever I'm telling you right now is just to save your time when you're solving the problem. Got my point here? Got my point here? Okay. Now you'll be able to do all these crisscross property. Okay. Now before we take a break, I have a very conceptual problem for you all. A little while ago we did sign of cos inverse x. Okay. And we figured out it may write it down something important. We figured out that sign of cos inverse x was under root of 1 minus x squared. Okay. Now this property is it true for all x belonging to the domain of cos inverse? I mean, can I say this holds for all x belonging to minus 1 to 1? Correct? No. You all agree with me? Or sir, when you ask the question in this way we all get slightly fearful about what we are going to say. Correct? No. This property is true for all x belonging to minus 1 to 1. No. Nobody is saying anything. Yes. Okay. Yes, it is true for all x belonging to minus 1 to 1. Okay. Now the same thing if I want to do this cos inverse is equal to sign inverse under root 1 minus x squared. Will it be true for all x belonging to minus 1 to 1? I am trying to say is the second step obvious from the first step? Can I say so? Anybody? Now let us try to analyze this situation. This guy is between 0 to pi always. Correct? Correct? Cos inverse of anything will be from 0 to pi only. And this guy this guy will be from minus pi by 2 to pi by 2. Correct? Now you are trying to relate two such quantities. One is between 0 to pi. Another is between minus pi by 2 to pi by 2. So if at all they are equal to each other they must happen in their overlap. Correct? So overlap is pi by 2 to pi. Sorry, 0 to pi. Am I right? Sorry, 0 to pi by 2. Yeah, these two are equated. That means even your cos inverse x has to be between 0 to pi by 2. Correct? Then only your overlap will happen. No. In other words your x has to be between 0 to 1. Correct? In other words this particular property will work only when x is between 0 to 1. No. 0 exclusive. Getting my point. So this is something where people get tricked very badly. So what I will do here, I will show you this graph and I will show you this graph. You realize that their graphs will be did to same only between 0 to 1. Between minus 1 to 0 they will not be the same. Okay, let's check it out. So minus 1 to 1 the other guy also. So here the deal breaker is not their domains. The deal breaker is their ranges. Getting my point. Okay, let me show you their graphs. Let me mute this. Cos inverse graph you have already seen. So I will just cos inverse x arc. Okay. And sine inverse arc sine under root of 1 minus x square to the power of 0.5. Okay. Now see this graph that you see I mean I'm just muting the graph of cos inverse. Okay. This is the graph. It is something like a flame top of a flame. This is the graph of sine inverse under root 1 minus x square. Correct. And this is the graph of cos inverse. Correct. Now if they are equal they can only be equal or they can only resemble each other in this part. You see that. So only in the intervals 0 to 1 these two functions will be equal to each other. So cos inverse x will be equal to sine inverse under root 1 minus x square only for x belonging to 0 to 1. Okay. See 0 to minus 1 this guy has gone on top and this guy is at the bottom. Okay. So just because cos sine of cos inverse x was under root of 1 minus x square please do not conclude that cos inverse x is sine inverse under root of 1 minus x square. That would be a wrong conclusion. That will be true only for certain intervals of x. Okay. Now the genesis of this disparity is in the next property which we are going to discuss after the break. Okay. So as of now we'll take a break time right now whenever my laptop is 6.17 we'll meet exactly at 6.32 p.m. Okay. On the other side of the break we'll discuss the mother of all properties in inverse trigonometry. Okay. See you on the other side of the break. Yeah. So we will now move into the next property which is property number 2 and this property let me tell you is the mother of all the properties that you're going to come across in the inverse trigonometry. Okay. So this property to a large extent is not justified well in schools. Okay. And because of that there are a lot of mistakes that people do when they start applying the school concepts to competitive level exam concepts. Okay. So I would request everybody to listen to the next this concept for the next one are very very carefully because this is the deal breaker for us to understand this you will be able to derive the subsequent properties completely on your right. And if you don't know this property or if you're not able to understand this property well then you will always make mistakes here and there while solving questions also. So this property is based on the reverse of what we did in the previous property. Okay. So this property says f inverse should be coming from the domain of f of x for which it was invertible. That means it should come from the principal value branch. Okay. So let me give you a small example here. See if somebody asks you what is sign inverse of sign let's