So welcome to session 2 of trigonometry. So let's start from where we left off in the previous session in the previous session. You would all recall I had done some negative angle identities with you. Okay, so last class we stopped at some negative angle negative angle. Identities negative angle identities, what about these identities?  Now from where do these identities come up, right? So we are going to discuss today something which is very very important part of our learning is the graph of these basic six trigonometric functions. So from the graph will come to know many things about these trigonometric functions, not only about a trigonometric function, but how they are also related to other trigonometric functions. So all these identities that you are seeing right now in front of your board, they have all come from the graph of these trigonometric functions. Now, of course they are ratios, but we can also treat them as functions. For example, sine of theta, sine of theta, you know that if you are putting in some input over here, it gives you some output, correct? It gives you some output, right? So why not treat this as a function itself? So why not make this as y is equal to sine x function? Okay, so we are going to give it a shape of a function and now we are going to study its graph. So one by one, I'll be exposing you to the graph of these functions. We'll be making a quick analysis on it and not only that, we'll also keep these graphs in our mind. Okay, so going forward, the graph which I'm going to show you should be kept in your mind. That is very, very important part. So let me start with the graph of y is equal to sine x function. So this is the first function that we'll talk about today. Okay, now we'll keep our x in radians. We'll keep our x in radians and now I'm going to show you the graph of y is equal to sine x function. No need to wrote down or because I'll be sharing you, sharing with you this PDF. So everything would be there in that PDF. Okay, even I'll not be plotting it. I'll be just showing you on the GeoGebra tool. So all of you please pay attention, stay tuned. So this is GeoGebra. So graph of sine x, let me first make a change in the dimension. So in GeoGebra, you can also change your unit in which you want to make your x-axis. For example, I've kept it in terms of radians. So as you can see, 0 pi 2 pi minus pi minus 2 pi, etc. Okay, I'll just show you the graph of sine x function. So y is equal to sine x. Okay, so this is a sinusoidal curve. This curve is called a sinusoidal curve. Now a million dollar question here is how do you plot such curves? What is the what is the reason that these curves come up? Okay, why is it like a wave? Okay, the answer to is very simple. What I'll do first is I will copy this graph first. I'll just copy this graph and just remove this. Yeah, so let's copy this graph. So here I'll be pasting this graph. You don't have to copy anything. Okay, this is the graph. What are the reason for this graph? What are the reasons for this graph? Let's go back to our previous class where we had divided angles into quadrants where we had divided in angles into quadrants. Okay, on this graph, let me make a unit circle for you. Okay, what do you mean by unit circle? What do you mean by unit circle? A circle of radius 1, correct? So let me just show you an angle over here. Let me show you an angle over here. Okay, let me say x radians. Fine. Now let me drop a perpendicular. Let me drop a perpendicular from this point to this point. Okay, so this is a perpendicular. So again, repeating what I have done so far, I have made a unit circle. Unit circle means this particular hypotenuse of this right angle triangle will be 1. This is x radians. Okay, so let me name it A O B. Okay, now what is AB length over here in terms of x? Can anybody tell me what is AB length over here in terms of x? Sign of x, absolutely. So AB is sign of x. Okay, now what are we plotting here? We are actually plotting. All of you please pay attention over here. This is very important. We are actually plotting AB length versus versus x versus x. So as you change your x, let's say if you start going in a anti-clockwise direction, okay, let's say you go 30 degrees, 30 degrees pi by 6. Okay, so when you rotate 30 degree, you know that AB length will become a half. AB length will become a half. Okay, so when x is pi by 6, AB length will become a half. So what do you do on this graph for pi by 6? Let me choose a color which you can see. Let me choose a green. So on this graph, at pi by 6, I will plot half. Okay, so basically you're plotting pi by 6, half. Okay, if you change it to pi by 3, AB length will become root 3 by 2. So as you can see here, this graph will show you that at an angle of 60 degree, your values, I mean, it's very important. Here, this graph will show you that at an angle of 60 degree, your values, I mean, it's too small for you to see here, but this value here would be root 3 by 2. Okay, so because at 60 degree, your AB length would slightly change. Now this would be your new AB length. Okay, so let me call it as new AB length. Okay, if you make it 90 degree, AB will come over here. AB will come over here. So as you can see at 90 degree, which is exactly the peak point, your value will become a 1, right?  AB length will start decreasing. So your graph will start coming down. And as you go to 180 degree, or you can say pi radians, AB would have again become a 0. So you are back to this position, right? Again, if you go to minus, or you can say 270 degree, your value of AB length will become minus 1. Okay, so what are you doing?  So this trend has been plotted. This trend will result into this graph. And this graph is basically called a sinusoidal graph. This graph is basically named as the sinusoidal graph. Is this fine? Okay, so any question, any concerns on how these graphs have actually come about, right? No problem with that. So for sin equal to pi by 2, AB will become a 1. Okay, and if you continue going, let's say it's x equal to pi, it will become again a 0. Okay, again x equal to 3 pi by 2, it will again become a 1 minus 1. So the value of AB has been plotted on the y-axis and your angles have been plotted on the x-axis. Okay, the same trend you will see, even if you are rotating clockwise. So clockwise is your, you know, other side of the y-axis. So that is your clockwise rotation values of AB. Any questions so far on this? Okay, now what we are going to do is, we are going to do a critical analysis of this graph. What does this graph actually convey about sin x? So what all informations can be, you know, derived from the graph of sin x function. Number one information that I can see here is the domain of the graph. Now what is domain? People who were not there with us in the bridge course, let me tell you domain are the valid inputs that you can put into the function. What is the meaning of valid input? Such real inputs which will make the output real, those are called the valid inputs because we are dealing with all these functions as real valued functions. So in order to know what is the domain of a graph from its sketch or from its, you know, figure, you just have to see what is the span of the function along the x-axis. Okay, watch my motion of my hand. So what all values of x is the graph present for? That is called the domain. In a similar way, what all values for y is the graph present for? That is called the range. So for domain you see this way, for range you see this way. So what do you think is the domain of the graph? Let's go back to this figure. Let's go back to this figure which we had sketch for sin. If you extend the graph to the right-hand side, you'll see this graph will never stop. It is going on and on and on. See, so far we reached. Same would be true even if you extend the graph on the negative side. Okay, so in light of this, we can claim that, we can claim that the domain of the graph is all real numbers, absolutely Anusha. So it is all real numbers. Okay, you can also call it as minus infinity to infinity. Is this fine? Any questions? What are the range of the graph? To see the range of the graph, you need to see the span of the function along the y-axis. Right. If you see your graph is extending along the y-axis from minus one to one. So only in that region, the graph is present above one below minus one. The graph is absent. Okay. So the range of this sin x graph will be minus one to one close interval. So under no condition, you should expect the graph to cross one or go below minus one. Okay, so sin of any angle cannot be two or one and a half like that. Or minus two or minus one and a half. Okay, they cannot go beyond one and minus one interval. Okay. Another thing I can see from the graph, that this graph is symmetric about origin. Symmetric about origin. What are the meaning of symmetric about origin? Symmetric about origin means whatever you have drawn, right, whatever you have drawn in the first quadrant, as you can see, whatever you have drawn in the first quadrant, you have drawn the same thing in the third quadrant. Okay. And whatever you have drawn in the second quadrant, the same thing you have drawn in the fourth quadrant. So think as if there is a very, very small mirror, which has been kept at the origin. Okay. So if you are on in the third quadrant, you can see its reflection coming from the first quadrant and same for the second and the fourth. Okay. Such functions whose graphs are symmetrical about origin, they are called as odd functions. So let me write it in green for it to be consistent. So sine X function is an odd function. Now this odd is a term that we normally use in functions. You will study more about it in your class 12. I think in 11th, we don't talk about odd function, even function. It's a subject matter of class 12. But just understand this odd functions are those functions which will satisfy, which will satisfy this equation. f of minus X is minus of f of X. That is, if you change your input sign, your output sign will also get changed. This is precisely why it followed the property sine minus theta is minus sine theta. Now I would like you all to go back to this graph and realize this. See, let me go back to the original original figure here. Yeah, okay. Now if you all see here, if you all see here, if I take an angle, let's say 90 degrees, I get a value 1, correct? But if I put a minus 90 degree, that is minus pi by 2, I'll get minus 1 as my answer. I will get minus 1 as my answer. That means if I'm changing my sign of the input, the sign of the output will also get changed. Okay. And this is the characteristic, which is shown by all the odd functions. Can somebody tell me any other odd functions that you have seen before? Anybody? Any odd functions you have seen before? X, very good. X cube, X cube. Okay. That's an odd function. In fact, any polynomial whose powers are always odd numbers will be an odd function. For example, X cube plus X, that will also be an odd function. Okay. Right. So any polynomial subjected to the variable subjected to all the variable X subjected to odd powers would be odd polynomials, odd functions. Is this fine? Any questions here? Okay. Another thing that we would like here to observe by all of you is that this function is periodic. This function is periodic. What are the meaning of a periodic function? What does common sense say about function being periodic? What do you understand by periodic function? Anybody? You can unmute yourself and talk. If somebody comes and says, hey, this is a periodic function. What do you mean by that? Okay. So Pradhan is saying its values repeat after fixed intervals. Absolutely. Okay. So I say that sine function is a periodic function because I can see the values repeating after certain interval. Okay. Now, let me show you something very interesting here. If I take, let me just zoom in the graph. Sorry. Yeah, zoom out of the graph. Okay. All of you please look at the graph. Look at the graph. If let's say I take a value zero. Okay. Sinex becomes zero at zero. When is the next occasion it becomes zero? So people are saying pi, people are saying two pi also. Both of you are right. Okay. But if I ask the immediate next interval where it is becoming zero, you will say pi. Correct. Correct. Again, two pi. Again, three pi. Okay. So you realize that after a gap of pi, sinex is becoming zero again starting from zero. So sine zero is same as sine pi. It's same as sine two pi and so on and so forth. Okay. Does this mean pi is the period of sine? Yes or no. Of course, there are two answers only and both of the answers I can see on the screen. Shrithish is saying yes. Aryan is saying Arya is saying no. Okay. Now, person who's saying no, can I hear from you why Arya this cannot be the period of sine. Try it out. Let it be long. No worries. Okay. Absolutely. Absolutely. She has written it very well. See my dear, try to understand here. How is a period defined? How is a period defined? Period is basically, yeah, try to understand this. If a function is periodic, if a function is periodic. Okay. With a period of T with a period of T. Okay. Then remember one important thing that I'm going to tell you now. If you change your input by T, you are going to get back the same value of the function. Okay. For all values of X belonging to the domain of the function. This is very important. Many people, they do not take this into consideration. This repetition must happen for all values of X in the domain of the function. So when you say, when I told you 0, 0 definitely repeats itself. Sine 0, sine pi, sine 2 pi. This repetition is happening after every 180 degree jump. What do you think? What do you think this will happen for other values of sine? For example, if I say sine 30 degree, that is half. Okay. Right. When is the next half coming up? Of course, next half is coming up at sine 150 degree. Okay. Again, when is the next half coming up? It is coming for 390 degrees. Okay. Again, when is the next half coming up? It is coming for if you just add 120 to this 510 degree. Correct. Are you getting more point? Okay. So now this is an example where you would realize that this is showing some different nature. You can see that sine 30, of course, this is 150 gap, but this gap is almost 240. But