 In these sessions I'm going to talk about, in this session I'm going to talk about first of all trigonometry as a function. Okay, so all the trigonometric ratios that we have learned sine, cos, tan, c, cos, cot. Last class we learned how to find values of these trigonometric ratios for several angles. Of course those angles were allied angles, but nevertheless we knew how to find out trigonometric ratios for several aligned angles. So we had a sign which we basically use ASTC rule and we had a magnitude part for which we use the table that we initially made in the session. Okay, so today I'm going to talk about trigonometry, trigonometric functions, trigonometric functions. Okay, and when we talk about functions it is obvious that we will also speak about its domain range. Okay, and these trigonometric functions you would be, you know, already aware that they're all periodic functions. That means they keep repeating themselves. So we'll also talk about their periodicity. Okay, these informations are very important for us because later on when we are going towards class 12 functions, these all concepts will be utilized there. Okay, so let's get started. 20 sessions means exactly 20 into 3.5 hours. There are five chronic sections. Every chronic takes four classes. Yes, Seethu, you are absolutely right. So many classes are required for chronic sections. Okay, anyways, so let's get started. I will start with my very first function, sine x function. Okay, so first of all, how does the graph of sine x function look like? Okay, from that graph, what all we can interpret, let's look at it. So sine x function graph is basically a sinusoidal graph which goes like this. Okay, now from where has this graph been plotted? If you recall, I mean, this is never ending, it goes on and on and on in both directions. So if you would recall, we had learned that sine x attains values from minus one to one, okay, for different values of the angles. So basically on the x-axis, I'm plotting the angles, of course, in radians. And on the y-axis, I'm plotting the sine of those values. Okay, now how does this particular wave come about? Now, you must have already done this in your junior classes. Let's take a unit circle. Okay, let's take a unit circle. On the unit circle, hypotenuse, I mean, any kind of distance from the center will be one only, right? So let's say for a particular angle, let's say theta, this is my hypotenuse. Okay, hypotenuse is one. And this is perpendicular. Now, if you change your theta, for that matter, x, I would say, because I have used x in my given graph. Okay, just see what happens to the p-value. Okay, so your sine x is nothing but the value of the perpendicular. So when your theta or x, in this case, I should say x, when your x is 0, p0, that's why your graph is passing through the origin. When you increase your theta, let's say it becomes 30 degree, then your p becomes half. So let's say this is my 30 degree, pi by 6. Okay, your p becomes half. Okay, just a second, guys. Okay, and when your p becomes, let's say 60 degrees, let's say 60 degrees, let's say pi by 3, it becomes root 3 by 2, root 3 by 2 is roughly 0.86. When it becomes 90 degree, it becomes a 1 because this perpendicular will actually become the radius of that given circle. And similarly, when I go beyond 90 degree, the p starts reducing. And that's why you will see that the graph starts falling here. And this trend continues if you keep on finding angles around this particular circle. And basically what you're plotting here is the graph of the length of p, directed length of p versus the angle. And this graph comes out to be of this shape. And this shape is actually called a sinusoidal. This is actually called a sinusoidal curve. Okay, so sine wave is actually a sinusoidal curve. Even cosine wave is also a sinusoidal curve. Okay, and this is basically used in a lot of concepts. You will be studying about simple harmonic motion oscillations. You'll be also talking about a lot of concepts selected to the electric fields, electromagnetic fields, they all basically move in a sinusoidal nature. Okay. Now, this is the graph of sine X. I will also show you the same graph on GeoGebra tool. So let's let's go to the GeoGebra tool. And we will plot sine X graph there. Y is equal to sine X. Okay. For the betterment of, you know, understanding, I will keep my X in radiance. So let me make my X in radiance. So small change I will make unit is in radiance. Okay. Okay. So this is the graph. And I would also take a snapshot of this and put it in your put it in your notes so that you can refer to it. Okay. So this is the graph of sine of X. Okay. Let me paste it. Now, look at this graph. And from the graph only, let us find out the domain and the range of this graph. So let's talk about the domain. Let's talk about the range. So what do you think is the domain of this graph? That means what is the extension of this function along the X axis? You say, sir, the function extends all the way from minus infinity to infinity along the X axis. So domain is all real numbers. Okay. Domain is all real numbers. That means sine X. Sine X function can be fed with any angle. There's no restriction on the angle. Sino-Soidal is the name of that particular curve, okay, which rises up and down like this. That is called a Sino-Soidal curve. Right. A name is given to it. Now there's a history behind it. It actually is an Arabic word. Okay. I don't want to go into the history. It is not required. Maybe Tushas, I would have told you a story last year. I don't know whether you were there with us. Okay. Anyway, there's a history behind how the name sinus has come into that. Okay. Anyways, what is the range of the function? The range of the function is from minus one to one inclusive. Okay. Now, is this function a periodic function? What's a periodic function for that matter? A function which basically repeats itself after a certain change in the input. But this repetition should happen for all the elements in the domain of the function. Again, I will repeat it and let me write it down also for the betterment of your notes as well. What is a periodic function? What's a periodic function? Any function f of x, which let's say repeats itself for a given change in the value of the input, let's say I change the input value by T. The value of the function comes out to be the same as what it has at x. Let's say you take x as A. Okay. So it had some value f of A. Okay. Let's say sine 30 degrees. 30 degrees sine is half. Okay. If I change this 30 degree to let's say 390 degree, that means I make a change of T in the input of the function. The answer will come out to be same sine 390 as half itself. So there is no change in the output value. Even if I change it by another 360 degree, let's say I ask you what is 250 degrees sine? You say again half. Right. So what are you observing here is that when the input is changed by a certain amount of T, okay, here T has to be a positive real number. T has to be a positive real number. Now why positive real number? Because if I don't keep it as a positive real number, then every function will become periodic because if I change it by zero, it will give the same value back, isn't it? But those are not called periodic functions. Right. So if I don't keep my T as a positive real number, then in that case, every function in this world will become periodic. Isn't it? Right. That is not our purpose. Right. Our purpose is to showcase that the function is repeating itself after a change of or after an every change of T in the function. Now this change has to be a positive change. See any interval is a positive interval, isn't it? Many people ask me, sir, could I subtract T also from X? Yes. See the change is positive only. So if you subtract T from here, you end up getting this part. Okay. And this goes on and on. It doesn't stop like this. Right. It goes backwards also. Are you getting my part? So a periodic function will exhibit this kind of a property. Is it fine? Now, one important thing to be kept in mind is this property should hold good for all X belonging to the domain of the function. It should not be valid only for one or two values of X in the domain. Okay. Like sine X function. This is what is my example. This itself is my example. If you see sine 90 degree. Okay. Look at this graph. Okay. This graph at 90 degree, the value is a one, isn't it? Right. Do I get one value