say I'll give you some angle by six. Okay. What will be the answer for this particular property? If you literally solve it by using your basic of you will get sign pi by six as half. So what is sign inverse half pi by six. Okay. But consider this situation when I ask you sign inverse of sign 5 pi by six. Right. Now you may be speculating right. So while this value x matched with this but this value did not match with this. Right. Why? Because here my pi by six was in the principal value branch. It was a part of that interval for which sign x was considered to be invertible. Right. Which was between minus the same answer came out from here as well. Because this x came from the principal value branch or that branch for which the function f in which even our case f is sign x. So the branch in which my f was considered to be invertible that was fed a value from that interval was fed and hence that value came out as my answer but in this way does not belong to minus pi by two to pi by two because minus pi by two to pi by two was the branch where it was considered to be invertible and I'm now putting a value which is beyond that branch so don't expect the same answer to come out here. Are you getting my one? Now you cannot put that restriction on a question center. He may give you a sign inverse x function and he may give you a sign x function that doesn't mean that sign x function that he has given you will be between minus pi by two to pi by two. He will consider the exhaustive domain of that function for sign inverse he will consider minus pi by two to pi by two that can be the scenario so in those cases of f of x is x only when this x that you are feeding to f comes from that interval where your f was considered to be invertible understood clear right so in light of this let us see what is the property set that we get so the property set as I already showed you which is minus pi by two to pi by two but here this x you cannot restrict anybody see if let's say I just ask you sign inverse sign of some angle and that x could be anything it could be like 1254 pi by 16 something like that that you cannot restrict but the answer that you are getting out of it from sign inverse of this black box should be such a value which will be in the principal value branch of minus pi by two to pi by two for that particular function got the point similarly cos inverse of cos x will be x but for this your x should be between 0 to pi if it is not in our next slide similarly tan inverse of tan x is x only when your x is sorry x is between minus pi by two to pi by two open I don't know why did I write r similarly cos inverse of cos x will be x only when your x lies between minus pi by two excluding 0 similarly sec inverse of sec x will be x when x lies between 0 to pi excluding pi by two and cot inverse cot x is x only when x lies between 0 to pi okay now of course j people they are not that stupid but but they will definitely give you such ś which are beyond these intervals that you see on the right side that is the most trickiest part so please note there are two different things this will give you a value right whereas this will give you an angle now both are called x over here okay but for this x has to be in minus one to one interval and for this your x has to be between minus pi by two to pi by two okay so many people say sir why don't you start writing theta in the second case okay you can call this as theta also if you want to okay it is up to you see x theta at the end of the day they are just names of variables I can use x for angle also I can use x for values also okay right so don't get confused between the two expressions they're different things one gives you a value other gives you an angle okay initially these confusions will remain right but as you practice more and more questions and you will understand things in a deeper way is there fine any questions okay so as as I discussed with you let us now try to figure out what will happen let's say in our first formula if your x is beyond this interval okay so let us take this first in our next slide so I'll just write down here please appreciate the difference in these two expressions okay they're not the same sign of sign inverse and sign inverse of sign they're two different things don't get confused between them so sir anyways it is not like they get cancelled out you know okay you have there are certain things which you have to take into consideration now let's talk about sign of sorry sign inverse of sign x in more detail okay now what we learned was it is x when your x is between minus pi by 2 to pi by 2 now let us try to figure out what happens to this answer or what will be this answer if I go beyond this interval that means let's say if I go from pi by 2 to 3 pi by 2 or if I go from 3 pi by 2 to 5 pi by 2 okay so what happens what happens what happens when I go beyond this interval even let's say I go backwards also what happens when I go between minus 3 pi by 2 to pi by 2 so what happens to this answer okay and I can keep on going let us try to figure out what really takes place