you can say that from sine 30 to sine 390, this gap is consistent. So you can see that the next value that you will get will be sine of 750. Okay. So this jump is consistent. Are you getting the point? So we have to take that period as our period for which your function will repeat itself for all x in the domain of the function. So you cannot take a call on the basis of one or two values of x. The best person, the best angle to take a call here is this guy 90 degree. So at 90, it is one. The next time it becomes one is at this position. Okay. Again, next time it becomes one is this position. By the way, this position is 5 pi by 2. This position is 9 pi by 2. Okay. So how much is this jump? How much is this jump that the function is taking? The function is taking a jump of 2 pi. Correct. Now, it could have taken a jump for 4 pi also, 6 pi also, 8 pi also, 10 pi also, 12 pi also. So there could be so many, you know, these possible as they can be so many periods possible for which the sine function or a particular function may repeat itself. Now here comes a very important fact. The smallest real value of t, the smallest real value, real positive value of t is what we call as the fundamental period. As what we call as the fundamental period. Okay. So sine x has a period of 2 pi, 4 pi, 6 pi, 8 pi, 12 pi, 20 pi, 22 pi, 100 pi. You will realize that it will repeat itself after that many jumps in the value of x. But what is the smallest of them? The smallest of them is 2 pi. Okay. So 2 pi is the smallest real positive t for which the function sine will repeat itself again. So sine function will be having a fundamental period of 2 pi. Are you getting my point? So this word fundamental is sometimes dropped. So if you just say period, you should always mention the smallest value of t for which the function is repeating itself. So if somebody says, hey, what's the period of sine? He actually means what is the smallest t. Okay. Now remember t is always a positive real number. It's always a positive real number. Okay. It cannot be zero. Forget about negative. It's a positive, always a positive real number. Okay. So it's periodic and its period is, or you can say fundamental period is 2 pi. Got the point? Got the point? This is very important. Okay. Now is constant function periodic? Because constant function will remain same throughout, right? f of x equal to 2. Will you call this as a periodic function? Just as a common sense question. Okay. Are you saying yes? So what is the period for that? Zero can't be. I told you it's positive. t has to be positive real number. Okay. So yes, the answer to that is it is a periodic function, but its period is not defined. Its period is not defined. Okay. Is this fine? Can you tell me other functions which you have come across as periodic? In the bridge course only we have talked about that function. No, Cossack's, we didn't talk about Cossack's in the bridge course. Nanana, Ansh. Nanana, Arya. Fractional part. Yes, Anusha is correct. Fraction part is another periodic function, not gif. Fraction part. So fraction part is another, you can say non trigonometric or you can say algebraic function which is periodic. So it repeats itself after every, what are the fundamental period there? One. Okay. We'll talk about it. No, floor function is not periodic. Floor function is not periodic. Okay. Fractional part is periodic. Okay. Now a couple of other things that we need to know is, what are the values when sine x becomes zero? So let's do this exercise. When you think sine x becomes zero, what are the values for which sine x becomes zero? Can we find a pattern? Okay, let me take you back to the graph. What are the values of x for which you think sine x becomes a zero? I can see it becomes a zero for zero. It becomes a zero for pi, 2 pi, 3 pi, 4 pi, 5 pi, minus pi, minus 2 pi, minus 3 pi. So if you have to generalize this result, what would you say? What would you say? All multiples of pi. Absolutely. This is something which you all need to remember. We'll be using this extensively in our tignometric equations chapter. Okay. So sine x becomes zero for all values of x which are multiples of pi. Okay. Where n is an integer. This is to be remembered. Okay. It will be used so heavily that you cannot afford to forget it. Okay. When you think sine x becomes one, let's do this as an exercise. All of you, just like we have stated a pattern over here, give me what is the, you know, pattern of the angle. Okay. Of course, in terms of, you know, n, n being an integer. So that sine x becomes one. If you want, you can look at this graph and figure that out pattern. I want that is one of the angles auto is pi by 2. The only angle for this sign is one. No, right? So I told you last time, while I was talking about the periodicity, pi by 2 is there, 5 pi by 2 is there, 9 pi by 2 is there. Can you give me a formula? Can you make a formula out of it? That is what I'm trying to say. No, no, no. How are n by 2 pi? What if you put n is 2? It'll become pi, you know, sine pi is zero. How it is one? Okay. So somebody has written pi by 2 plus 2 pi to n pi. That is very, very good answer. I like that answer actually.  Is this correct? Or you can say 4n plus 1 pi by 2. Is this correct? Let's check. Let's check. Let's go back to our figure. Okay. As you can see here, the first angle where it is showing 1 is pi by 2. Next jump is 5 pi by 2. Next is 9 pi by 2. If you go further, it will be 13 pi by 2 and all. So I can see that this formula which the student has told rightly fits into this scenario because if you put n as 0, you'll end up getting this. If you put n as 1, you'll end up getting 5 pi by 2, which is this. When you put n as equal to 2, you'll end up getting 9 pi by 2, which is this. If you put a minus 1, you'll end up getting minus 3 pi by 2. Minus 3 pi by 2 is this part. That's absolutely correct. That is also 1. Okay. So I think it was... Who gave that answer? Yeah, Anusha. Absolutely correct. Okay. So this is going to be the pattern for the angle such that sine x becomes 1. By the way, these type of expressions are called general solution. What it is called? General solution. So if I say tomorrow, give me the general solution for sine x equal to 0. What will you say? X equal to n pi and being integer. Give me general solution for sine x equal to 1. What will you say? X equal to 4n plus 1 pi by 2 and being integer. Okay. This also should be there in your mind. Please remember this. I'm bubbling it for you so that you don't forget it. Is this fine? Okay. Let's play one more game. When do you think sine x becomes minus 1? Give me the general solution. That is the word I'll be using now. Give me the general solution for which sine x becomes a minus 1. Think carefully and answer. If you want, I will take you to the graph minus 1. So minus 1 I can see here. So 3 pi by 2 is the first position. Next is this 7 pi by 2. Next is 11 pi by 2. If I go back also minus pi by 2. Can you make a pattern out of this? So pi by 2 if you remove, can you make a pattern out of these numbers? Next one will be 15 pi by 2. Do you see a pattern in this number? Are these numbers recognizable? Have you seen these numbers before? It's an AP. Absolutely. It's an AP. In an AP, if you write down the nth term, what is the nth term of an AP? It's the first term plus n minus 1 into the common difference. So you can say 3 plus n minus 1 into 4. 4 is the common difference. So your answer will become 4n, 4n, 4n. How much is this? 4n minus 1, right? Isn't it? So it becomes 4n minus 1 pi by 2. So you can say for every 4n minus 1 pi by 2 value of the angle, sin x is going to throw out minus 1 at you. So here it was 4n plus 1 pi by 2. Now here it is 4n minus 1 pi by 2. So this is going to be the general solution for sin x equal to minus 1. Again, this is to be remembered. Don't forget this. Don't worry. We'll have a full-fledged subtopic on this, which is trigonometric equations. We'll come back again to these concepts. So I thought since we were talking about the graph, we'll have a good analysis of it. So I think we have done a complete bio-data of this graph. Why is this graph like this? We have understood that. What is the domain of the graph? We have understood that. What is the range of the graph? We have understood that. What is the symmetricity of the graph? What is the periodicity of the graph? And what are the special values for some angles for which sin shows certain characteristics? Can you do this for other values? Yes, we can do this for other values. That is what the entire chapter is about. So I can tell you the general solution for which sin x is half. I can give you a general solution for which sin x is equal to root 3 by 2. I can give you a general solution for which sin x is equal to minus 1 by root 2. I can definitely give you. When the right time comes, we'll talk about it. This was the beginning of the graph. It was a little slow. I was explaining everything to you, but as the time progresses, I'll be slightly faster. So without much ado, my dear students, let's move on to the graph of cos x, to the graph of cos x. So the second trig function, trig function, not trig ratio anymore. Of course it's a ratio, but we have made a function out of it. So let us see the graph of cos function. Graph of cos function, I'll be just plotting in the same diagram. So y is equal to cos x. So red graph that you can see is your cos x graph. If you see very clearly, or you can say critically, you realize that both these functions are of the same nature. Both are sinusoidal in nature. The only difference is, in one you are plotting the perpendicular versus x, in the other you are plotting the base versus x. So how will this base of that reference triangle going to fluctuate from 1 to 0, then minus 1 to 0 to minus 1? So this oscillation is what you are capturing on this graph. Another thing that is very evident is that, one of the graph, or you can say there is a phase difference between these graphs. What do you mean by phase difference? Phase is a very heavy word that I have used over here. Probably you will hear that in simple harmonic motion. As you can see, this achieves its peak at let's say 0, that is cos achieves its peak value at 0, sin achieves its peak at pi by 2. You can see this peak at these places. So there is a phase difference you can say between these two graphs of pi by 2. This is the reason for the complementary property that you had studied in class 9th or 10th. Sin 90 minus theta is cos theta. Or cos 90 minus theta is sin theta. So this property that you have learned my dear, this property is basically, you know, has come from this phase difference between these two graphs. Are you getting my point? Are you getting my point? Now, let's say I want to ask you a physics question. Okay. Siddhas has a question, if two functions are phase differences, can I say f of, do you read this kind of word phase difference in functions? Where have you read a word phase difference in functions? What is phase? It's a physics related term. Okay. No, no, no, that cannot be generalized. I'll talk about it in, you know, when we do functional equations with you in functions, but don't use that word phrase there. That would be an incorrect word. Okay. Anusha has a question, sir, but how does the graph even have a negative side because cos minus theta is cos theta only. That's a very deep question you have asked, Anusha. Cos minus theta is cos theta. That doesn't mean both of them can't be negative. Cos 120 is minus half. Cos of minus 120 is also minus half, but both are minus half, right? So why can't the graph have a negative side? Can't both negatives be equal to each other? Unsure the question. So what about if the graph is odd at some point or even at some? See, the nature of the graph is not odd and even at points. It is the generic nature of the graph. It is not a nature of the graph at a given point. If a graph is odd, it is odd throughout. That means it will be symmetrical about origin. If a graph is even, it will be even throughout. I'll talk about even as of now. We have not talked about it. It is even odd nature is not calculated at a point. It is the characteristic of the graph, right? I am, let's say, 180 centimeter tall if I'm at my house also and if I go out also. My height doesn't decrease if I go out, correct? So the function characteristic will not change with different, different points. Of course, the value of the function may change. Characteristic, why is it not changed? It is sinusoidal. You don't say it is sinusoidal at pi by 2 or it is sinusoidal at zero. Okay. Now, I would like you to ask a physics question. No, it is not necessary that a function has to be even or odd. This is not a complementary division. It can be neither also. For example, x plus x square is neither. Okay. Let me come to it when I talk about this thing, even odd function. All functions in this world need not be, you know, categorized under two buckets, right? It can be neither also getting my point. So if a graph is not symmetrical about origin, right? And if it is not symmetrical about y axis, it is neither getting the point. Okay. Let me come to my question. I had a question for all of you. I think I let me ask you that question. My question to all of you is let me hide this. Okay. Which of the two function is leading? You understand leading if there's a face difference. One is leading from the other, right? Which is leading cause is leading or sign is leading. Okay. The only two options either cause or sign. Okay. Let us say my x axis was time axis. Okay. Let's say my x axis was time axis. Okay. So this was let's say time axis. Okay. Right. Now we all know that cause of 90 degree minus T. Now I will call it as T. Okay. Let me write it as pi by two. Let me write it as pi by two. This is going to be sign of T. Right now since we know cause is unaffected even if you flip the position. If you flip T minus pi by two it will be sign T. Now everybody just have a close look at it and tell me and tell me which is the leading function. Okay. Everybody says cause. That's absolutely correct. Okay. It's very simple. What T achieves at zero. This achieves at pi by two. Right. Sorry. What what this fellow. Now let me write it like this. Let me write it like this. Okay. Now sign