again later on? Yes. You will say, sir, now here also you will get a value of one only five pi by two. Okay. Does it also give me a value on this side of one? Yes. Maybe at minus three pi by two. Okay. So again, I mean, I've just shown you a part of the graph. So this gap that you see, this gap that you see, this gap is called the period of the function. Okay. This gap is called the period of the function. Okay. So in this case, the period of the function is actually two pi. Okay. Now, if I take the same graph and I say, sir, can I say the period is also four pi? Okay. Let me take you to the bigger graph here. Let me just zoom in a bit. Okay. And let us take the same value of, let us take the same value of pi by two. All of you, please pay attention. Okay. So at pi by two, its value is one. Correct. Now, let's say I change this value by let's say four pi. That means, let's say I reach here. Okay. This is actually nine pi by two. So this gap is four pi. Is it repeating its value again? Yes. Again, let me change it by four pi. So if I change it by four pi, maybe I will reach 17 pi by two somewhere over here. Okay. Again, its value is one. Correct. So even four pi should be the period. Absolutely right. Two pi is also the period. Four pi is also the period. Even six pi will also be the period. Even eight pi will also be the period. Ten pi will also be the period. Basically, all even multiples of pi will be period of sine x. But here is a catch. When we mention the period, normally, or when the question asks you, find the period of a function, they actually mean to ask you the smallest t possible for which the function is repeating itself. So in all the examples which I gave you, two pi, four pi, six pi, eight pi, ten pi, twelve pi, fourteen pi, which is the smallest value. What will you say? Obviously, you'll say two pi. So that is actually called the period of the function. Maybe the word fundamental is not mentioned every time. Okay. So the fundamental, as of now I will write it fundamental, but the word fundamental may be missing many a times. Okay. So if a question in your, let's say DPP says, what's the period of sine x? Okay. They actually mean fundamental period, but they don't mention it. So what is fundamental period? It is the shortest t, t for which the function repeats itself for every x belonging to the domain of the function. And t must be a positive real quantity. Okay. For sine x, that value is two pi radians. So if you change your x value by every two pi, right, your value of sine x will come back to the same value. Is it fine? Any doubt related to the period of the function? Okay. All the six trigonometric ratios, which we will be talking, we'll be calling as functions in today's class. So those same ratios I will be now, you know, renaming as functions, they are all periodic with a period of two pi. So they're all periodic with a period of two pi and pi. We'll see which is periodic with pi and which is 30 with two pi. What is this upturned a right now you are? Okay. This is, this is called for all, for all, this is a short form for saying for all. I hope I have given you some list of every, you know, symbols. Okay. This is for all, for all x. Okay. This is there exists. There exists. This is belongs. I think we had done it in our sets chapter. Isn't it? Okay. Any questions? All right. So we are not going to waste too much time on it, but I would like you to, you know, answer this question from your observation related to the graphs. Tell me, tell me for what value of x does sin x actually become a zero? This is very important because you'll be using this in the domain of other technometric functions, which we are going to see in some time. Okay. So one value that say two has given say two has said, sir, it is at pi. Okay. Why not zero set? Okay. So at zero, definitely sin x is zero. At pi it is zero. Why not two pi? Why not three pi? Why not four pi? Why not minus pi? Why not minus two pi? So they can be all answers possible minus pi, minus two pi, da, da, da, da. Here also two pi, three pi, da, da, da, da. So can I say sin x will be zero for all x equal to a multiple of pi n being a integer? Okay. Now many of you would be thinking, why is sir so interested in sin x becoming zero? Okay. You'll see in some time why I'm so much interested. Okay. So I'll be using it in finding the domain of the reciprocals of sine and, you know, other functions involved. Okay. So please remember this result. It is very useful for us that we know for what values of x my sin x will always become a zero. So for any multiple of pi sin x will become a zero. Okay. Very good. Now without waste of time, we will now go to the other function boss x. So today will be slightly fast dear students, but of course, not at the risk of you not understanding things. So I'll be fast, but not that fast that you will not understand things today. Okay. So I will keep this graph on here and I would now make the graph of Cossack's. So why is the equal to Cossack's? Okay. If you see Cossack's graph, which is in red right now, I hope if you can see the color change, one is green, other is red. So Cossack's graph is actually the sinusoidal curve, same as what we had for sine x, but at a certain phase difference with sine x. Okay. So both are waves, both are sinusoidal waves, but they have a certain phase difference between them. Now you cannot say which is ahead of the other. Okay. So because there's no starting and the ending point. Okay. So they have some phase difference. So sine x and Cossack's graph, please note down that they have a phase difference of 90 degrees. If you see the crest position or if you see the trough position, if you see the crest position of both these graphs, you'll realize that this gap is of five by two. Okay. And this has something to do with their complementary properties. We all know sine and cosine. The name itself has the word cosine. Right. I hope everybody knows why the word cosine was given to it. Go for complimentary. Complimentary of sine. Complimentary means 90 degrees. You all know that, right? Complimentary angles. Right. So sine and cosine are basically given these names because they are having this particular phenomena that their phase difference is of 90 degrees. Okay. So I will right now mute the graph of sine x and I will just, I will just take Cossack's and take a snapshot of it and paste it on your notes. In fact, I will slightly zoom in. No, sorry, zoom out. I always make this confused thing. Okay. So let me take this. Okay. And put it on here. Okay. Instead of me drawing with my ugly drawing, let us, let us borrow it from the tool itself. Yeah. There you go. This is the graph of Cossack's. So this is a function idea. Now, all these signometric functions, we are looking, all these signometric ratios, we are now calling it as functions, right? Just like we had functions in our functions chapter. Okay. So yes, look at it and tell me, what do you think is the domain of this function? What do you think is the range of this function? Is this function periodic? If yes, what do you think is the fundamental period? Let's answer these all things and I would request you to leave your response on the chat box. I want to see your response. Awesome. Awesome. Very good. Exactly. So domain is all real numbers, minus infinity to infinity. Range is minus one to one. And Cossack's is basically made of the same wave as Sinex is made, is just that there is a lag. So whatever was the period of Sinex, the same will be the period for Cossack's as well. And that is stupid. Okay. Now, before moving on to the next one, I would like to ask you a quick question. When do you think Cossack's becomes zero, my dear? Quick, quick, quick. Give me a generic answer. Don't give me a few values here and there. Okay. So Manu says pi by two. Is that only value Manu? Okay. So this is your pi by two Manu. This is your pi by two. I can see it, three pi by two also it becomes zero. See, these graphs will tell you everything. Graph is like the bio data of that particular function. If you know the graph, you can answer everything you want to about the graph. Okay. Yes. Nikhil, be careful when you're making that generalization. See, pi by two, three pi by two, five pi by two, da-da-da-da-da, back also minus pi by two, minus three pi by two, minus five pi by two, da-da-da-da-da. What is the pattern that you see here? The pattern that you see here is