let's try to figure out these question marks okay and once we are able to figure out this question mark we'll be able to see some pattern okay so which one would you like to start with let us start with this guy first okay what happens to the result of sign inverse sign x or what answer comes out from it when your x is between pi by 2 to 3 pi by 2 so when it is between 90 degree to 270 degree of course everybody agrees that the answer will not be x correct up till now this thing should be you know clear to everybody that I cannot write an x over here right because if I write an x over here means you're trying to say that sign inverse is throwing out a value which is beyond the principal value branch which is not possible so my answer should be this you know between minus pi by 2 to pi by 2 only it cannot exceed it right it cannot go beyond that that restriction of minus pi by 2 to pi by 2 correct so what should be this answer okay of course it will come out in terms of x only let's try to figure it out let's try to figure it out now there are three ways to figure it out I'll discuss all the three ways the first way is okay the simplest of all way is by choosing a special value okay I call this as special value approach or special value method okay and many people like this method a lot that's why this is the first thing I'm discussing with you so do something do one thing take an x which is between minus sorry take an x which is between pi by 2 to 3 pi by 2 and that x should be known to you I mean it should be a known angle and avoid taking the extreme points tell me any angle between 90 to 270 any allied angle that you know of all right tell me any angle between 90 to 270 any allied angle okay 135 135 I'm writing it in degrees only just for you to understand it properly okay yeah let's write 135 degrees okay now let's say if somebody asked you what is sign inverse of sign 135 degrees how will you evaluate it you'll say okay I will first find out sign 135 sign 135 is 1 by root 2 correct 1 by root 2 gives you 45 degrees correct now tell me what would you do with 135 now this is your x okay so that you end up getting 45 degrees what would you do with this okay now you have to only you know use multiples of pi to make that change why multiples of pi to make that change now this is very important see what are you what are you doing here you're trying to find out sign inverse of sign 135 degree you want to find some value for it right let's say I call that value as alpha now you're trying to do actually this let's take sign on both the sides let's take sign on both the sides okay now treat this as sign of sign inverse of some x so isn't this as good as sign 135 which is x correct so now you're trying to find out such an alpha of course that alpha should be between minus 90 to 90 I will talk about that also but essentially what are you doing you're trying to solve this kind of an equation sign 135 is equal to sign alpha right so now that alpha could either be pi minus 135 or we can say 180 minus 135 correct or it could be 180 or sorry 360 360 minus 1 plus 135 correct or it could be 540 minus 135 and so on and so forth now here out of all these operations which operation will you apply to 135 so as to get 45 you will say obviously set the first one I will apply okay so you'll get 180 degree minus 135 to get 45 isn't it in short what you have done you have done pi minus x isn't it so let me write here so you have actually done pi minus x to get your answer isn't it so the answer to this question the answer to this question what would be the answer of sign inverse sign x when your x lies between pi by 2 to 3 pi by 2 your answer will be pi minus x okay now many of you would be thinking sir try taking some other value right so I want to see whether the answer is consistent irrespective of whatever value you take okay now you take any other value I'm requesting you to give me some other value for x 150 degree okay so Harshita is taking x to be 150 degree okay so let's see so this is only done in the previous stage okay so sign inverse sign 150 degree okay it gives you sign inverse half which is 30 degree I'm just writing it in terms of degrees even though you should be writing in in radians so isn't this 30 degrees coming from 180 minus 150 which is just to find the fact that your answer should be again pi minus x so whatever I took it is falling in line with this also okay so this is the simplest approach that you can use in order to get the result over here in fact towards the end of this exercise I will give you a graph also of sign inverse sign x okay is it fine now this is method number one which many people appreciate because of its simplicity method number two method number two is the graph approach or the graph method now see everybody please pay attention when we are trying to find out sign inverse sign x okay and your x is let's say between pi by 2 to 3 pi by 2 you are looking out for an angle alpha you are looking out for some angle alpha where this alpha should be coming from minus pi by 2 to pi by 2 because as I told you sign inverse