when you put sign T and cause T if you see sign T function and cause T function. What signs achieve what sign achieves at zero cause achieves that cause achieves that minus pi by two. Correct. Yes or no. Okay. But now many people ask me. You can also say this is cause pi by two also. Correct. That means what sign achieves at zero T equal to zero cause will achieve at T equal to pi by two. So sign is coming first right at that position sign is coming first then cause function. Correct. So wouldn't the answer be won't the answer be cause function sign function then how do you answer this? So those who are saying cause nobody would like to change your answer are you getting my point what I'm trying to say the function which is leading right that function will achieve a particular value before the other one right. Okay. So what sign achieves at zero cause may achieve at pi by two or can also achieve at minus pi by two if it is achieving at minus pi by two right then cause is leading function. But if it's achieving at pi by two it is a lagging function are you getting my what is my you know situation here. So Anusha is not saying so neither is leading. Okay. Again see this what signs what sign achieves at 30 degree. Okay. So let's say 30 degree is a you know parameter of time I make it as pi by six. Okay cause will achieve that it got cause will achieve at that at pi by three but it'll also achieve the same at minus pi by three so it is very difficult to tell which function is leading the other one are you getting my point. So please do not unnecessarily use this you know term phase here we can just say we can just say there is a you know difference in the two functions phases by 90 degree. So we will not use the word leading and lagging over here. Fine. This is just a question that no you may have in your mind. Okay. So that's why I thought I would discuss about it now. I forgot to copy the graph. Sorry. Let me go back again. I forgot to copy the graph. So let me copy the cost graph and we'll do a similar analysis on this as we did for sign. So let me just take a snapshot. Let me put it over here. Okay. Now, let us try to study this graph in more detail. The first thing that you would all notice about this graph is its domain is the same as what we had for sign. So domain is all real numbers. Okay, I'll be slightly faster range here would be again minus one to one. Okay. This graph shows symmetricity or it is symmetric about the y axis. Now people who don't know about even function, let me tell you even functions are those functions whose graph are symmetric about y axis. So this is an even function. Correct. This is a even function. Fine. Even function satisfy this functional equation. That means they are not affected by the negativity of the angle. So whether you put sign, sorry, cost 30 or cost minus 30, it will give you the same result. That means even if you change the sign of the input, output is not going to change its sign. You can see it from the graph. Let's say this is your 60 degree. Okay. Sorry. 60 degree. This value is half. So even for a minus 60 degree, it will show you a value of half. It is not going to change its value. Okay. Another thing just like your sign function. This graph is periodic. It is periodic and the period is 2 pi. Period is 2 pi. Now I'm not saying fundamental over here. It is automatically understood. It's period is 2 pi. Okay. Some things willing will inquire about this graph. The first being when you think Cossack's becomes zero. What is the general solution? Now look at this graph. Cossack's will become zero at pi by 2 at 3 pi by 2 at 5 pi by 2 minus pi by 2 minus 3 pi by 2 minus 5 pi by 2 and so on. If you generalize this, you can say it is becoming zero for odd multiples of pi by 2. For odd multiples of pi by 2. Correct? So 1 pi by 2, 3 pi by 2, 5 pi by 2, minus 1 pi by 2 and so on and so forth. Okay. So we can say here this is zero for all x equal to 2n plus 1 or you can say 2n minus 1 pi by 2. Okay. And belonging to integer. Okay. Both are fine. 2n plus 1, 2n minus 1 both are fine. This is to be remembered my dear. Don't forget this. This is one of the most important general solution that you are going to see. This is one of the most important general solutions that you are going to see. Okay. Next. When do you think Cossack's becomes a one? Let me take you to the graph. Cossack's becomes one for zero. Okay. For 2 pi. Okay. If you drag the graph a little bit or let me just zoom in a bit so that we can cover all the values. So far. Yeah. Yeah. So if you see look at the graph for zero for 2 pi for 4 pi 6 pi 8 pi minus 2 pi minus 4 pi. So what can you generalize this as even multiples of pi? Absolutely Ansh. So for all even multiples of pi, you can say 2n pi. Okay. And belonging to integers. Again, this is a result which has to be kept in mind. Okay. When does Cossack's become a minus one? Let's go back again to the graph. Let's go back again to the graph. So minus one it first becomes at pi. Okay. Then 3 pi then 5 pi then 7 pi then 9 pi minus pi minus pi minus 3 pi. So how can you generalize this? How will you generalize this odd multiples of pi? Absolutely odd multiples of pi. So for all x equal to 2n plus one or you can say 2n minus 1 pi. Okay. And belonging to integer. So these results have to be, you know, kept in mind. Is this fine? Okay. So everything is now known about Cossack's no problem with Cossack's. So without much ado, we can now move on to the graph of Tanix. Okay. Tanix function graph is basically obtained from Sinex by Cossack's. Right. Of course, we don't there's no active. There's no like way you can divide two graphs to get the answer. Of course, we plot it in this we are going to plot in case of Tanix. We basically plot the ratio of perpendicular by base. It is not as simple as your perpendicular and the base in case of sign and cost. So let me show you the graph in this case as well. Okay. So all of you please pay attention. The graph of Tanix is slightly different from the others. It is not sinusoidal. Okay. It is basically a graph like this. Okay. You can see this graph will extend indefinitely up as well as indefinitely down. Okay. And there is a big break in the function. Where at 90 degree 270 degree 450 degree minus 90 degree etc. Okay. Why is that break is because tan of these angles for example, let's have a look over here. Let's have a look over here 90 degree. Okay. At 90 degree tan is not defined. Please note this down. Don't say tan 90 degree is infinity. You should say tan 90 degree is undefined. Okay. Why does undefined is because it is sign of 90 by cost of 90 and cost of 90, you know is going to become 0 at 90. So it is an undefined expression. Never say it is infinity and all. Okay. That's the wrong way to say. If you want to say infinity minus infinity, then you should say like this tan of theta limit on theta tending to 90 minus or 90 degree minus. This is infinity. Okay. This value will go towards infinity. Okay. And similarly limit of theta tending to 90 degree plus 90 degree plus. Okay. This is going to minus infinity. As you can see the moment you cross 90 degree, the graph will zoom go down just before 90 degree will be at infinity. Okay. After 90 degree will go down to minus infinity. Okay. So you can say 89.9999 architecture. That will go towards infinity. But 90.01001 degree that will go towards minus infinity. Okay, but exactly at 10.90 the function is undefined. Okay. So you can see the range is all the way covers the entire y axis. Okay. Now this is a question that I would like you to answer. Let me take a photograph of this so that we can paste it. No need to copy anything. Everything is there in this PDF. Okay. Now, look at this graph and answer the following question that I'm going to ask you. Yeah, it resembles, but they're not the same, right? It's like horse and a mule. Both resemble, but they are not the same. Okay, their characteristic is very different. The rise in tan is very, very fast. Okay, but in the rise of x cube is not that fast. Okay. Yeah, so now everybody answer this question number one. What do you think is the domain of this function? What do you think is the domain of the function fast? Absolutely, aria. Brilliant. It is all real numbers, but not at odd multiples of pi by two. So remove a set of all points. Remove a set of all points, which are odd multiples of pi by two. Now this curly bracket means set of points. So those set of points would be removed from the domain. That means 90 will be removed. 270 will be removed. 450 will be removed. 630 will be removed. Minus 90 will be removed. All those points will be removed. So those set of points, the function is not defined for. You cannot put that as your input. Okay. Range as we can see, it is all real numbers. So I'll not waste much time. It is all real numbers. Okay. Is this graph periodic? I can definitely see this periodic. And what are the period? What are the period? Pi, absolutely. So period is pi. As you can see, after a jump of pi, its value is going to get repeated for all x's in the domain. Okay. So what do you get at pi by four? The same you will get at five pi by four. The same you will get for, you know, nine pi by four. So on. Is this fine? Even function or odd function? What is that? Even or odd? Even or odd? Is the graph symmetrical about origin or symmetrical about y axis? Is symmetrical about origin? Yes. So please write this down. It is symmetric about origin. About origin. Okay. That means it is a odd function. Okay. And that's precisely why tan followed that, you know, property which I had discussed the other day. Tan of minus theta is minus of tan theta. Okay. Just a quick question for you. What do you think is the general solution for tan x equal to zero? When do you think tan x becomes zero? Now, this is no question at all because tan x will become zero but precisely the same points where sin x will become zero because tan x is after all sin x by cos x. So yes, absolutely correct, Shatish, for all n pi. Okay. And belonging to integer. So please just remember this. Rest of the values, we are not going to talk about it right now. But yes, we are going to definitely talk about them in trig equations. Trig equations. Okay. Anything that you would like to know about this graph, I think we have known everything about it. Let's now move on to the graph of... Oh, sorry. Let's move on to the graph of the fourth function which is your cosec function. So y is equal to cosec function. Cosec we all know is the reciprocal of sin. Okay. It is the reciprocal of sin. And I'll show you the graph as well. And we'll try to understand why the graph of cosec is as it comes. Okay. This is the graph of cosec. All of you please have a good look at it. Up also I'm dragging. It is going all the way till infinity up. And down also it is going all the way till minus infinity. Okay. Now all of you have a look at the graph. I'll ask you a few questions. You know, it'll look cooler if I draw the graph of sin x along with it. They exactly fit each other. So where sin is present, cosec is not interfering except for those points where it is touching. So they are like in a fitment for each other. Okay. Now, can you explain me the reason for these long use? Yes, y-axis is a vertical asymptote. There are actually so many vertical asymptotes. Okay. Now, can somebody explain me why the graph goes to infinity comes down and again goes to infinity. Anybody? Absolutely Ansh. Absolutely. Okay. So the reason is precisely very simple. You know, cosec is equal to one by a sign. Okay. Now, of course at zero, you cannot define cosec. Okay. But the moment you come to zero plus, the moment you come to zero plus, this will become a very small positive number, right? Sign will become almost like 0.0000001. Zero plus angle, sign will also be a very small, you can say zero plus quantity. So one by zero plus is a huge number. Imagine doing one divided by 0.0000001. What will be the answer? A very large number. That's why close to zero plus, cosec graph will be almost going towards infinity. Will be going towards infinity. As you increase the value of the angle, let's say as you are going this way, right? Sign is increasing. If sign is increasing, overall this value will decrease. So this graph of cosec starts decreasing, starts falling down. Okay. So it starts, this has increases, it falls down. So both will now touch each other at pi by two. So at pi by two, both will become one one each. Okay. And after pi by two, as sign falls, again cosec will be on the rise. Okay. Till when will it be at the rise? Till sign become zero and at zero, it suddenly becomes undefined. Now the moment you cross 180 degree, the moment you cross 180 degree, let's say<PHONE_NUMBER>000001. Okay. Sign will become a very small negative value. Okay. Reciprocal of a small negative value is a large negative value. That's why down this graph will go towards minus infinity on the right side of pi. Okay. And the same trend continues. This is why this graph looks like this. One simple way to remember it is wherever you have drawn a bump sorry for using the word bump. There would be a bump over here. Okay, so the bumps are clashing with each other. Okay, that's a way to draw the graph of Coseq in case you forget, in case you forget this. So easy to forget these graphs. Is this fine? Is this fine everybody? So what I'm going to do is I'm going to just take a photograph for this. I don't need a sine graph.  