basically all odd multiples of pi by two, all odd multiples of pi by two. You can write it in a fancy way as two n plus one pi by two or two n minus one pi by two. Both are fine. That doesn't make any difference and being some integer. Okay. So please remember this is very useful in finding the domain of some of the subsequent functions, trigonometric functions that we are going to see. Is there a sign? Any questions? Any concerns? Okay. All right. So we'll now move on to the next in line. Next in line is your tan x function. Tan x function is basically sine x by cos x. Okay. Its graph looks considerably different from the graph of its counterparts, which is sine and cos. Okay. So I'll show you the graph of tan x. Okay. This is how the graph of tan x actually looks like. Okay. And you'll see that this graph has a number of vertical asymptotes. What is vertical asymptotes? Basically those vertical lines, which the function tries to come very close to but never achieves it. Okay. And what are those vertical asymptotes? I will write a few of them. One is pi by two. Other is three pi by two. And you will tell me the reason for having these vertical asymptotes. I'll make one more x is equal to minus of pi by two. Okay. So you can see these are your vertical asymptotes, which tan x graph has. That means it is never able to touch those lines. It comes very close to them, but never touches those lines. So as to say at pi by two, three pi by two, five pi by two, minus pi by two, minus three pi by two, et cetera, et cetera, et cetera, the function actually becomes undefined. Okay. Anyways, I will hide these three. Oh, let me, let me keep them no issues. And I will take the snapshot of it and put it in your notes. Okay. Let me put this on your notes. Okay. Now let us try to understand what makes this graph different from a sinusoidal curve. Can you show the previous screen? Oh, why not? Guys, if I don't read your message, speak out. Okay. Don't wait for me to read it because I may have skipped it because I get so many messages. Okay. Pratej done. Okay. Okay. Yes. Now see, what are the, what is the reason for this graph? Why does the graph start growing so fast from zero till it reaches or tries to reach pi by two? Can anybody explain me the reason for this, you know, sudden increase in the value of the function? Of course it goes all the way to plus infinity. But what are the reason for this? But the reason for this is very simple. It is made up of sin x and cos x. Sin x is in the numerator, cos x is in the denominator. As you increase from zero to pi by two, sin x is a function which rises. Recall the graph. I hope you have already drawn it on your notebooks and cos x falls. So the numerator is increasing and the denominator is decreasing. Of course, both are positive in the same, you know, in this particular interval of the angle. So if a fraction's numerator is increasing and denominator is decreasing, overall what will happen to the fraction? You'll see it will increase at a very fast rate. And that is precisely what is happening to this function. But unfortunately, by the time x reaches 90 degree or tries to reach 90 degree, your cos x starts becoming zero. And if your cos x becomes zero, overall the function becomes undefined. So now try to ask yourself, this undefined nature would be shown at how many x's. And what is the pattern of those x's? Because that is going to be the answer for your next question which I am going to ask you. What do you think is the domain of the function? What do you think is the domain of the function? Just answer that. Anybody? Quick, quick, quick. Very good. Very good, Nikhil. All real numbers except those values which are odd multiples of pi by two. Why? Why these values are not allowed because they will make cos zero. And that's the reason why I asked you in the previous slide, when does cos become zero? So that you're able to write down the domain of tan x without any hesitation. So tan x value will be or tan x input will be only those real numbers which are not multiples of or you can say odd multiples of let me correct myself odd multiples of pi by two. Clear? Now look at the graph and tell me what is the range of the function. So the range is everything from minus infinity to infinity. Exactly. So tan x basically can take any values. Okay. I think the other day somebody was asking me, how does tan x become 20? Yes, tan x can become 20. Tan x can become 5 lakh also. Right? Don't don't compare it with sine and cos whose values are restricted between minus one to one. Tan x, any real number it can take, tan x can become one billion also. Okay. So it can take all real numbers. Now look at this graph and tell me, do you see that this function is periodic? And if yes, what do you think should be the fundamental period of this function? Excellent. So yes, this function is periodic and the period is going to be pi. Now very simple to find out the period. See, at zero it became a zero. Right? Now see, when does it become zero next? You'll say, sir, pi again, two pi. So what is the gap between these? You'll say, sir, pi, pi. Okay. So like that, you can say this is your fundamental period of tan x. Okay. So sine was periodic with two pi, cos was periodic with two pi, but tan is periodic with pi. Don't forget that. Okay. So let's move on now to the other parts, the other functions, which is go seek, seek and caught quickly without much waste of time. Any questions here? Anybody? Any questions? Should I move on copied, Pratej, Setu, Holland? Okay. So far, so clear. Guys, one more thing. Going forward, you should be as familiar with these graphs as you are with, let's say a straight lines graph or a quadratic polynomial graph. Right? So they are to be included in your problem solving strategies. Right? Start using them in solving problems. Keep them in mind. If at all they are helpful in solving a problem. So it should be imbibed in your mind and not just in your notes. Okay. 1112th, you have to please ensure that graphs are used, whatever places you can to solve your questions. They're not optional anymore. Graph sometimes will help you solve a problem. Next one is the graph of Coseek. Coseek is what? Coseek is reciprocal of sine. Okay. How does the graph look like? This graph is again different from the other three graphs that we have seen. This graph is like elongated use. Let me show that to you. Meanwhile, I'll mute this. Okay. And I will also like to switch on the sine x graph for you. Okay. Just in case you figure out some observation. Okay. Now see, the graph of Coseek x are these elongated use. Now these use are elongated means they are going all the way up to infinity, going all the way down to minus infinity. Right? Now see how these two graphs have beautifully fit in the space. Okay. So one thing that you will see is that where sine x exists, Coseek actually doesn't accept for a few locations. Okay. And another observation which actually will help you remember the graph is wherever sine shows a hump or you can say a crest, there Coseek graph shows a trough. Are you seeing that? And vice versa. Where sine shows a trough, Coseek will show a crest. Okay. You can say there is a, no, bump to bump matching. Okay. So this bump, this bump is matching with this bump. Okay. This bump is matching with this bump. So when I was learning these, I always used to forget Coseek x graph. The easiest graph to remember is of course, sine x. Okay. Sine x and Cosek. I always used to forget Coseek x graph. Then basically I figured out that the best way to remember Coseek x graph is make the sine x graph and wherever there is a, you can say a crest draw opposite you on that. Wherever there is a trough also draw an opposite you down at those positions. Okay. So this is how you can also remember the graph of, of Coseek x. Now, what are the reasons for this elongated use? Very simple. At zero, you can see that sine x becomes zero. Correct. So one by that will not be defined for sure. So at zero Coseek x is not defined. Same will be true for pi. Same will be true for two pi, three pi, four pi, minus pi, minus two pi, and so on and so forth. By the way, I'm giving you some order points for telling me the domain of Coseek x. Anyways, so when sine x is, when x is zero, sine x is zero. So one by sine x is undefined. But when you slightly move towards the positive side of zero, let's say zero plus. Sine zero plus will be a very, very