of anything will give you an output which is in the interval minus pi by 2 to pi by 2 only are you getting my point in short you are trying to figure out an angle which is between minus pi by 2 to pi by 2 for which sign is same as the sign of that given angle x right so what we do is we will make a regular sign x graph okay so let me make a regular sign x graph so this is my regular sign x graph okay and basically your x is somewhere between pi by 2 to 3 pi by 2 okay take a dummy x let's say you take a x over here okay so this whole thing is your x this whole thing is your x fine now what are you doing here you are trying to see which angle which angle in this branch which I am bubbling gives you the same answer as that of sign x that means which is this angle alpha are you getting my point so you're trying to see that which angle alpha between 0 to between minus pi by 2 to pi by 2 is giving you the same sign as sign of x okay which is clearly from the diagram this value so I'm looking for this value I hope you can see that in the gray color correct now looking at the symmetricity of the figure can I say alpha this gap alpha and this gap should be the same correct so can I say from that figure that alpha is same as this gap and this gap is the gap between pi and x correct so the answer to this question becomes pi minus x same answer as what we got first step for approach with approach one as well correct but this method many people find it slightly difficult correct another method I'll tell you which is the quadrant method third method quadrant method see it is up to you to use any method you want quadrant method okay in this what people do they take an angle which is between pi by 2 to 3 pi by 2 take any angle which is between pi by 2 to 3 pi by 2 let's say I take this angle okay so this is my x okay now look for such an angle which should lie between minus pi by 2 to pi by 2 that means it should be in this zone for which sign is same as sign of x right so I'm looking for such an angle alpha which is in this gray zone whose sign is same as sign of x so it says that obviously that angle must be this guy which I'm showing you pink this guy yes or no this is my required alpha now you tell me how is this alpha related to this x how is this alpha related to this x tell me tell me look at the figure they're so symmetrical this gap and this gap will be the same tell me how is the alpha related to that x most of people are thinking sir this time I'm getting x minus pi please note that x minus pi this angle is a negative angle as per the diagram you you're looking at a clockwise angle of alpha so it cannot be x minus pi it will be negative of this correct so it'll give you pi minus x back as your answer getting my point getting my point so whichever of the three approaches you are more convenient with you can use that to find out the answers to such questions okay however at the end of this particular discussion you will get a graph of sine inverse sine x that will be a one-stop solution for all such problems so you don't have to worry about remembering all these values but as a matter of exercise I would request everybody to tell me what is this second question mark please figure this out and tell me your response on the chat box so what is sine inverse sine x when your x is between 270 degrees to 450 degrees give me a response on the chat box you can use any of the three methods you feel like either you choose a special value and figure out the process or you choose a graphical approach or you choose the quadrant approach it is up to you okay sathyaam okay now I'm getting two types of answers I'm waiting for more types yes just two people able to figure this out okay see everybody wants to take a shortcut route okay so even I will take an angle okay let's take an angle between 270 and 450 degree which is very familiar to me maybe I'll take let's take excess 300 degrees okay that's the simplest of angles okay now somebody is asking you what is sine inverse sine 300 degrees okay so as a matter of fact I will actually find sine 300 degrees which is negative root 3 by 2 okay and this gives me the answer as minus 60 degrees correct now obviously in order to get minus uh 600 minus 60 degree from 300 degree you will do this operation correct yes or no that's how you get minus 60 correct in other words you are doing x minus 2 pi so this becomes your this question double question mark okay so this is x minus 2 pi okay so only person who got this right is Aravind Aravind Aran sathya man Harshita got your mistake okay I'll give more opportunities don't worry I'll give more opportunities okay tell me what should be the answer if my x is between 5 pi by 2 to 7 pi by 2 tell me the answer and whatever answer you are getting my dear always do this quick check is my sine x for example some people here said 2 pi minus x right so see what are you trying to find what are you trying to say you're trying to say that let's say I consider your wrong answer to be right okay so I will do quickly this check I will quickly check whether these two are equal or