Now again, we'll not stop here. We'll just do the quick analysis. Number one. What do you think is the domain of the function? So is this as simple it is going to take all your numbers but not the multiples of pipe but not the multiples of pipe because it is going to make sine X zero and if sine X become zero things will become undefined. So absolutely correct r minus n pi r minus n pi absolutely correct. Okay, what is going to be the range of this function? Look at the graph. The graph will suggest that the graph is absent. If you see this graph clearly, you'll realize that the graph is absent in this zone. There's no graph in this zone. This is no man's land. Okay, so be careful while you are telling me the range. So the range is going to be. Absolutely. So I'm just that is not correct. So I'll tell you why it is not correct. See the range is going to be minus infinity to minus one inclusive of minus one union one to infinity inclusive of one. Okay. Many many students will write this as real numbers minus this interval. But when you're when you're subtracting the interval, please ensure minus one and one are subjected to round brackets because you don't want to remove minus one and one because they should be there in your answer. Okay, so when you write a square bracket Unch, it is it is like saying that you are you are not allowing one and minus one to be taken by Cosey, but that is not the case. Okay. What is this union? Okay, don't worry union means this or this. Okay, we'll talk about it in sets chapter. So this at least you understand R minus that gap minus one doing. Okay, don't worry or we'll talk about union intersection when the right time comes. Okay. Third thing is this graph symmetric about origin or is a symmetric about y-axis? Absolutely. It is symmetric about origin. Okay, that makes this function and odd function. That makes this function is an odd function and this is precisely why Cosey of minus theta used to follow minus Cosey theta formula. Okay, is this periodic? Is this periodic? Is this periodic? Absolutely. It is periodic, right? It's periodic with a period same as that of sign. So it is periodic with 2 pi as you can see here, whatever value it gets at let's say pi by 2 the same value will get at 5 by 5 pi by 2 the same value will get at you know minus of 3 pi by 2 and so on and so forth. So it is having a period as 2 pi. Okay, we'll not talk about the general solution and all because that is something which again we are going to talk about later on. So there's no point repeating the stuff again in the other chapter. Okay, so we'll do a quick analysis now of C graph. The fifth one y is equal to seek of x. Seek of x is basically reciprocal of course and I will show you the the property here. I'll show you the graph here y is equal to seek of x and I'll put the both the graphs on C green graph and blue graph. Again, you can see that there is a phase difference between these graph of pi by 2. Okay, that is precisely why we had learned this in our class 10 Coseek 90 minus theta is seek theta and seek of 90 degree minus theta was Coseek theta because these graphs these graphs are suffering from a phase difference of pi by 2. This gap is pi by 2 gap. Is this fine? Okay, so now I will hide this and I'll show you something interesting in this as well. Even this behaves the same way as cos sign was behaving with Coseek. So I'll just draw the graph off. Okay, see that bump thing is again coming over here. Fine. Okay, so let me take a photograph of it. So same exactly same structure. No difference. They're all use extended use like this. Okay, they're all extended use but not parabola. Don't call it as a parabola. That's a completely different thing. So let me paste it over here. Let's do a quick analysis of this graph. Okay, you can refer to this graph and answer these questions. What are the domain of this function? What is going to be the domain of this function? Anybody? Yeah, hello. Can you all can you all hear me class? Can you all hear me? Okay, sorry, there was a power cut. I got disconnected. Okay now I'm back. Yeah, so what are the domain? I think most of you would have answered because I got disconnected. Absolutely correct. So all real numbers minus odd multiples of pi by two odd multiples of pi by two because you don't want your cost to become zero. The moment your cost become zero, those values are not permitted inside seek. Okay, what is going to be the range of this function? You'll say, sir, the very same range that Cosec had, which is minus infinity to minus one union one to infinity.  It just says you can come from minus infinity all the way till minus one or you can go from one to infinity that or word is written as a union word. Okay, you can also mention it as all real numbers except minus one to one interval. Okay, this function is symmetric about y axis symmetric about y axis and therefore this is an even function, right? And this is precisely why it follows this property. Cosec of minus theta is Cosec theta. That means it is not affected by the negativity of the angle. Okay, so any even function will follow this particular characteristic. Okay, fourth property. It is periodic. It is periodic and its period is period is two pi. Period is two pi. Same as that of the period of cos. So up till now, sine was periodic with period two pi, cos was periodic with period two pi, tan was periodic with period pi. Okay, so tan is pi period. Okay, again, Cosec was periodic with two pi to pi. Okay, any question regarding this? Any question regarding this so far? What is this twice? But this is pi. How it is pi? Look at the graph. Zero, the value is one and again at two pi, the value is one. So this gap is two pi, you know, next. So without much ado, we'll now talk about, sorry, we'll not talk about the graph of cot. That is the sixth and the final basic trigonometric function that we are going to study y is equal to cot of x. Cortex is basically of course one by one by tan, but it is also cos x by sine x. Okay, this will give you a brief idea about its domain and all when we are discussing it. So let me show you the domain. Let me show you the graph of Cortex y is equal to Cortex. Okay. Okay, now this graph resembles the graph of tan, but of course the orientation of this curving is different. Why does difference see? You would realize that it is because of your transformation that you had learned in your bridge course. People who have done the bridge course. Okay. This is a question for you from tan x graph. If you want to achieve Cortex graph, okay, what will you do? Of course, you know that tan of 90 degree minus theta or tan pi by two minus x is Cortex. Okay, this is a complimentary angle properties. See because of these properties, you would notice that the name has got changed. Co cost is cosine co means complimentary of sign. Right? Co seek is complimentary of seek. That's why the name co seek is given. Cort is cotangent. That is why it is called complimentary of tangent cotangent. The short form is caught. Okay. So from this particular, you know, property can we reach to the graph of cot? It's a very simple. The first thing that I would do in order to reach to this graph is all of you please watch me doing on the Geo Jebra. So I'll first hide this. I'll first hide this. Okay. Let's say I just know the graph of tan. Can I get the graph of cot from this? I'll see what I'm going to do. First of all, I'm going to change the sign of X. And we just saw the graph of Cortex. Are you want to copy something from the graph of seek X? That's what you're saying. Just a second. I'll show you the graph. The last thing that we did was. Yeah, this was the last thing that we did. We wrote down the range symmetricity and the periodicity and the periodic period of the graph. Let me know once you're done. Everybody's waiting for you then. Okay. Thanks. So let's focus on this graph. Now what I'm going to do is first step as I'm going to chain the sign of X if I change the sign of X. This is what will happen. This is precisely the reason why caught got you know tilted this way tan was tilted this way caught got tilted this way. Okay. Next what I'm going to do is of course we all know changing the sign of X means reflecting a graph about y-axis. Our people who are not there in the bridge course. We did a very very interesting concept in the bridge course called transformation of graphs. I would request all of you who were not a part of the bridge course you please get the learners log in. Okay, I think the reserve would be the right person to guide you on that and watch all the bridge course videos in your free time. So changing X with minus X reflected the graph of tan about the y-axis again, and I'm doing this reflection. I'll show you once again. Okay, just watch the nature just have a eye on the graph. Okay. I'm changing X to minus X. Okay. Okay. So DC got reflected about y-axis now. What I'm going to do is I'm going to change X with X minus pi by 2 X minus pi by 2 means the graph will get shifted pi by 2 or 90 degrees to the right. If you see this was actually the graph of Cortexy both the graphs are same. Right Cortex is now on I'm not offering Cortex and owning this both are same. You see that so basically these transformations comply with your complimentary angle properties that you have learned in your class 10th. Okay. Oh, sorry, I forgot to copy that graph. So let me copy this. So this is the graph of Cortex. So we'll do a quick analysis of this as well and we'll move on to few questions. So yes, let's answer a few questions with respect to this graph. What do you think is the what do you think is the domain of this graph domain very good. So you'll say you don't want any values which will make sine X zero. So just remove those values for which sine X is zero. It doesn't matter numerator may become zero. I don't care denominator should not become zero. Okay. So you'll say all real numbers except n pi and being a integer what are the range of this function all real numbers absolutely or you can say from minus infinity to infinity. Just like that. Just like that is this function symmetric about origin or symmetric about y axis in short. Is it an odd function or an even function? What do you think odd function? Absolutely. It is symmetric about origin. Okay. That makes this function and odd function that makes this function and odd function. That is the reason why caught of minus theta would be minus of caught theta. So if you change the sign of the input the sign of the output also gets affected. Okay. Next is this function is periodic just like tan it was it is periodic with a period of pi tan cortex is again a periodic function as you can see zero is obtained at pi by 2 it is again obtained at 3 pi by 2. Okay. So this gap is a pie gap. So it is periodic with a period of pie. Is this fine any questions with respect to these graphs fine. So now let us move on to some problem solving. So what type of problems can you expect on in a such graphical nature of these dignometric functions? So let me start with the question think and answer. Do not be in a hurry. Do not be in a hurry. The question is there on your screen right now is says which of the following is the least which of the following is the least I'm putting the poll on. Okay. Take your time once you have done it. Please press on the poll button. You don't have a calculator. You don't have a lock table with you. There's there's no option for you to know their values. What do we need to know is which of the which of them is the least that I would not tell shitish is this in radiance or degrees I will not tell I've already talked about this in the previous lesson if this is the question you're asking now means you have not understood the previous class itself. Of course it is radiance my dear degrees. If it is they'll put a degree symbol. No, what did I tell you if it is degree? They'll put that small circle over there. If it is great, they'll put a G if they put nothing it is radiance by default. Okay, so when somebody says sign one, he means sign one radiant. No problem Anusha, but you think it is that okay fine. Your answer is registered. So I think 11 of you have already voted. Okay now 13 of you have voted. So what do you think Ansh is the answer? You can type it out. Okay fine option. Okay fine. Okay one person remains to vote. One person remains to vote. Is that me? No, I don't think so. I'm the host. I'm not supposed to vote. Okay, anyways, most of you voted. I am ending the poll right now. Okay, as you can see maximum Janta has gone with D 36% of you say option D is the right option. A and C have got equal voting. Okay, and B is the least of them. Okay fine now let's discuss it. Or did I close the poll? Oh, yeah, stop the share. Let's discuss this. Now the only way you can solve this is can you all see me? Am I audible? Can you all see me? Hello. Yeah, now is proper. Okay. Okay. Again, there was a power fluctuation. Now this weather is very bad. This year itself is very bad. Yeah, sign function. So let me just draw it since all of them are positive angles. I'll not go to the negative side. I just draw this part. Okay. Now try to understand here. This is zero. This is pie pie is 3.14. Okay. This is 2 pie 2 pie is 6.28. Okay. Now sign 3 sign 3 3 will be almost here. Let me show you 3 will be almost here very close. This gap is almost point one for gap. Right. So sign 3 value will be this. This will be sign 3 value. Okay. Sign 2. See this is 1.57. This is 1.57. 