small value, which is positive. Let's say plus 0.0000001. If you reciprocate it, you would realize that the reciprocal will be a very large positive number. And that's why this graph reaches all the way to plus infinity. Right. Thumb up. Coseek x is more than one, but also less than minus one. Right. Let me complete it. Sethu, you will come to know about the full picture. What you were asking me about, Coseek x square. Coseek x square will be more than one. Don't get confused. Okay. Now, what happens as sine rises? As sine rises means one by sine will fall because the denominator is becoming larger. So this graph starts falling down. Okay. And this will increase and this will fall and they will meet at pi by two again. So at pi by two, both of them will actually attain a value of one. Okay. And after that, sine again starts falling down. So Coseek becomes higher or Coseek starts rising up. Right. It rises till you reach 180 minus or pi minus. That means little less than pi because exactly at pi, it will become undefined. So at pi minus, let me call this position as pi minus. At pi minus, it will again go to plus infinity. Okay. And this is the trend that it will follow for rest of the intervals. I hope you can figure out why the graph is oriented or shaped in this manner. Any question related to what is the, what is the reason for this shape to be taken by, oh, I'm sorry. What is the reason for this shape to be taken by Coseek X? Everybody's fine with that? Okay. I'll take a snapshot of this and fit it in your notes, just in case you need it. Let me put it in your notes. Okay. So this is your graph of Coseek X. Now look at the graph and tell me what do you think should be the domain of the function? What do you think should be the range of the function? And is this function periodic? If yes, what do you think is the fundamental period? Now domain, I would request you to give me your answer. All real numbers except multiples of pi, beautiful, Karthik. Very good. So can I say all real numbers except multiples of pi because at multiples of pi, sine X will be unhappy. It will become zero. Correct. And dividing by zero is not allowed. It is undefined. What is the range of the function? Now see the range of the function is all values from minus infinity to minus one. Let me write it in white. I'm writing it yellow. Yeah. All values from minus infinity to minus one union one to infinity. Okay. So Coseek X is greater than equal to one, but it can also be less than equal to minus one, right? But it cannot be between minus one to one. So we sometimes also write this interval as all real numbers except open interval minus one to one. Why open interval? Why not close interval? Because if I remove close interval means I'm removing minus one and one from the range of Coseek X, which is not true because it can become a one. It can become a one. It can become a minus one. Okay. Right. So you can have several analysis out of it. Many people will say, sir, can I say mod of Coseek X is greater than equal to one? Yes. Can I say Coseek and square X is greater than equal to one? Yes. Okay. But Coseek X is between minus infinity to minus one inclusive union one to infinity, one inclusive. Is it clear? Setu, especially are you clear now? You're not making any mistakes. Okay. Now Coseek has the same fundamental period that sine has, which is actually 2 pi. So Coseek will repeat itself for those changes in X, which is going to repeat your sine X. And what repeats sine X? A change of 2 pi. Okay. A change of 2 pi will repeat the sine X values back. In other words, fundamental period of sine X and fundamental period of Coseek X will be the same and that will be 2 pi. Is it fine? Any questions? Any questions? All right. So can I now move on to the next one, which is Ckeks quickly without much problem? Okay. All right. So let's now move on to move on to the next one. The next one is our Ckeks graph. Ckeks is nothing but reciprocal of Coseks. Now Ckeks and Coseeks graph look very much the same. So what I will do, I will draw the graph of Ckeks, keeping Coseek X also as on. Okay. Now C, appreciate the graphs. What do you see? The shape is exactly the same. As you can see, the orange graph and the blue graph have exactly the same shapes. But yes, there is a face difference between them, just like sine and cos had. And since Coseek X is one by sine, Ckeks is one by cos. The face difference between this function together will be same as what we had between sine and cos as well. And that is actually a pi by 2. So this gap, this gap is a pi by 2 again. And this is precisely why they get their name as Cke and Coseek. I hope everybody knows the full form. Cke means seek it. Coseek means we're talking about the graph of Ckeks, which is actually reciprocal of Coseks. And I had also taken a snapshot of the graph. Okay. This was for the graph of this thing. So I'll draw it again. Graph of Ckeks. Yeah. Okay. And let me just make the original settings back. X axis is in terms of pi. There you go. Okay. Let me take a snapshot. Now, let's have a look at this graph. Which graph is this? Let's have a look at this graph and try to answer what is the domain, what is the domain, range, and fundamental period of Ckeks. Now, keep in mind that those values of X will not be permitted, which will make your Coseks become zero. So you'll say, sir, okay, then in that case, I can have every value, but I should not have odd multiples of pi by two, because at odd multiples of pi by two, cost becomes zero, which is not allowed. What about the range? This is the range seems to be the same as what we had for Coseek. So Coseek range was minus infinity to minus one union one to infinity. Or you can also write it as all real numbers except open interval one to one. Okay. So Coseek and Cke share the same range. Okay. And does that phenomena of that, you know, cost and Ckek having their bumps at the same place? Does it work? Yes, it works there as well. So I'll show you that as well. So I'll draw Y is equal to, Y is equal to cost of X. And I would like you to see that phenomena. So wherever there is a crest, there is a trough of Coseek there, sorry, Ckek there. And wherever there is a, you can say trough of cost, there is a, you can say a crest of Ckeks coming there. Okay. Another way that will help you to remind the graph of Ckeks. Okay. And what about the period? The period will be the same as what we have for cost. Cost period is 2 pi. So Ckek period will also be 2 pi. Is it fine? Any questions? Any concerns? Any questions? Any concerns? Okay. Now, the last function that we are going to talk about is, is cortex. Okay. And here is an exercise which I want you to do. Very simple exercise. Get me the graph of, get me the graph of, get me the graph of cortex by using the property that cortex is tan pi by 2 minus X. If you all know the graph of tan X, right? We all know the graph of tan X. Okay. So this is our graph of tan X. I'll just draw one of the arms. Okay. This one of the arms. I don't want to draw multiple arms. In fact, in reality, it goes on and on like this. Okay. Can you use this? Can you use this to sketch, to sketch the graph of cortex? If yes, how? In fact, I've already given you a hint. This property is known to every 10th grader, right? All of you would have done this property. Okay. So tell me, how do you get the graph of cortex using the graph of tan X? Any recommendation? Any kind of, you can say reasoning that comes in your mind. Please share with me. I'll be happy to know that. Shifted by pi. Sethu, you want to copy something on the previous one? Don't worry about copying here. Samjo, Samajipi, focus on copying. You can do also when I circulate the notes. Don't buy what sir? Your note vanished. That shifted by pi by two. Okay. If you shift it by pi by two, my dear, then you get the graph of tan X minus pi by two. But I want tan pi by two minus X. I know that you're going to say that. Those who are saying, sir, shifted by pi by two to the right. You know what you're doing? You're getting the graph of tan X minus pi by two. But that is not what I need. I need the graph of tan pi by two minus X. Right. Very good. So we have to do a step before we shift it. We have to change our first of all X with a minus X. Okay. Now see why I'm doing that. If you change X with a minus X, first of all, what impact does it have on the graph? You'll say, sir, if you're changing X with minus X, we