not no they are not this guy is minus of sine x I hope you remember your basic class 11 thing so this answer cannot be correct okay so when you're giving me your answer do this quick check okay so whatever answer I'm saying is the sign of that answer same as sine of x then only it will work then only it may work okay so just see this and then tell me the answer very good Harshita that is correct they're taking a lot of time again Tanvi sine of whatever you said is same as sine x are you sure okay take an angle between see what is this angle 450 and this angle is 6 so take an angle between 450 and 630 degrees take any angle let's take 600 okay that's a familiar angle towards okay now now what is sine inverse sine is 600 degrees okay now sine 600 600 will be 540 600 will fall in the third quadrant right so this will be 60 so this will be negative negative root 3 by 2 correct okay so this will give you negative 60 okay now in order to get in order to get minus 60 from 600 what will I do I will do 540 minus 600 which is nothing but 3 pi minus x yes or no correct so this answer here will be 3 pi minus x okay and you can check sine x and sine 3 pi minus x will be same correct and this can go on and on okay let me give you one more opportunity let's do this triple question mark let's find this out everybody please find out that triple question mark this one huh Vishal are you sure okay one more one more checkpoint whatever you're trying to say as the answer okay take any any number take any x in this interval okay and apply that formula that you have said is your answer between minus pi by 2 to pi by 2 always for example the answer that you said Vishal would lead your result to be between minus 11 pi by 2 2 I think minus 7 pi by 2 so that is not going to be the answer that cannot be the answer okay Arvind it's okay it's okay it's this very important property that's why I'm spending time on it because it's so important to get it right because later on all the properties will be related to this indirectly okay so up till now three people have answered and all their answers are different so Satyam has a different answer Arvind they gave a different answer and of course Vishal gave incorrect answer which I already told him one of you is right no I think none of you is right see guys again let's try to figure it out let's try to figure out again let's use a special value approach ah Satyam now you are on the right track so I'll use a special value approach let me erase unwanted stuff from here between minus 3 pi by 2 to minus pi by 2 take a lot familiar angle or maybe let's say I take a minus 120 degrees I'm keeping it in radiance because radiance is more easy to connect in terms of simple arithmetic operations oh sorry degrees is more easy to connect okay now let me ask this question what is sine inverse sine minus 120 degree first of all sine minus 120 degree is itself negative root 3 by 2 okay coincidentally I'm getting negative root 3 by 2 for the third time right now now this guy is going to give you minus 60 degrees take a a minus 120 how what will I do what operation should I subject this to to get minus 160 sorry to get minus 60 if I add a pi I mean this is my x okay if I add a pi I will get plus 60 correct so what I'll do I'll add a pi but that will give me plus 60 and in order to get minus 60 I'll do a minus here isn't it isn't it so I have to do 180 plus x and minus of the whole thing that means the answer that I should be getting is minus pi minus x got the point okay so this answer is this answer is minus pi minus x okay anyways even if you're making a mistake consistently over here don't worry when you know the graph of sine inverse sine x you will no longer make such mistakes so let us quickly jump to the graph is this is this values I mean can I can I see a pattern in which these answers are coming so let us see whether there's a pattern or not and that pattern will be evident from the graph okay okay so let's go let's go to let's go to the next page let's go to the next page okay in fact what I I will do is I will copy the previous page result so that I don't have to write this thing once again remove my camera yeah so this was the previous slide uh discussion that we had now let's try to graph it let's try to graph it now all of you please pay attention very very important so on the y-axis I am plotting sine inverse sine x and on the x-axis on the x-axis I'm plotting x okay of course now all of you see this you are calling this as y okay so treat as if you are plotting y equal to x y equal to x when when your x is between minus pi by 2 to pi by 2 okay so this is as good as you plotting y equal to x when your x is between minus pi by 2 to pi by 2 correct so on the x-axis between minus pi by 2 to pi by 2 you will be getting a part of y equal to x line which is like this maybe I should write this slightly smaller so you're going to get a part of this line like this y equal to x line everybody knows how y equal to x line looks okay now when you are looking at pi by 2 to 3 pi by 2 when you're looking at pi by 2 to 3 pi by 2 what