2 will be somewhere between these two fellows. Correct. So 2 will be somewhere over here. Let's say roughly. Okay. Rough. It is just an estimation that I'm doing. So this value will be signed 2. Of course sign 2 is greater than sign 3. So sign 2 is out of the picture. So people who said be. Oh, why you're gone. You're out of the picture now. Okay. Now sign 1. Now one is as close to 1.57 as 2 was. So let's say one is almost here. I mean roughly speaking. Okay. So this gap will be sign 1 sign 1 is still greater than sign 3. Okay. So sign 1 will have a higher value than sign 3 sign 3 and sign 0.14 will have the same value. So 0.14 is almost here. Okay. So sign 3 and sign 0.14 radians will have the same value. One is definitely greater than 1.14. So sign 1 is still lesser greater than sign 3. So sign 3 is leading as of now. So people who said option be wrong sign 7 sign 7 means you are almost you can say 0.72 gap. This is 0.72 gap. So 0.72 gap will again be somewhere over here. Okay. Slightly less than sign 1 but still more than sign 3. So still the winner is the winner means the least value is sign 1 option a is correct option is correct. This is just playing with the estimation. We are not playing with we don't know the actual values but by just looking at their graph we know their relative positionings and by relative positionings because they are symmetrical so we can know which is more than the other not. I would not generalize this every time I cannot say but yes if you feel there is a need for example in this case how would you solve it right graph or probably you'll try to make a you know reference triangle and try to compare their perpendicular lens and all okay. So graph is obtained from the same fact. So why don't we use that correct? Okay, so sign 1 will be sorry sign 3 will be the least of these three. Okay, let's try another one. Good question to start with now. This is a multi-correct question for real values of theta which of the following is or are positive. So any real theta if you put which of the two or which of the three or which of the four will always be positive. Yes, Ansh what's your question sign 0 and sign Pi was repeating right was the period Pi does it hold for all values of the domain that defines the period right and she won't confuse period with a value some intermediate value should repeat consistently for all the values that is period. As I told you sign 30 sign 150 is also the same. So will I say 120 is the period? No, because it will fail for 90 degree. Well, it will fail for 60 degree. Okay, so does it make you say okay, got it fine. Okay, let me I'll not put the poll for this because multiple options are correct. So I can see off answers coming from shittish and Anusha so far. Oh no shittish has changed his answer. None of these are correct. Yeah, very good shittish was challenged DJ board National Testing Agency. Okay, what if I say some of them are correct? Okay, Gurman very good. Everybody should participate. Are we yes? Okay, let's discuss this this is not a rocket science. It's a very simple concept see you know that course of theta or sign of theta, whatever you talk about. You always lie between minus one to one. Okay, this is the range of these functions, right? Sine theta also no matter whatever theta you put one million one trillion minus one million one billion the range of these functions will always be minus one to one. Okay, now if I ask you the simple question cause of any angle Phi which lies in this interval. Let's say now this is your angle which lies in between minus one to one. Now when nothing is mentioned it is radiance. So cause of let me let me show you the graph of cause again. Okay, this is the graph of cause correct. So this is 1.57 this is minus 1.57. I'm just writing pi by 2 and minus pi by 2 in terms of decimals approximately. Of course, we cannot write it completely. One is here my dear almost here. Okay minus one is here. You can see cause of one and cause of minus one will have or between them all the values would be positive. Do you see this? Okay, so cause of cause theta or cause of sine theta will always be positive. We cannot say the same about sine of cost theta because sine graph is like this sine graph is like this. Okay, so let's say one is here then minus one is here. So you can see from zero to minus one. You can have negative values. So I cannot say definitely they'll always be positive. So these two are ruled out option A and D and B can only be the answer. Correct. Yeah, yeah, yeah. Absolutely. Or is this fine? Okay, this could be a simple question for you. Okay, next thing that we are going to talk about is a graph based question. Okay, I hope you can read this question. The question says the question says which of the following is or are correct. There are multiple options that can be correct. Okay. Now what I'm what I'm suggesting you is let us try to solve this option by option. So option number a do you think is this correct? Just write your answer as a dot true or false whatever you feel. Okay, then we'll discuss. Read the question very carefully. Tan X to the power Ellen. This is Ellen by the way. Ellen Ellen means log to the base e. Tan X to the power Ellen cos X is less than cortex to the power Ellen cos X for all X lying between pi by 4 to pi by 2. Is this information correct? Or is this information not correct? Take your time. Don't be in a hurry. Shrita and Aditya have replied. Okay. Take your time. Don't be in a hurry. This is a J advance question. Okay. Okay. So as of now I can see four responses and all of you have given the same answer. Okay. Can I expect answers from others or you have given up? Oh, no problem Aditya. Okay, let's discuss this guys. This let's discuss this is a very good, you know, question that we have with us. See look at the exponents. They are same in both of them. Okay. Bases are of course different. One is tan other is caught. Okay. Now in the interval 45 degree to 90 degree, which is more tan is more or caught is more. Now the answer to this can be easily obtained if you draw their graphs on the same xy coordinate system. For example, this is tan and this is caught very accurate, but I just yeah. So this is your pi by 2. This is your pi by 4. They'll all meet at pi by 4. Now which graph is at a higher level more means at a higher level, which is at a higher level only in this gap only in this gap, which is at a higher level. You'll say of course tan of course tan is at the higher level, right? So now many people just because they have the same power and this base here is higher. They think it is false, but wait a minute the game is not over. Look at the powers nature. This is basically a negative quantity by negative quantity because in this interval because when your x is in the interval pi by 4 to pi by 2, your cost 6 will be in the interval 0 to 1 by root 2. In this interval, Ellen of something is known to be negative. Ellen first becomes a zero at one. Right? So before one that is between zero to one gap. It is negative, right? So you are dealing with a negative power my dear understand this. You're leaning with a negative power. Okay. Now I'll take a dummy scenario. Let's say tan x is 3 and let's say cortex is I just take a dummy value. Let's say 2. Okay. If I raise it to a power negative half even this I pay to power negative half. I'm just taking a dummy sample example just to test myself. Which one do you think is bigger? Now this is a grave mistake that most of you have done. Actually cortex to the power Ellen cost x would be bigger. As you can see this is 1 by root 3 and this is 1 by root 2. Since root 2 is lesser than root 3 over all its value is higher than 1 by root 3. So the first statement is actually true gone.  That's why think don't be in a hurry to answer. Don't be in a hurry to answer. Okay. So even though tan x is bigger than cortex basis base is bigger for tan as compared to caught but the power is negative. If the power is negative that will change the game all together. Right 4 to the power minus 1 is more or 3 to the power minus 1 is more of course 3 to the power minus 1 is more got it. Okay, none of you said true. So everybody got a false. No worries try B1 not tell for the be I think this discussion will you know open up your mind while you are answering it. Look at B and tell me B is true or false again. LN is also there LN seek LN seek look at the interval you are dealing with 0 to pi by 4 everybody is taking time. Okay, that's good. Should all think and then answer. Okay, she did preside your answer is be dot so that I know you are answering for be okay. Okay, Aditya good she did okay. We'll see nobody else wants to answer only she did and Aditya so far. Okay, Gurman, okay, Ansh, okay, Parvati, okay, Aditya. Okay, fine. Good. I liked the way you are trying very good. Everybody is giving response as far as possible. Some of you are very consistent in answering right or wrong. It's a different thing trying is important here. See now you again see the powers are the same. Okay, but this time these powers are positive quantities. Why why because seek from 0 to pi by 4 if you just recall the graph of seek it is like this so from 0 to let's say this is pi by 4 you are from 1 to root 2 you are from 1 to root 2 so you are from 1 to root 2 so the moment you have crossed one LN of that will be positive. Okay, so let it be you know anything. Let it let's say it is you know 0.5 only. Okay, let's say this is 0.5. Now in the base in the base sign and cost which is more of course costs will come down like this sign will go up like this so they meet at pi by 4. So from 0 to pi by 4 cost is more. Okay, so the base here is more. Okay, so you tell me when I'm talking about let's say x to the power half. Okay, the more is the x the more is the value, right? Yes or no yes or no. Just try to just try to recall the graph of root x. I'm just giving a dummy example. This is a graph of root x the more is the value of x the more is the value of x to the power half. It doesn't matter even if this is less than one because the increasing function. It's an increasing function. Yeah, sure. Yeah, thank you so much. Okay, so when you say sign x is greater than cost x sorry sign x to the power ln of ckx is greater than this is a false statement because cost x has a higher value. If you don't believe me, we can just plot it also and check. Let me just go and plot it. Okay, so y is equal to sign of x to the power of ln ckx ln ckx. Okay, and the other is y is equal to cost. This we have to put on the bracket as a whole. I think that takes that only anyways. Yeah, so cost cost x to the power of ln ckx. Okay, now it's a very very close call. I'll just put zero and pi by four. So x equal to pi by four. Okay, if you see this interval, I'm just zooming in. I'm just zooming in now. This was the orange one was the graph of sign. If I'm not wrong. Yeah, which one do you see is higher? I can see cost x to the power ln x been definitely higher. You just see this upper part. I'm zooming in more. I'm zooming in more. Now see this everybody this guy is at a higher level. This is higher. Higher means it is more correct. This is pi by four and this is zero, right? So this guy is the winner. This guy is higher, right? And what does your option say? Your option says sin x is higher, which is wrong. So so this is the false option. This is a false option. I got it. Can you plot it Anusha? If you can plot it, then take a bow. If you can plot it, then I mean nothing like it, nothing in comparison to that, right? But can you plot it? That's the million dollar question. I cannot sign x to the power ln ck.  If we can, we can definitely go for the plotting. But the problem is we cannot plot it. I can just verify it on the geojibra, but you will not have a geojibra in your, you know, accommodative exams, right? Okay, if you know a bit of calculus per litme, it may help you calculus may help you. We'll talk about it when the right time comes when we learn calculus, when you learn about the increasing decreasing nature of functions. Meanwhile, see what do you think for C? Is it true or false? Just write your answer as C dot. It is not by the slope that we judge. It is by the positioning. One was below and the other was above. So what is above is higher, right? What is below is lower. C, what do you think? True or false? Now, remember the bases are same. Powers are different. Okay. Again, there's a mix of spawns coming from the class, but most of you are saying true. Okay. Oh, that should not, that should not influence your, your thought process. Okay, Priyam, Shitesh, Parvati, Oshek, Aditya. Very good. Now, this is a sitter. This is actually a sitter. You should not drop it. Seek pi by three is to everybody knows it. Okay. Now this guy, this guy is a positive number. Okay, let's say 0.5. Okay, but this guy is a negative number. Let's say minus 0.5. Let's say I'm just taking a rough estimation, which is more. Of course, this is more. Correct. In other words, if you take graph of two to the power x as your reference point, it's an increasing function. If I put more input, then my output will also be more. So what I get for putting minus 0.5 will be lesser than what I get for putting 0.5. So more the x value. So this is basically two to the power x graph. We have brought it more. The x value more will be the value of two to the power x. That's that's very obvious. We all know the exponential function graph. So this option is a true option. Okay, so here you realize that how surely the question setter has mixed the basic idea of your graph along with your trigonometric ratios along with your trigonometric functions. D1 is it true or false? Again, put your answer as D dot true or D dot false. Okay, Parvati very good. Okay, again, this is a one second answer. Power is negative. Of course, powers are same also, but they're negative also, right? When powers are negative. The one with the least base will have the higher value. So half and three by four, half is the least of them. Half is the lower, you know, lesser of them. So this will have more value. So this statement is again a true statement. Why false? Why false? Any question here? When the power is negative, you know the one with the least base will have a more value. Okay, no problem. So this is a very good exercise that we had. Now we are moving towards. Now we are moving towards supplementary and complementary angle properties. Now we are moving towards supplementary and complementary angle properties. Then we'll take a break. Complementary. And supplementary angle properties or supplementary. Angle identities now we all have learned in our class 9th and 10th that sign 90 minus theta is cost theta cost 90 minus theta is sin theta tan 90 minus theta is got theta. That's the main reason why these trigonometric ratios were called complementary to each other caught 90 minus theta is tan theta go seek 90 degree minus theta is seek theta and seek 90 minus theta was go seek theta. This is what we had learned in our class 9th and 10th. Okay. Now what I'm going to do here is I'm going to scale up on these properties. I'm going to scale up on these properties and I'm now going to ask you the next question which think very carefully and answer. What if I ask you these following set of questions? What do you think is sign 90 plus theta and cost 90 plus theta tan 90 plus theta caught 90 plus theta go seek 90 plus theta don't answer right now. We'll take up one by one. Okay. What do you think is sign 90 plus theta in terms of theta in terms of theta don't say sin 90 plus theta sin 90 plus theta only in terms of theta only. Okay. Of course something romantic ratio and theta. What is the answer in this case? What do you think and how do you get this now? Let me tell you it is not as the previous class exercise that we did because if you want to know this value from your previous class experience, you would need to know your theta here. So we can't know the quadrant because we don't know theta. Right when you don't know theta, you don't know the quadrant hence you don't know the sign. Correct and you also don't know the reference angle. So how do you solve such questions? It is just a theta in general. It can be anything it can be 100 degrees. It can be 5000 degrees. It can be 2 million degrees. Right. So how do you solve such questions minus pie is not as I told you Anusha period can never be a negative number. You can subtract a period from it. Right, but period is always 2 pie. We never say period is minus pie but period of sin is 2 pie. So how would 180 degree lead to this? Shruthi how how will how did you get that figure? Then theta is 90 minus and then okay. You have you have actually caught your ear like this.  