learned in our bridge course that the graph will get reflected about Y axis. So when I reflect it about Y axis, the graph becomes like this. I use my pen is going here. Yeah. Okay. Hey, Karthik, if you shift it by pi by two to the left, you will get the graph of tan X plus pi by two. But that still doesn't solve the purpose, no? Getting the point, Karthik. What is my end goal? Pi by two minus X I want, not X plus pi by two. Neither X minus pi by two. So for that to get pi by two minus X, I have to first make this change. Are you getting my point? Okay. Once you make this change, the next thing that you will do is what you people were saying earlier, change X with X minus pi by two. So let me write that in blue. So change your, I hope you can all read this, change your X with X minus pi by two. So now when you're doing it, your graph will shift by pi by two to the right. So now the graph will come like this. Okay. Of course, other arms will also start coming in the same way, which maybe I can draw like this. Okay. Here also, it will be like this and so on. Okay. This dotted line just shows vertical as it goes. Okay. Like this will continue in both directions. So this ultimately is what is your tan of, let me write it in green color. This ultimately is what tan of pi by two minus X, which is actually, which is actually caught off X. Okay. So if you know tan X, you can automatically be drawn from there by your basics of graph transformations, which you have already learned two weeks, two months back. Okay. And nevertheless, I will show you on the GeoGebra as well, how exactly it looks like. Okay. So can you go to the GeoGebra screen and see the graph of cortex? So let's see the graph of cortex. So Y is equal to cot of X. There you go. This is what we had predicted. Correct. So this graph is, I'm going to put it on your, this thing as well. Let me paste it here. This one. Yeah. Any questions, any concerns with respect to the graph of cortex? Yes. Yes. We can do the same thing also. So Shargui is suggesting, could we first reflect the graph about the X axis and then can we change our X with X minus pi by two? Yes. We can do that. Both are fine. Okay. All right. Now look at this graph and answer the following questions. What do you think should be the domain of this graph? What do you think should be the range of this graph? What do you think should be the fundamental period of this graph? Let's try to answer this quickly. What's the domain? What's the domain? Or you can better write it as cos X by sine X. So you'll say, sir, I can put everything, but I should not put such a thing which will make sine X become zero because overall things will become undefined. So I will say domain is going to be all real numbers except multiples of pi. Excellent. See, there is no recta business here. You get this all from logic. Logic will stay with you for two years, of course, for the lifetime. Memory will ditch you. Wherever logic fits, fit in logic. Where logic doesn't fit, fit mnemonics. And if mnemonics also doesn't fit, then we have to basically mark up, which ultimately you do to a certain extent in biology or chemistry. What are the range of this function? Could you repeat the domain again? See, domain will be, Karthik, all values of X, but those values will not be entertained, which will make sine X become zero. Now, remember, I asked you when does sine X become zero early on in this session today? And we all learned that sine X become zero for multiples of pi. So those multiples of pi have to be removed. Same with me, by the way, Setu. I can't remember things for long. Unless until there's a logic there. So what are the range of this function? The range of this function is all real numbers, which you can see. Is it periodic? Yes. You can just take a sample value, let's say pi by 2. At pi by 2, it becomes zero. Again, at 3 pi by 2, it will become a zero. In fact, minus pi by 2, it was also zero. 5 pi by 2, it was also zero. So what is this gap? What is this gap? This gap is pi. This gap is a pi. So after every change of pi, the function cortex will repeat itself. So it is periodic with pi. So let's summarize with respect to periodicity. Sine X, cos X, cos C kicks, these are the four functions which are periodic with 2 pi. Tan X, cot X, they are periodic functions with a period of pi. By the way, trivia for all of you. Do you know of any non-trignometric function which is also periodic? If yes, what is that function? And tell me its period. Trivia. Do you know of a function which is non-trignometric and it is periodic? Any function? You have already learned that function by the way. No modulus function? Fractional part. There you go. Say to bang on target. Fractional part. Yes, this is a periodic function. This is a periodic function. Recall the graph. The graph used to look like this. It is repeating itself. So what you had, let's say at 0.5, the same you'll have at 1.5, the same you'll have at 2.5, the same you'll have at 3.5. So which tells you that its fundamental period is actually 1. So it may come as a question to you later on when we do functions in our class 12. Which of the following function is periodic and one of them would be your fractional part. Fractional part is also a periodic function. So many people think that only trigonometric functions are periodic. No. This is an algebraic function which is actually periodic. Its period is 1. Okay. Anyways, the concept of period itself is a long topic. Normally, I cover it up in 1.5 and 2 hours when I do it in class 12. So in 12th, we are going to talk more about it. Okay. As of now, you just have to know what's, what is basically period of these six trigonometric functions. Okay. Now, what I'm going to do is based on the graphs that we have learned, just based on the graphs that we have learned, we are now going to look into some set of identities. The first identity I'm going to talk about is negative angle identities. Negative angle identities. Okay. Very important. What are these identities? Number one, sign negative theta is negative sign theta. Can somebody tell me why? From the graph, from the graph, I don't want you to take examples and prove me. From the graph, can you show that sign of negative theta is negative sign theta? Why is that happening? Let me sketch a miniature graph for you. Let's say like this. Let's say I take an angle theta. Okay. I get an answer from it, which is sign theta. If you just negate the sign of this input, let's say minus theta, you realize that your output will actually become negative sign theta means if your input changes its sign, the output will also change its sign. And it is true for any theta lying in the domain of the function. Are you getting my point? Such functions in the language of function is called odd function. I think I have discussed about this when I was doing it in the bridge course. Okay. Odd functions have a special characteristic that if you change your input sign, even the output sign will get changed. Okay. So such graphs are symmetric about origin. Symmetric about origin. What is the meaning of symmetric about origin? Think as if there is a very, very, very, very, very, very, very, very small mirror kept at the origin. Okay. And that mirror is reflecting whatever is on this side to this side. So basically it reflects the part you have drawn in the first quadrant in the third quadrant. Okay. Whatever it is there in the second quadrant, it reflects it in the fourth quadrant and vice versa. Right. So any function which basically satisfies this functional equation is called an odd function. Odd function, the biggest characteristic is their graph is always symmetrical in opposite quadrant. So you can say symmetrical about origin. I don't mean to say symmetric about y axis. Don't get me wrong. There's symmetrical about a point and that point is origin. Are you getting my point? In such functions, if you change the sign of the input, the overall sign of the output will also get changed. Sign 30, half sign minus 30, minus half sign. Let's say 270 minus one sign minus 270 plus one. Are you getting my point? So if the angle sign changes or the input that you're giving to this function, the sign of that input is changed, then the output sign will also get changed. Are you getting my point? Okay. Next in this particular series is your cost function. Cost function basically is a Bindas function. It doesn't care about the negativity