are you plotting you're plotting y equal to pi minus x correct how does y equal to pi minus x look like again it's a straight line with a slope of minus 1 correct so can I say it is going to look like this please note it will be cutting the x-axis at pi correct now when you're looking at 3 pi by 2 to 5 pi by 2 interval what are you plotting you're plotting y equal to x minus 2 pi correct so x minus 2 pi is again a positively slope line which will go like this correct so this is between this is this way 3 pi by 2 to 5 pi by 2 correct and in between you look at 2 pi okay if you see there's a trend coming up in the way this graph is shaping up so it will be like this zigzag zigzag fashion okay and you can keep drawing this on and on but I'm just saving your and my time by drawing it quickly okay so this graph will keep on going in both directions all the way okay because you can put any real x okay so you can treat as if you can treat as if your sin x graph has become sawtooth like this or triangular like this okay so please remember this graph this is very very important graph which will be useful in solving many problems right and each of these lines that you see they are your given expression that you already found out this is y equal to x this is y equal to pi minus x this is y equal to 2 pi plus x minus 2 pi this is y equal to 3 pi minus x in fact I can always predict this line equation this line equation will be x minus 4 pi this line will be 5 pi minus x and here also this line that you see this is y equal to minus pi minus x this this point will be minus 2 pi so this line will be y equal to x plus 2 pi okay yes or no easy to figure out everything from this particular you know graph okay so few things that I would like you to note down about this even though it is very obvious from the graph number one sin inverse of sin x is an odd function okay it is a periodic function this is periodic with period who will tell me what is the period of sin inverse sin x with fundamental period or period equal to right 2 pi very good next domain of sin inverse sin x any real number but a range of sin inverse sin x will be as good as range of sin x which is minus pi by 2 to pi by 2 okay so as you can see the graph is not going beyond minus pi by 2 to pi by 2 so this graph is continuous for all x belonging to real number it is differentiable for all x belonging to real number except odd multiples of pi by 2 okay as you can see at pi by 2 at 3 pi by 2 at 5 pi by 2 at 7 pi by 2 etc it has got sharp corners or it has got corners so there it cannot be differentiable is it fine any questions any concerns any questions any concerns okay now look at this graph look at this graph and only from the graph tell me what is sin inverse of sin 27 pi by 8 just from this graph you can answer your answer this question what is sin inverse of sin 27 pi by 8 give me a response on the chat box Satya are you sure knock her jiay or you want to not sure see make use of this graph make use of this graph this is your x okay now if you see sorry your x is 3 pi plus 3 pi by 8 correct by the way 3 pi by 8 is lesser than pi by 2 correct so you are somewhere over here this is your x correct means the answer that you should get will come from this line's equation y equal to 3 pi minus x correct so the answer that you will get will come from 3 pi minus x and x is 27 pi by 8 which makes the answer as minus 3 pi by 8 got it okay so it is not 3 pi by 8 is it fine any questions okay i will not allow you to move till you answer these questions of mine so let us take this find the following find the following okay number one in fact we'll go one by one sin inverse of sin 2 sin inverse of sin 2 give me a response on the chat box dot dot dot so this is 0 this is pi by 2 this is pi 3 pi by 2 2 pi 2.5 pi 3 pi i'm just writing in terms of pi only look at this graph and try to answer this question pi k terms may i will appreciate in terms of pi only so tell me in terms of pi only don't give me absolute value okay okay okay anybody else see what is this 2 is this 2 in degrees or is it in radiance first of all let me ask this question maybe something that i will ask the 11th grader now yes it is in radiance correct so don't never consider just a 2 written as 2 degrees right if it was in degrees i would have mentioned a small o on the top of it but it is not that correct now two radiance means you are somewhere over here that means you are more than pi by 2 but lesser than pi correct so when you go up you hit this line isn't it so this this is the line that you hit so what is the equation of that line that will help you to find the value correct so the equation of this line is clearly x minus pi i'm sorry pi minus x pi minus correct so the answer to this question will be pi minus 2 clear everybody okay so we'll take more questions those who could not solve this one so more opportunities what is sign inverse sign 5 sign inverse sign 5 okay vishal okay satyam come on i want answer to come from everybody please this is a mother of all inverse telemetric function properties if you understand this