Many people did not see what I wrote over here. Identity this word itself speaks volumes. Identity means you can play with that angle theta. So you can do anything to that theta identity means this is like a box here. Whatever you want to put in this box. You have to put the same thing here in place of theta. If I want to put 1 million degree also. I can put I can put minus theta also. If I want I can put 5 theta also if I want this is the meaning of identity now correct. So if you change your theta with minus theta in all the left hand side expressions whatever is the right side is what is your answer so your right hand side only cost minus theta cost minus theta is just cost theta. So answer is cost theta for this nothing else. Okay, so by this now you can answer all of them. So let's answer quickly. So the next one will be minus sign theta absolutely this one will be minus cot theta. Now remember your negative angle identity that will help you out over here. This will be minus tan theta. This will be again seek theta. This will be again minus go seek theta. Okay, is this fine? Okay, let me scale this up. Let me scale this up. Let's see whether we are able to tackle that. What about if I just include 180 minus theta? Okay, cost 180 minus theta now that you know the trick you should be able to tell me fast. Okay, the faster you tell me the faster we can have a break. I'm sure most of you would be hungry also one simple trick you have to tell me from where we can get the answer don't have to you know, I give me the answer for all of them. What is that simple trick that you will follow by which we can get the answer for this pretty fast pretty fast hint is lying in the second set of identities. I mean just opening it up for you to use anything you want. What would you do? What is that? Anusha, I didn't get your question. Replace theta with theta. Yes, we can do that. Why not? But is it helping you? Is this does this help you? No, no, no, no, no. No sign to theta is not to sign data. Definitely not in that case sign 60 would have been 2 into sign 30 so sign 60 would have been one. It is not right. Okay. Yes. Who will tell me a simple act of replacement?   It's not striking you. Absolutely good man. That's the answer. I was looking for in this theta my dear students if you replace this theta with a 90 degree minus theta won't it become 90 plus 90 minus theta? Won't it become 180 minus theta? So this series can be taken up care by this. So it'll become cause of 90 minus theta, which is sign theta back. Okay. You can do the same for others. I'll I'm just quickly writing down the results for you. This will be minus cause theta. This will a minus tan theta. This is just the reciprocal of the first one.  Okay, this is the reciprocal of the second one. So minus c theta and this is the reciprocal of the third one. So minus cot theta. Okay, so you don't have to remember anything. Everything is coming from retrospections. Nectar inspection means whatever you learned in the past is helping you to solve the present. You don't it's not like a rocket science. Okay. Now I can continue with this exercise on and on and on and on. Right. I can give you now 180 plus theta.  But till how much will I go? There is some, you know, limitation where I have to stop myself. Right. I can't sit. I cannot, you know, tell you everything. So I would not tell you a trick here to get the answer very fast. It's just a trick. Okay. Don't apply logic to this trick. I'm just giving you a trick to get these answer fast. See the problem here that is arising is we don't know theta. We don't know in which quadrant it lies. Right. It's when nothing is known about theta. There is very much uncertainty about the sign and the expression both sign in the magnitude both. Okay. If theta is known to you. There is nothing to worry. Then you can apply your previous class concept and sail through right. But the problem here is boss theta is unknown. Theta got the law. What is theta? How will I solve this? Okay. So how do I crack such problems? So for that I have figured out a simple trick. Okay. And let's say even if you're going by this approach, then you have to always rely on the previous result to see the next. So let's say if I asked you this question, what is sign of 630 degree plus theta? How will you answer this? Then will you start your journey from sign 90 minus theta and make your way up to sign 60 plus theta that would be like, you know, half an hour you'll waste there. Right. So how do you solve such questions? I'll solve I'll tell you a trick for that trick is very simple. All of you please listen to this. Don't apply logic to it. Okay. Assume your theta to be some acute angle. Assume it to be an acute angle. Now here itself. Most people will stop me sir. Theta can be anything. Why acute it can be it's just a trick just a trick. So calm down just a trick. Okay. Just assume it to be acute. Else you'll not be able to figure out anything. Okay. So just a trick or you can say a fast way to get to the answer. Okay. Assume it to be acute. Let's say 60 degree. Okay. Say 60 degree. Now you tell me 366 30 plus 60. Of course 690 which quadrant are you in? Which quadrant are you in? Fast fast fast quadrant. You should tell me fast. 690 which quadrant 4th quadrant correction. So you're in the 4th quadrant in 4th quadrant sign is negative. Right. So put a negative here. Half the work is done. Okay. Put negative here. Okay. Now listen. 630. 630 is an odd multiple of 90 degree. It is basically 7 into 90 right odd multiple. If this is odd then make the complementary of sign and right over here. So complimentary of sign is cause cause of what of this angle theta. So this is going to be an answer. Done got the point. Okay. I'll apply it to this one the one which you have just now seen let's say I want to find out what is tan 180 minus theta. Okay. You already know the answer so you can verify it. So first of all I will assume theta to be acute. Let's say this is some acute angle. Let's say 30 degree. So 150 degree will lie in the second quadrant. Second quadrant tan is negative. Now this is 2 into 90. Now remember sorry. I forgot to tell you this when if this is even then you have to retain the same tignometric ratio. So it will remain tan of this angle theta same simple. So minus tan theta is an answer. Don't include minus here. Okay, this minus is not to be included inside only theta has to be included. Okay, so this is a working rule which I have told you if you follow this you will never make a mistake. If theta is known nothing like that, right? I'll give more examples Aditya. If theta is known then you follow your first class method that the method which we discussed in the previous class but if theta is not known, okay, it's an identity or kind of the thing that you want to figure out. Then you can follow this. Okay, let me give you one more scenario. Let's say go seek off. Let me give you 810 degree plus theta. Okay, fine. So see how will I tackle this? First of all, I'll take theta to be some acute angle. Okay, let's say 45 degrees anything anything you want. It's your call now 845 sorry 855 will lie in which quadrant? Tell me fast 855 will lie in which quadrant? Second quadrant very good second quadrant in second quadrant. We know we know Cossack is positive. So write a positive. Okay now now this fellow 810 is 9 into 90. So what is 9 odd number? Correct. So what is the complimentary of Cossack seek? So your answer will become seek theta done over. Isn't it very cool? No, so it saves a lot of time for you. You don't have to start your journey from 90 minus theta and then you know, keep moving forward and you can test it out. It'll work. It is definitely going to work. Right? So if you if somebody asked you what is Cossack 855 degree? Okay, let's let's do this as an exercise Cossack 855 degree. You know, it's the second quadrant. So positive. What are the reference angle over here? Okay, let me now go back to my first class approach. What is my reference angle here? So for reference angle I have to see as a nearest multiple of 180 degree so 855 degree if I'm not wrong. It is 5 into 180 degree minus 45. Okay, so 45 is the reference angle. So the answer will be same as Cossack 45. Okay, so your answer would be the same as the answer that you have got over here seek 45 and Cossack 45 have the same values 1 by root 2 1 by root getting the point. Okay, let's say I taken 30 degrees. Let's say I taken 30 degrees. So let's say I want to figure out what is Cossack 840. What is Cossack 840 as per my identity? My answer should be seek 30 seek 30 is to correct. Let me figure out from here. So in this case 830 could be written as 5 into 180 degree minus 60 degree correct. If I'm not wrong 840 sorry 840 my bad 840. Okay, so your answer would be of course positive sign but positive Cossack 60 positive Cossack 60. Oh, sorry. This is this is not 30. This is root 2 by root 3 2 by root my mistake. So this will be also 2 by root. Okay, both answers would be the same. Both answers would be the same. Is this fine?  This is not a rocket science. They come from the reference angle point of view. Okay. So when you are dealing with an odd multiple of 90 degree, you are always in these positions this position or this position. Correct. So your reference angle always looks like this. Right. So what is sign becomes cost? What is cost becomes sign? What is seek becomes Cossack? What is Cossack becomes seek? What is caught becomes tan? What is tan becomes caught? Right. So it is basically that these two triangles are similar. See whatever is this angle. This is the same angle. Okay, so whenever you are realizing that you are dealing with an odd multiple of 5 by 2 your reference triangle will be basically I know attached with the y-axis. That's why these complimentary switching happens. That is why this complimentary switching will happen. Is this fine? So this working rule. Everybody's clear. Let's take few questions. Let's take few questions. Let's take this question. This is a proof that question. So I think just tell me done. Yeah, I'll give it. Unch. Let me just have few questions then we'll take a break. Here you can follow any of the two methods that you have learned you have already learned a trig ratios for no any angle given to you. Of course, it should be co-terminal and allied to your known angles. So you may treat this as minus sign for 420. Okay, you can first apply your basic negative angle identities. This will remain as good as cost 660 and this is sign 330. This is your LHS just I've done once you're done now. You can treat as if you have written sign 360 plus 60. Okay, this you can treat it as if cost 360 plus 90. This you can treat it as cost 720 minus 60. This you can treat it as sign 360 minus 30. Okay, then you can apply it to your complimentary and supplementary angle properties. So sign 360 plus 60. Of course, this is even multiple of 90. So it will remain minus sign of 60 degrees. Okay, again, this is even multiple all of them. I think are even multiple of 90. So you will retain the same trigonometric ratio for this angle. But be careful about the sign. Okay, be care. Sorry, sorry, I made a mistake here. This is 30 slip of pen. Okay. So you are in the first quadrant and this is even multiple of 90 so cost of 30 again. This is in your fourth quadrant even multiple of 90. So it will remain cost 60. This is again fourth quadrant, but sign is negative there. So you'll put a negative over here. So this will become a negative value. Okay, and it will be negative sign 30. Okay, putting the values it becomes minus root 3 by 2 into root 3 by 2 minus half into half. So this is minus 3 by 4 minus 1 by 4 answer is minus 1. Okay, that's how you get your RHS. Okay, now we'll take a break. I think most of you would need a break now. Let's break now and we'll resume at 626 p.m. Is that fine? 626 p.m. All right. So the next thing that we're going to talk about are your identities. Now we have actually started with the main part which is slightly complicated. You would have already done identities in class 10th, right? So I'll just test you on your basic identities. Now identities are a very, very big concept in trigonometry. You learn about compound angle identities where more than one angle is involved. You'll talk about multiple, sub-multiple angle identities. We'll talk about conditional identities. There's so many things which are involved with identities in trigonometry for you in class 11th. So I'm going to just begin the session with your basic identities. I want to test whether you all are, you know, absolutely fine with those identities or not. So the parent of all identity is this Pythagorean identity, right? And they are derived Pythagorean identities. So when you divide it by let's say cost square, you end up getting 1 plus tan square x is equal to cx square. This is another identity. All of you are aware of it. And if you divide the first one by sine square, you end up getting cot square plus one is equal to cosec square. Okay, so this is what you had already done in your class 10th class 10th basic identities, right? Now I would begin the session by asking some questions to you just to see whether you are fine with those identities or not. Okay, just to test you whether your understanding is proper with respect to these identities. So let me just ask you some questions. Okay, let me start with this question. It's a very simple question. Okay, question is