of the sign. If you put cost 30, it will give you root 3 by 2, cost minus 30 also root 3 by 2. Okay. So this function gives or doesn't care. I was about to use some different word. This function gives, I mean, no care to the sign of the input. Okay. So let me show you in the graph. Okay. So this is the graph of, this is the graph of cost. There you go. Okay. Now here, what will happen if you put, let's say theta, it'll give you some value cost theta, correct? Even if you put minus theta, it will still give you cost theta. It doesn't change the answer. It doesn't change the answer. Right? So let's say if you put this as your theta, it gives you some value. Even if you put minus theta also, it gives you the same value. It doesn't change the answer. Okay. See, it's not like minus sign comes out. Say to what I wanted to say, in case of sign, if you change the input sign of the angle, the output will also change its sign. Are you getting the point? Because if I put some angle theta and I get some answer out of it, and if I reverse the sign of that angle, even the output will reverse its sign. That is the reason why this property is true. That is what I explained in the class. Got it? Okay. Now this behavior, which is shown by cost function is very typical of a function which we call as even function in mathematics. So even function is a function which basically doesn't care about the change of the sign of the input. Okay. Few examples I can give you from your, you know, whatever you have known so far, x square function. Does x square function change if I change my x to minus x? You'll say no, sir. It still remains x square. Correct. So x square is an example of that. Model is function. That's also an example of even function. In fact, any polynomial having only even powers on x, that's an even function. Okay. Such graphs are symmetrical about the y-axis. Such graphs are symmetrical about the y-axis. So as you can see, the graph of cos x is exactly symmetrical about y-axis. So whatever you have drawn on the left side, the same thing is there on the right side as well. Okay. I would like all of you to give me one, one example of each other than trigonometric function. Give me an example of odd function other than sin x. Quickly, everybody should give me one, one example. Any function which exhibits this criteria or any function whose graph, you know, is symmetrical in opposite quadrants. XQ, beautiful example, beautiful. Right. Nikhil Shaduli, very good. Okay. Can I say x plus x to the power of 5? Yeah, x to the power of odd number. Exactly. Okay. Okay. Give me one example other than mod function and x square of an even function. Yes, y is equal to or f of x equal to x. Yes, give me an example of even function other than the two which I have already given. No. Fractional part is not symmetrical about y-axis. Is it symmetrical? I guess symmetrical hota asadikta. See, it would have been like this. Okay. Anyways, good try Shaduli. Tell me, tell me another example. X to the power of 4. Okay. Or let's say a polynomial which is made up of, you know, let's say x square plus 2 itself. Okay. So these are all examples of even function. Mod function already I told you. Okay. Anyways, continuing with this property list. So this guy is odd. Let me write odd next to it. This guy is even. Tan x function is also odd. That's why if you change your, if you change your sign of the input, the function gives you negative answer. Okay. So this is also odd. Okay. Next, Coseek. Coseek is also odd. Coseek is also odd. So odd are in actually favor. So odd are actually in favor. There are more odds than evens. Coseek is even. Coseek is even. Coseek doesn't care about the change of the sign of the input. Guys, let me tell you one thing. There is a very innocent question that came last year. When I write seek minus theta, seek theta, I don't claim that seek theta will always be positive. I hope none of you are getting that kind of a doubt. What I mean to say, whatever output will come from here, the same output will come even if you change the sign of the angle. That output can be negative also. Don't be like, oh, sir, seek theta is always positive since you wrote a positive sign here. No, I don't mean to say that. I don't mean to say that. Okay. Least integer function is symmetrical about y axis. No, y axis symmetrical. Make the graph. You will not get that. Okay, surely. Okay. Yeah. Cortex is also odd function. Please note that. So there are four odd functions in all and there are two even functions. Okay. So only cost in seek, they don't care about the negativity in the angle. They will treat it as if there's a positive angle given to it. So if somebody says cost minus 120 and cost 120, both will give the same answer, which is minus half each. Are you getting my point? But sign for seek can cut. They are odd functions. So if you change the sign of the angle given inside, the output will also change its sign. Got the point? Any questions, any concerns with that? Okay. So with this, we'll now move on to complimentary and supplementary angle properties. Guys, please pardon because I think I'm going slightly fast. But this is the previous slide once. Who is that? Sir, Pratij. You want to see previous slide, Pratij? I will show you. Because I have to at least cover compound angle identities today. That's my target. Any question other than copying this? Anybody, please do let me know. So what is complimentary and supplementary angle identities? Now, these identities are very important because every nook and corner we get to use them. I think today, I think Tejaswini asked me some some doubts which had this particular identity involved. So maybe Tejaswini, after I do this concept, you will be able to solve those questions. In fact, one of the questions which you asked me, it is based on complimentary and supplementary angle identities. Now, one of these identities, remember when you were young, childhood days, you learned about these identities, right? So sine 90 minus theta is cos theta, cos 90 minus theta is sine theta, correct? Do you remember that? Do you remember that? Similarly, tan 90 minus theta is cot theta, cot 90 minus theta is tan theta, correct? Yes. So these are identities are very well known to you, correct? So what I am going to do is I am going to complicate this, okay? Complicate this means I am going to add more identities to this type of sets, okay? Was it deleted? Okay. Nevertheless, by the way, you can prove them easily by using your graphs of sine cos tan c cos c cot. Just like we figured out the graph of cortex by using graph of tan x. So they can easily be proved, okay? Now, many people say, sir, why to go to graph? Let's say I want to prove this, right? Make a right angle triangle, okay? Let's say I call this as theta. Let me name it as a b c, okay? So this angle automatically becomes 90 minus theta, okay? So if somebody says what's cos theta, you'll say b c by a c, right? And same is true if somebody asks you sine 90 degree minus theta, you'll say opposite, which is b c by hypotenuse, which is a c. So since these two are same, since these two are same, even these two should be same, correct? I agree. This is also a good proof, but there is a problem with this proof. What is the problem with this proof? Can anyway tell me? This is a very limited proof. It works only for theta lying within a triangle. Then this proof will go for a toss, right? But this is identity, my dear. What are the meaning of identity? Identity means all such angles for which these functions exist. It will be true. I can put theta as 5 lakh also, it will be true. I can put theta as minus 1 lakh also, it will be true because it's an identity. But those 5 lakh or minus 1 lakh, etc., angle cannot be justified within a triangle. So this is good for a youngster, maybe 9th, 10th grader, if he does it like this, we'll say, okay, good. But as a grown-ups, now there's immediate transition from 10 to 11, and all of a sudden you are now grown-ups. So now you have to understand that, okay, this is by using our graphical transformations, which we had learned in our functions, trigonometric functions. Is it fine? Now, let us make some small, small, small, small changes and see how does these identities or how does further identities evolve. Now instead of 90 minus theta series, I'll call it as 90 minus theta series, let me ask you 90 plus theta series, okay? So let me ask you this