well your life will be easy in this chapter else it will be equally difficult good harshita others should i start naming him smithy nithya pramithi rahul rohan rahul vihan areshri shri venkat 5 where do you think will 5 lie of course it will be lesser than 2 pi correct in fact in fact if i if i ask you 5 will be will it be more than 1.5 will it be more than 3 by 2 pi yes because 3 by 2 pi is 3 into 1.57 i think this will be lesser than 5 okay and of course it will be lesser than 2 pi so you are somewhere over here you will be somewhere over here 5 correct so you are basically dealing with this line okay this line equation is y equal to x minus 2 pi so the answer to this question will be 5 minus 2 pi because your x is 5 clear now people are not responding so hence i have to ask more questions okay tell me sign inverse sign 10 sign inverse sign 10 okay vishal okay arvind satyam very good yes so 10 see this is this is 3.5 pi i think 3.5 pi will exceed 10 so 10 is somewhere between 3 pi to 3.5 right so 10 is somewhere over here so you will be dealing with this line okay what are the equation of this line y equal to 3 pi minus x so the answer to this question will be 3 pi minus x x here is 10 clear any questions okay so i would not like to start with cos inverse cos x but i will definitely take few questions with you on so let's take few questions on this topic so let me just this question has been asked in ncrt exemplar sign inverse sign inverse of cos 33 pi by 5 in fact let me just take a snapshot of this let me put the poll on in fact if you want to answer on the poll or through the poll very good one person has answered miraculously some people are also giving me answers which are beyond minus pi by 2 to pi by 2 that is not possible sign inverse of anything has to be between minus pi by 2 to pi by 2 only so few of the options you can outrightly reject they cannot be your answer but i will not tell which one because some people are since some people were answering with the wrong options see guys you're taking this chapter in the same way as you took your trigonometry of class 11 now it needs deep thought okay logical way of answering don't don't just take this chapter like any other chapter it is one of the trickiest chapter in mathematics okay let me conclude the poll in the next 15 seconds five four three one go okay so i got responses from nine of you so far of which four of you say option number d now first thing what i will do is this expression cost 33 pi by 5 i will write i'll try to simplify it see anyways it's going it's a value that i have to find out okay so why not write it as 6 pi plus 3 pi by 5 okay so this as good as cost 3 pi by 5 cost 3 pi by 5 is sine pi by 2 minus 3 pi by 5 correct and that is clearly sine minus pi by 10 so now this question is as good as sine inverse sine minus pi by 10 please note that minus pi by 10 very much belongs to minus pi by 2 to pi by 2 right right so it belongs to that y equal to x line so this answer will be minus pi by 2 sorry minus pi by 10 so option number d is correct now people who said a this cannot be the answer because this is 0.6 pi correct so this does not belong to minus pi by 2 to pi by 2 it cannot be your answer this also cannot be your answer because this also does not belong to minus pi by 2 right your answer has to be between minus pi by 2 to pi by 2 okay so either c or d could have been your answer but we figured out that d is the answer getting this clear everyone okay let's try to do the next one okay again a simple question which of the following is correct poll is on okay arvind okay satyam we'll see we'll see just waiting for everybody to give it a try and give their responses on the poll next class would be offline right in your school promises as we discussed 315 to 645 okay we'll continue with this same property this sunday's test of j main whatever we will cover whatever we have covered in inverse tignometry till today that will be that can be asked okay okay should we start the discussion because it's already time two minutes have gone and it's 730 also so let me stop the poll at the count of five five four three two one and go okay most of you have gone with option number a so out of six people five people say option a and only one person i think it's satyam who has gone with b okay see tan one radian okay now see one radian is approximately i mean i'm just taking a rough picture it's 57 degrees but i'm taking it as 60 degrees right so this will be very close to root three root three is 1.73 something very close to it and tan inverse one is pi by four pi by four is roughly 0.76 so option number a is for sure correct okay so this cannot be correct and of course this two cannot be correct none of these cannot be correct so option number a is correct okay good so we'll stop our discussion here this chapter has much more to it we have only covered let's say 20% of the topic 80% is left to be covered so we'll do the remaining topics in the offline mode in your school till then bye bye take care see you very soon right in your school premises bye bye good night