tan theta plus seek theta is 1.5. Find the value of sine theta tan theta and seek theta. Just let me know once you're done. If tan theta plus seek theta is 1.5 find the value of sine theta tan theta and seek theta individually. Okay, wonderful. Okay, Anusha. Oro very good. Okay, let's discuss this is a very easy question. I think everybody should get the answer for this. Now, if you know tan theta plus seek theta. One thing for sure we all know from the second identity is seek in square theta minus tan square theta is 1. Okay, this actually can be factorized as seek theta plus tan theta times seek theta minus tan theta correct. Now this value is already 1.5. So this is 3 by 2 so 3 by 2 seek theta minus tan theta is equal to 1 which means seek theta minus tan theta is 2 upon 3. Now, this is let's say your first equation and this is let's say your second equation. Can we use these two results to find the value of the other trigonometric ratios? Of course, let us let me write that down also along with the same term. Okay, let's add them. Let's add them. If you add them you get to seek theta is equal to 2 by 3 plus 3 by 2. If I'm not mistaken, that's going to be 13 by 6. So seek of theta is going to be 13 by 12. All of you have got this value. Excellent very good very good. So seek theta is 13 by 12. We can use any one of them to get the value of tan theta. So tan of theta would be nothing but 2 by 3 sorry 3 by 2 minus seek theta 3 by 2 minus seek theta minus seek theta will be this. Correct? Right? So what does this give you 36 minus 26 that is 10 by 24 that is 5 by 12 so tan theta is 5 5 by 12. Okay. Now all of you please understand here the moment you are getting tan theta as a positive quantity and seek theta also as a positive quantity. It clearly indicates that your theta is in the first quadrant, right? If it is in the if it is not in the first quadrant, you cannot get both of them positive. There's no other quadrant where both of them are positive together. Okay. Now why I'm saying this is because I've seen many people that disregard the sign that disregard the that quadrant in which your angle lies now to get the value of let's say sin theta many people will do this. So sin theta what is sin theta sin theta in this case would be tan theta into seek theta correct. Sorry tan theta by seek theta. Sorry my mistake tan theta by seek theta correct. So your answer will be 5 by 12 5 by 12 divided by 13 by 12. Okay, but however, I've seen many people doing this way. They'll start with the formula of Coseq square theta. Coseq square theta is 1 plus cot square theta. Okay, that is absolutely fine. So you can use this formula and cot square will be 12 square by 5 square. So if you just take the LCM will get 13 square by 5 square. Okay. Now when Coseq square theta is 13 by 5, let me write it 13 by 5 the whole square there could be two possible values of Coseq theta one could be 13 by 5 other could be minus 13 by 5 many people simply disregard this that leads to a faulty result that leads to a faulty result here. You are safe because you're dealing with first quadrant. Okay, many people ask what is cost theta cost theta is sir under root of 1 minus sign square theta that means you're always saying cost theta is positive always cost theta is positive because if it is under root of something, it must always be positive right? Is this the right way to say you are actually confusing between these two expressions. They are not the same idea. So it could give you both plus minus you could give you both plus minus. Yes, that is the right way to deal with it people always is that cost theta is under root 1 minus sign square theta under root. See if you're saying that my dear you are labeling that cost theta can never be negative. Cost it I can never be negative. That's the wrong thing to know state. Okay, so in this case you are safe because theta is in the first quadrant. So from here we can get sign theta is equal to 3 by sorry 5 by 13 5 by 30. Okay, so this this process is good enough, but if you're using this process be aware of the quadrant. Okay, so sign theta is known seek theta is known. So cost theta will also be known 12 by 13 not issue. Any questions here? Let's take another one. I hope you can all see the question. Okay, so this is a single option correct question. I would run the poll for this. Okay, please mark the poll after you have solved it. Take your time. Don't be in a hurry.  I hope you can all see the poll right? Okay, then in that case Parvati, you can write down the answer on the chat box itself privately to me. Okay, three of you have already voted. Okay, Anusha. Okay, how many boards exam how many board exams are pending or for you for others? Oh, okay. I think Raja is going to have it on June 15th or something. I'm not very sure others are having to have it in July.  You are having at home. Yeah, no school and all only sent them class. You're doing because it is in the evening. You're saying it or because of the lockdown. But nowadays I'm I'm seeing you know children have started coming out in the common area playing and all. Yeah, a few grounds have opened up. Correct. That's correct. Yes, I can see eight of you have voted. Please please fast. Two and a half minutes have already gone in another 20 seconds. I will close the poll 20 seconds. Okay at the count of 5 now I'll close 5 4 3 2 1. Okay. So maximum Janta has voted for option B. Okay, and then the next highest has gone to a. Okay, so let's see whether Janta is correct or not. Now this is a expression which you would have seen million number of times in your class 10th. So there is a stereotypical approach for this. What do we do in this case inside the bracket? We multiply and divide with this. Okay, same thing will do or in fact, you can say similar thing will do with this we will multiply and divided 1 plus cos alpha. Now this will lead to 1 minus cos alpha the whole square and lead to sine square alpha in the denominator. Okay, so you can just say. You can just say it is basically square of this under root. Okay in a similar way here, you can also say it is 1 plus cos alpha by sine alpha square under root. Okay, now remember my dear students. This is from the bridge course again. Okay, just recall. Not very far away when we did this concept under root of X square is actually mod X people who have a habit of cancelling square and square root. This is for you when you're when you're doing a square root of a positive of a square your answer will always come out to be mod X is because square root will always give out the principal root that is a positive value. So even if you're doing even if you're doing this operation, your answer will come out to be mod of minus 4 that is for itself. It will not come out as minus 4. Okay, so get rid of this habit of cancellation of square root with square.  Okay now mod has a property that mod of X by Y is mod of X by mod Y mod of X by Y is same as mod X by mod Y. So you can write it as mod of 1 minus cos alpha. Let me write equal to equal to here. Okay by mod of sine alpha similarly, this will be mod of 1 plus cos alpha by mod of sine alpha. Okay now 1 minus cos alpha is a greater than equal to 0 quantity similarly 1 plus cos alpha is also greater than equal to 0 quantity correct minimum value is 0. It can always be 0 or more than 0. So this mod becomes irrelevant for the numerators.  Remember mod is irrelevant for already positive quantity. So mod of X when X is positive is X itself. Okay, but what about sine alpha? Dear students, I would like to bring to your notice this fact. If you have if you would have ignored this fact, then you will get a wrong answer. This means you are in the third quadrant in third quadrant sine alpha is negative. So mod of a negative quantity as we all have seen in the bridge course will come out to be minus of that quantity. So if this is a mistake which you have done, you will end up getting or if you've ignored this, you will get end up getting option a so people with option a you are wrong. Okay, anyway, let me complete it.  So the answer will be minus 2 by sine alpha. That's option B and not a okay. So Janta was correct actually, but still many of you gave a wrong answer to this. Many of you gave a wrong answer to this. Okay, be aware of such things. How third sir it is given no pi to 3 pi by 2 inch 182 to 270. Where are you when you're looking at 180 to 270 interval? Third quadrant know. Correct. Unch next question. Okay, let's see who is able to answer this again. I'm just testing your class 10th. I have not this is not class 12 11 stuff, but this is a prerequisite for this. So if you are not doing well in this fairly, it'll create a problem for you. So I'm putting the pole on for you.  Oh, okay. Okay. Take your time. No hurry. Let's say three minutes have been awarded for this. If you're not able to see the pole. No worries. Don't panic. Just put it on the chat box, whichever option you think is correct. Okay, somebody has answered. Nice. By the way, June mid would be your first monthly test. Okay. Topics will be whatever we have covered till date, including the bridge course topics and official communication will be sent to you regarding that along with the syllabus. It would be on the learners platform. I think learners platform is already known to you. You have already taken tests in class 10th on that. Yes, yes for maths. I will include everything graphs limits derivatives bit of integration. Technometry whatever has been done so far. Those all would be included J main pattern. That means 75 questions. Three hours would be allotted for it. Five questions towards the end of every subject would be integer type. You have to put some integer value. Only two of you have responded so far. Come on. Come on. Buck up students. Please fast. Good Oshik very good. Correct Parvati. Okay in another one minute. I'm going to stop the poll last 20 seconds. Last 20 seconds last five. Okay five four three two one stop. Okay. So nine of you sorry 10 of you voted. This is the result that you have given me mostly Janta is saying C option. Okay, that means it will remain X. Okay, let's check. See I can see this type of expression on the top. Okay and a similar expression is there here also. Okay, so what I'm going to do is I'm just going to shuffle. I'm going to produce this kind of a term that is one plus sign theta minus cost it over here. So let me do one thing. Just a small internal change. I'm making in the expression. I'm just clubbing these two together and I'm keeping cost theta separate. So now what I'm going to do is I'm going to multiply with one plus sign theta minus cost theta. Both in the numerator and denominator. Okay, so my attempt is to create this term on the numerator. Okay, so let's see what happens in the denominator. You can see it is something like all of you please see here. It is something like a plus B into a minus B. So that would become a square minus B square a square minus B square right now. What is this cost square cost square is again one minus sign square sorry for making a change here itself. So that I have to write less. Okay, and one minus sign square is actually one minus sign into one plus sign. Let me do this thing. Let me take one plus sign theta is common. So I end up getting one plus sign theta minus this which is minus one plus sign theta now you can see that one and minus one will cancel off sign theta and sign theta will make to sign theta which will cancel off from here. So this entire thing is gone. So you're left with one plus sign theta minus cost theta by one plus sign theta which is actually your same expression that you wanted to find out. So this expression is same as that of X. So option C is the right option option C is the right option. Now if I were you I would solve this question by putting a special value of theta because I wanted to see what it is coming up to. So if I were to solve this question, I would probably put theta as 90 degree or something. Okay, so if I put this put theta as 90 degree, but they also be careful. They also be careful because you should be able to distinguish between the options. Okay, 90 degree or probably you know, you can put 0 degree or 30 degree, whichever you think will conveniently get to the figure. So in your examination condition, do not solve it in a rigorous way. Do not solve it in a rigorous way. Okay for learning go with a rigorous way. Right? It's like when you are it's like when you are practicing batting right? Of course, you'll play all the you know short switches in the copy book right? Cover drive flake, whatever, but when you are in the ground, you have to score runs anyhow. Okay, you don't have to always you know, put a cover drive for a 4. So the best way to solve this question is smart Substitutions for theta period if sin theta plus cost theta is 1 by 5 and theta lies between 0 to pi then tan theta is people options may be correct. Multiple options may be correct tomorrow early morning a 9 to 10 30 would be KVP by session on number theory. Those who are interested please join that session very short session of one and a half hours. Number theory is a big big concept right? We cannot do of course in one and a half hours. I will have multiple sessions on number theory. So tomorrow is the first one. Please type out your response on the chat box. Okay, Aditi. Okay, Siddhich. Okay, Gurman. Okay. Okay, Anusha. Others. What do you think? Oshik Pradyun Priyama Rubav Shanmukh Arabi Parvati. Shiddhich has changed his answer. Okay, Shiddhich acknowledged. Okay, last 30 seconds. Please wrap this up. Then we'll discuss it. Okay, let's discuss this. So the first thing that we have in front of us is sin theta plus cost theta is 1 by 5. Okay. So what I can do is I can bring this 5 to the left inside. Fine. Let me square both the sides. Okay. So when you square both the sides, you end up getting 25 sin square theta. You get 25 cos square theta. And you get 50 sin theta cost theta. And remember this one square is actually sin square theta plus cos square theta. So you can make use of trigonometric identity there also. So this one square, I wrote it up as sin square plus cos square. Now bring it to the other side. You'll end up getting 24 sin square theta plus 50 sin theta cost theta plus 24 cos square theta equal to zero. Okay. Now divide this by cos square theta throughout and divided by in fact 2 cos square theta throughout. So you'll end up getting 12 tan square theta. Plus 25 tan theta plus again 12 equal to zero. Right. Is this factorizable 144 144 is 16 into 9. Yeah, so I can write this as 16 tan theta plus 9 tan theta plus 12 equal to zero. If I take a 4 tan theta common here, I'll end up getting 3 tan theta plus 4 and here also if I take 3 common, I'll get 3 tan theta plus 4 equal to zero. So I end up getting 4 tan theta plus 3 times 3 tan theta plus 4 equal to zero. Okay. Now this leads to two conclusion either tan theta is equal to minus 3 by 4 or tan theta is equal to minus 4 by 3. Okay.  