question. What do you think is sin 90 plus theta or cos 90 plus theta or tan 90 plus theta or you can say cos 90 plus theta? I'm putting ditto symbol, I hope you are able to understand that, I don't want to write it over and over again. And let's say cot 90 plus theta, how will you figure out? What is the answer to these or what is this identity? Let's complete these identities. Okay, who will complete it? Who will tell me the first one? Let me write 90 plus theta series. Yeah, what is the answer to the first one? Maybe I've done it in school. Okay, now I'm getting different types of answers. Some of you are saying minus cos theta, some of you are saying cos theta, some of you are saying sin theta also. Guys, this is something which you can answer with your prior knowledge. Okay, you are equipped enough to answer this question by whatever knowledge has been given to you little bit, little while ago. Answer this. How will you answer this? That is up to you. Meanwhile, let me take your attendance. Okay, let's say I want to answer this question. How will I answer this? Now, see, I will recall that I had done sin 90 minus theta a little while ago. What is that cos theta? Correct. Now, this is an identity, my dear. Identity means what? I can play with my angles. I can play with my angles. Right? So what I'm going to do in this particular identity, I'm going to replace theta with minus theta. So when you do that, on the left hand side, you'll get 90 minus of minus theta. On the right side, you'll get cos of minus theta. One important thing to be kept in mind that whatever you do with theta on the left, the same thing you have to do the theta on the right as well. Okay. Don't give a different treatment to the the the thes. So if you're changing your theta on the left with minus theta, you have to change the theta on the right also with minus theta. So when you do that, automatically the left hand side becomes our required expression. And what is cos minus theta, my dear. Just now we had learned that cos is an even function. It doesn't care about the negativity of the angle. So your answer automatically evolves from here. And this answer is cos theta. Is it fine? Any question? Any concerns? Now use the same logic and try to answer what is cos 90 plus theta? What is cos 90 plus theta? Please do that. Everybody. And I should get the same answer from everybody. Yeah. So take take your sine theta graph say two and shift it by 90 degrees to the left. Whatever graph is obtained, you will automatically obtain the graph of cos theta from there. So it proves sine 90 plus theta is cos theta. Well done. Most of you have got it right. It's minus sine theta. See again, same history. Use your past. We learned that cos 90 degree minus theta is sine theta. What do you do here? Replace theta with a minus theta replace theta with a minus theta. So when you do that, you get cos 90 minus minus theta. Okay. What is cos 90 minus minus theta? It's cos 90 plus theta. And that's what I need, isn't it? So sine negative theta is negative sine theta. Why? Because sine is an odd function. So this becomes minus sine theta. Now the other results you can easily obtain. For example, if you want tan 90 plus theta, take the ratio of these two. So this will be the ratio of these two guys. So this will be minus cot theta. Correct. Cosic is the reciprocal of the first one. So it will become seek theta. Seek is the reciprocal of the second one. So minus cos theta. So cot is the reciprocal of the third one. So it's minus tan theta. Okay. Again, guys and girls, please, please, please do not misinterpret it as if I'm saying this thing will always be negative. No. The external negative sign doesn't mean this entire term is negative. Stop thinking in that way. Okay. Right. The other day, somebody was saying, sir, cos 90 plus theta is always negative because you wrote a negative sign here. No, it depends on theta. Okay. So if you sign theta itself is negative, then it will become a positive quantity. Are you getting my point? And this is identity. And please understand this is true for any angle theta for which these functions are defined. This is not only limited to an acute angle or something like that. You can put any theta you want, but provided that function must exist for that theta. Now, for example, tan 90 plus theta, don't put theta as zero because neither tan 90 is defined nor cot zero is defined. Okay. So in that case, your theta cannot be zero. Identity doesn't mean you feel whatever you feel like you're putting inside. No, you can put only those things for which those functions exist. Isn't it? Okay. Now, this particular thing can go on and on forever. Now I can ask you 180 minus theta series. So let's do that. By the way, can I switch the slide? Anybody who wants to copy this? Any questions? Ask. Sir, can you explain the sign 90 minus theta is equal to cos theta? Sine 90 minus theta is cos theta. Yes, sir. This you want me to explain? Yes, sir. See, sine 90 minus theta is cos theta. If you look at the graphs, let's come from the graph point of you, Pratich. Sine x graph. Sir, not this, sir. The right side one only. You said 90 minus theta, no? Sir, there it was written 90 minus theta. Sine 90 plus theta is equal to cos theta. You want me to get this result, right? Yes. This result is convinced, you know, you're aware of this result? Yes, sir. No doubt? No, sir. Change theta with minus theta. Can we do that? Yes, sir. When we do that here, what does it become? 90 minus minus theta, what does it become plus theta? Yes, sir. Here it will become cos of minus theta, which is cos theta. Cos of minus theta is cos theta, right? Just turn your page. I think one page to the left, you'll see that I'd given a negative angle identities. Yes, sir. That's how this result comes. Got it for this. Got it? Okay. Yes, sir. Siddharth, Karthik and Karthik Sanoj, they're two different people. Are you? I thought both of them were same. And both of you are in Kormangala, NB's Kormangala. Are you mistaken identity? Siddharth, Karthik is in SSR view. Okay, okay. I gave so many attendance to Siddharth, Karthik thinking it is, sorry, I gave so many attendance to Karthik Sanoj thinking your Siddharth, Karthik. Okay, so we'll move on to the 180 minus theta series. 180 minus theta series. Okay, same set of questions I would like to ask you my dear. What do you think is sine 180 minus theta? What do you think is cos 180 minus theta? What do you think is tan 180 minus theta? What do you think is cosik 180 minus theta? You're writing, it is a painful task. Okay, ditto, ditto, ditto. Now again, no need to remember anything. Many people they sit and remember these complementary and supplementary angle identity, no need. If you remember the past, in fact, I'll give you a trick also in some time, which I've kept it as a secret as of now. If you remember the results that you have already derived, you can actually come up with these results easily. So now I would request you all apply your prior knowledge and tell me what you think is the answer for this and why. Excellent, Siddharth, Karthik. S.K. Excellent, excellent. Many of you have already got the idea. Beautiful. So we have already seen sine 90 degree plus theta is cos theta. Now what I'm going to do, I'm going to replace. Tell you what should I replace theta with? You say, sir, replace theta with 90 minus theta because if I do that, I end up getting 90 plus 90 minus theta and I have to make the same replacement on the right side as well. I can't do only at one place. So if this theta I'm replacing, this theta also has to be accordingly replaced. So this automatically becomes 180 minus theta and this automatically becomes sine theta, which you already know, you already know. So this result is sine theta. Okay, give me the second one quickly, quickly, everybody fast. Yes, somebody was saying something. Cos theta, sir. Cos theta, illa illa illa, it will be minus cos theta. Why? We'll figure it out also. Not a big deal. Cos 90 plus theta is minus sine theta. Change your theta with 90 degree minus theta. So it'll become cos 90 plus 90 minus theta, which is negative sine 90 minus theta and this is automatically what you need and this is minus of cos theta. So answer is minus cos theta. Clear? I think it was Tejaswini who said cos theta. Now it is clear why it is minus cos theta. Sorry. If I've taken the wrong name, please forgive me. Is it clear now