Now wait a minute. Wait a minute guys. It's a very important thing that I would like to tell you over here at this stage. You have squared both the sides the moment you do that there is actually no distinction between you solving this. There is actually no distinction between you solving this and you solving for this both are covered under this guy. Correct. So there's a possibility that you have introduced extra roots into the system doesn't mean I should not square it. No, it doesn't mean that when there is a necessity to square it you cannot bypass it you have to square it, but when you are squaring it you you have to bear the consequences of it. The consequences you will introduce extra roots into the system. So when I am solving a question by squaring both the sides or by cubing both the sides or whatever power I'm raising on both the sides I need to check whether any extra roots what is what what is called as extraneous roots has any extraneous route kept into my system. That is what we need to verify and rule out. Let's check minus 3 by 4 first is it an extra root? Is it a root that belongs to this guy and not the original one? Right? Let's check now tan theta is minus 3 by 4 sin theta will be equal to 3 by 5 why not minus 3 by 5 why not minus 3 by 5. This information will help us you are in the first and second quadrant only where sign is positive in both first and second sign is positive. Okay, but cost theta would be minus 4 by 5. If you add it my dear you will get minus 1 by 5. So as I guessed this is an extra root. So this is to be rejected. So people who marked be just because I said there is multiple option correct. You got trapped actually it has only one answer and that is minus 4 by 3. Still I'm let me know confirm this as well. So when tan theta is minus 4 by 3 sin theta is going to be 4 by 5 and cost theta is going to be minus 3 by 5 because you are in the first and the second quadrant only. So sin has to be positive but cost has to be negative. Now when you add this you'll get 1 by 5. So that is fitting in our original equation. So option a is the only right option. Option a is the only right option.  Is that fine? So never been a hurry to answer. Okay. Okay, let's take a few more questions. I had few questions here. Let's let's solve this. Let's take this up find the value of this expression value means value. Don't give me an expression itself. It should have some numeric value. So whenever any question says find the value, it means some numeric value assigned to it. That's fast. Very good. Yes, yes, yes. Unch. It is 90 degree only. Okay, Auro. So as of now shitage and Auro has and have answered. Let's see. Okay, Unch. Very good, Gurman. Okay, Pradin. Okay, so most of you have tried to answer few of you are correct. Few of you are not. See, look at this angle and this angle. What do they add up to? Any person can guess they add up to give you pi by 2. Correct. So cos square 7 pi by 16 is as good as sine square pi by 16. Correct. Okay, complimentary angle property. Right. So what is cos square? It just means square of this term. Correct. So this term inside could be written as cos or sine 90 minus this. So sine 90 minus 7 pi by 16 square, right? So that is actually this term. Okay, in the same way this term, let me write it over here. In the same way this term could be written as sine square 3 pi by 16. Okay, so this plus this will become a 1, this plus this will become a 1. That means your answer for this question is 2. The answer for this question is 2. Okay, not 0, not 1, it's 2. Sorry, I didn't get you. See, cos theta is sine 90 minus theta. That's what I used. So this is your theta. Okay. So I wrote it as sine 90 degree minus theta. So that will give you sine pi by 16. So this will be as good as sine square pi by 16. Cos 2 pi by 16 common. Why? How do you take that common, Auro? Cos square you are taking common. So if I take cos square pi by 16 common, what will it help me with? No, I didn't get you. You are saying we have to do this. Can you unmute yourself and say that? I don't get how taking cos square pi by 16 is going to help you in any way.  Okay. See, most of you are applying your basic, you know, taking common and all. See, cos square is an operator. It is not a term. It's an operator. Operator cannot be taken common. Getting my point. So I cannot say sine 30 plus sine 60. Okay. Take sine common and it will become 30 plus 60. Okay. This doesn't work. Absolute absolute wrong thing. Okay. Don't do all these things. Okay. Next question. I hope you can read this question. Let me put the image down. This is super easy question. I think you would have done questions of this nature. See, she did give me the answer. The moment you look at it, probably you will be able to get the answer. Very good. This is Sai. Sai Sai. Okay. Very easy. Most of you have given the answer to this. See, if you multiply, let's say this expression is equal to some value K. Okay. If you multiply K with this given expression to you. Okay. Which is actually equal to tan theta tan Phi tan Sai. Okay. You realize that this into this will produce a one. This into this will produce a one. And this into this will produce a one. So you end up getting a one. That means K value is cot theta, cot Phi, cot Sai. So option number A becomes the right option. Very simple question. Is this fine? Any questions here? Okay. Now, let us start with the concept of compound angle identities. Compound angle identities. First of all, what is the meaning of a compound angle? That is something which we need to understand. Compound angle is nothing but more than two angle involved in sense of addition or subtraction is a compound angle. A simple example will be A plus B. It's a compound angle. It has got two angles in it. Okay. You could have A minus B also. You could have A plus B plus C also. You could have different different signs also. Let's say A minus B plus C. Okay. So these are examples of compound angles. So these are compound angles. So in simple terms, if more than one angle is added, subtracted or a mix of operations between them, you would call them as a compound angle. Okay. So now we are going to learn. Now we are going to learn some compound angle identities. Okay. So I am going to start with the derivation of this compound angle identity cos of A plus B. Okay. Now there are several ways to prove it. There are several ways to prove it. Now I will be taking that way which has been prescribed in your NCRT syllabus. Okay. What is that way? Let me make a diagram for you. Okay. All of you please look at this diagram. Let's say there is a circle. It's a unit circle. It's a unit circle. Okay. On this unit circle, I take an angle A. Let's say this is angle A. Now if this is one, of course, this point will also be 1, 0. Now who will tell me what is the coordinate of this point? Let me call this as P for the time being. What is the coordinate of this point P? Can somebody tell me what is the coordinate of that point P? Absolutely correct. That will be cos A, sin A. No doubt about it. Okay. Now from this A, I increase an angle B and I reach a point Q. Who will tell me what is the coordinate of Q? What is the coordinate of Q? Very good. That's impressive.  Now I'll do the same thing now by going B down. So let's say I go B down from here. Okay. And I reach this position R. Okay. Now if I go B down, it is basically taken as a negative angle. Correct. Okay. So what do you think? B is the magnitude by which I am going down. So basically B is the magnitude which I am going down. So let me call it as minus B. Okay. So what will be the coordinates of R? No, no, no, no, no, no, no, no. Sitesh, that's wrong. Ansh wrong. Anusha wrong. Same as Q. If it is same as Q, they should be at the same point. No. Why they are different? Two points have the same coordinates. They should be at the same location. No. So it's going to be cos minus B and sin minus B. Okay. Simple. Fine. Yeah, correct. Correct Anusha. Now cos minus B is as good as cos B and sin, and sin minus B is as good as minus sin B. So we all know that. So let me make that correction over here. I just restructure this as cos B, minus sin B. Okay. Now I would like you to all focus on these three triangles, OPQ and OQR, OPR. I'm so sorry. My bad. I want you to focus on this triangle. Let me name it as, let me name this point as point S. I want you to focus on this triangle. OQS and OPR. Okay. So everybody, focus on triangle OQS. Focus on triangle OR. What can you comment about these two triangles? They are congruent. Absolutely. They are congruent. They are congruent. Okay. Why?  And the adjacent sides to that angle, okay, are both equal to the radius of the circle, which is one each. Okay. So by SAS, you can say they are congruent. Okay. Now if they're congruent, you would be agreeing to the fact that PR should be equal to QS. That means this length should be equal to this length. PR should be equal to QS. Agreed. No doubt about it. So what is PR? Or you can say PR square will be equal to QS square. Everybody's happy till this stage. Any question? Correct? Okay. Now in order to find QS, I will use the distance between two point formula. So Q is this. S is this. What are the distance between them? So you'll say cos A plus B minus 1 the whole square. Sin A plus B minus 0 the whole square. Okay.  Is equal to QS square. QS square will be cos A minus cos B the whole square. And sin A plus sin B the whole square. Hope there is no doubt about this step. Okay. So PR square is equal to QS square. Correct? Anybody has any problem with this? Now let me expand it.  On the right side you'll see cos square A plus cos square B minus 2 cos A cos B and you'll see sin square A sin square B plus 2 sin A sin B. Okay. Now pay attention. This plus this will become a 1. Pythagorean identity. Okay. Cos square something plus sin square the same thing will become a 1. Okay. 1 plus 1 is 2. So it will be 2 minus 2 cos A plus B. Correct? Is that fine? On the right side what will you see? Let me remove this. On the right side you'll see cos square A sin square A that is also 1 cos square B sin square B that is also 1 and you'll end up seeing minus 2 cos A cos B minus sin A sin B. Right? Now 2 and 2 goes off.  Now this is an identity for you. It will work for any AB combination. You can take A and B to be any real number. It will work for this. Okay. So this identity is the first of your compound angle identity. This identity is your first of your compound angle identity and the coming identities would all be dependent on this. So I will not derive them. I will just use them to find all other identities. Okay. Now is this the only way to prove it? No. There is one more way to prove it. I will just take the next 4 minutes to prove it in another way. But this way has some kind of questions raised. That's why we don't tell this way to the students in the class initially. Let's say there is a rectangle. Okay. Let's say there is a rectangle. Fine. In this rectangle let me make a right-angled triangle inside. Like this. Okay. So this is a rectangle and inside a rectangle there is a right-angled triangle. Right. So let's prove it as P, Q, R, S, T and let's say M. Okay.  Now even though I am proving it for a very constrained situation like this. Right. That is why many people raise a question. Let's say you are restricting it to be within the first quadrant. A and B could be in any quadrant. That's why we don't tell this proof in most of the cases. Yeah, you can write this out if you want. Okay. Now all of you please notice that this length is going to be 1. So please note that I have taken this length to be 1. Okay. It is my call. So I have constructed a triangle in this way that hypotenuse length is 1. Okay. So now let us try to find out the other dimensions in the interest of time. I will write it down. This will become cos B. Okay. This will become sin B. Correct. Now since this is angle A and this is cos B, this will become cos B sin A. Correct.  Yes or no? Any doubt so far? Now if this is angle A, can I say this will also be angle A? And if this is angle A and this is sin B, then this side will be, I am just writing it over here. This side will be sin A sin B. And this side will be cos A sin B. Correct. Yes or no? Now this is A, this is B.  So this angle will also be A plus B. Correct. So it is very obvious that now this will become cos A plus B. And this will become sin A plus B. Okay. If you look at this right angle triangle over here. Right. Now see all the identities, most of the identities will come out from here. You can see that PQ is equal to SR. PQ is equal to SR. So you can say cos A plus B plus sin A sin B is equal to SR.   The same figure that we got in the previous derivation. Okay. So keep this in mind. These formulas have now to be remembered. Okay. These formulas are now to be remembered. Meanwhile, we'll stop over here. We'll stop over here. And next class would be very important class for you because I'll be dumping many, many identities on you.  And what if the angles are more than to what to do. All those things will be dealt in the next class. Okay. So next class, please do not miss out. Yeah. This has a limitation because many people can raise a question. How are you calling this as an identity because it is limited to angles within a 90 degree, right? Okay. So that's why we don't tell this proof many a times. But this is another way how you can look at it. There's another way. The coordinate geometry is more robust, more robust. Okay. Okay guys. Thank you so much. Bye-bye. Good night. Stay safe. Thank you so much.