everybody? Okay. Now getting the other values are very easy. You already know sine 180 minus theta, cos 180 minus theta. So for tan 180 minus theta, what you have to do? You say, sir, take the ratio. So if you take the ratio of sine and minus cos, what do you get? Minus tan theta. For cosec, you take the reciprocal of the first one. So you'll get cosec theta. For sec, you take the reciprocal of the second one. You'll get minus sec theta. For cot, you take the reciprocal of the tan one. You'll get minus cot theta. Is it fine? Could you go to the right? Okay. So now we can go on and on and on and on and on and on. That means I can now give you 180 plus theta series to 70 minus theta series to 70 plus theta series, 360 minus theta series, 360 plus theta series, 450 minus theta series, 450 plus there's no end to it. And of course, it is not possible to sit and write all those identities. They are never ending. So what I will do now, I will give you a trick to get these identities on your fingertips. You don't have to memorize anything other than the trick. Yes, yes, all these, see graph, graph doesn't give you a different picture, right? Say two, there is a linkage in all these concepts. And this linkage is very, very strong. Whatever you can get with these identities, the same thing you can get with graphs as well. In fact, one or two questions you should try looking at the graphical transformation to see whether those identities actually come. Correct? For example, I just now told you sine 180 minus theta is sine theta. Now, what do you do with your sine theta graph? Change your theta with minus theta means reflected about y axis. Then change your theta with theta minus 180 means shifted by 180 degree to the right. You'll see you'll get back the sine theta graph again. So this can easily be proved by graph as well. Graph doesn't give a different picture, right? Graph go upload outsider ke tera mud treat karo. Okay, that is not my outsider. It is there to help you out and it doesn't sync with whatever you know. Okay, so definitely this can be proved using graph as well. Okay. Now, as I told you, this identity can go on and on. So I can give you 180 plus theta series, I can give you 270 minus theta series, I can give you 270 plus theta series, I can give you 360 minus theta series, I can give you 360 plus theta series, I can give you 450 minus. But giving these series, you will end up forgetting it. So this can go on and on, on and on. Okay. So let's do one thing. Let's try to figure out a trick which will help you to remember these identities no matter whatever in a complimentary or supplementary angle identity you are going to take up. Okay, so let's go to the next page. Can I go to the next page? If you have copied everything from here, can I go to the next page? Done. Okay, great. Now, what is the trick? Guys, the word here is trick. Okay, it's a trick. Don't apply logic to it. It's a trick. Okay, of course, it will be logical to a certain extent, but many people start debating against it. So this is just a trick. Okay. A shortcut way to remember the identities or complimentary and supplementary angle identities. Okay, let's say the question setter has given you a trigonometric ratio and he has given you a multiple of 90 degree plus or minus theta and you want to know what is this equal to or you want to complete this identity. Okay, you want to know what is this equal to. Okay, so this equal to this answer, it has got two parts to it. It will have a sign. If you go back, if you go back, if you go back, you can see that there were signs in front of it. It was either plus sign theta or minus cos theta, etc. So there were some signs involved. Okay, so it has two components. It has got a sign to it and it has it has got a trigonometric ratio for that angle theta. Yes or no. Okay, so the answer or you can say that the identities which I have discussed to you, it has got a sign component to it and it has got some expression in theta, something like this. Okay, this equal to be any of the six trigonometric functions, right? So how do you decide sign and how do you decide E? That is what I'm going to tell you right now. So please everybody pay attention. In fact, stop writing anything. Listen to me. I'll give you ample time to copy down. Okay, no need to write down anything right now. Okay, the first thing is sign. How do we figure out sign? What is E? E means sign, cos, tan, c, c. Now how do you figure out that what will come there? Sign will come, cos will come, tan will come. Okay, that is your E. Okay, E I basically use E for expression. Okay, so what what trigonometric ratio will come there? Okay, now everybody please pay attention. The first part is the sign part. Guys, again, this is a trick. Don't start arguing here. Okay, for the trick, what do I tell students is that you assume your, let me write it in yellow, you assume your theta to be any acute angle. Okay, now here itself, many people will say, sir, I have an objection. You said it's an identity. It can work for any angle theta. I still stand by it. Yes, it's an identity. It can work for any angle theta. But what did I tell you just now? I'm using this as a trick. Okay. Okay, just for a trick just for purpose of remembering the formula. Yes, in reality, theta could be beyond an acute angle also. Okay, but this is just to get my answer in a faster way. Are you getting my point? So theta could be 500 degrees also. Theta could be one lakh degree also. Theta could be, you know, minus one lakh degree also, right? So it need not be acute. It is just for getting my result in a faster way. That's why I wrote trick and I circled it also. Okay. Now, once you get, once you assume your theta to be acute, just figure out the quadrant of, quadrant of that given angle to you this, what are the quadrant of this guy? Okay, you all are now expert at finding the quadrants. We had a good amount of problem practice in the last class as well. Okay. And then figure out what is the sign of that trigonometric ratio given to you, this trigonometric ratio given to you for an angle lying in, angle lying, lying in this quadrant, whatever quadrant you have figured out, that will be the sign. Are you getting my point? Okay, I'll take an example. Cos 180 minus theta. I want to know what is the sign that will come over here. You already know the result. So you can match your answer with what I'm going to get down. See, if I assume my theta to be acute, assume it to be acute and you can assume to be anything like assume theta to be five degree. Let's say I use an acute angle. So 180 minus five, which is 175, which quadrant will it lie? You'll say second quadrant. In second quadrant, cost is positive or negative? Cost is positive or negative? You'll say it's a negative. So negative will come over here in your formula. Go and check the previous slide. Let me take you to the previous slide. Do you see there's a negative sign coming here? See here, negative sign is coming or not? Yes. So this trick helps you to get what external sign will come in the identity. So once you've got it, the sign part is taken care of. Any question, any concern related to sign? Any question, any concern related to sign? So sign is taken care of. Now, what will be your E coming over here? So what will be the trigonometric ratio after that sign that will come over here? So for E, the rule is quite simple. The rule is if N is even, that means this N, this N, if this is an even number, then your E will be same as the trigonometric ratio given to you in the question. For example, this is 2 into 90 degrees. 2 is even number. So this will be cos of theta itself. The same trigonometric ratio will come over here. Now again, go and check your result, which I derived in the previous page. See here, cos theta comes. Getting my point. But if N is odd, then your E will become the complementary of that trigonometric ratio. The complementary of that trigonometric ratio. So if it was odd, then cos will become sin. Sin will become cos. Tan will become pot. Pot will become tan. Seek will become cosik. Cosik will become seek. Are you getting my point? Are these two things clear? What does your E become and what does your sign become depending upon whatever is the question given to you? Let us take few more examples to get this fundamental clear. Let us say, let us say, the question setter has asked me, sin 360 degree minus theta. He wants me to complete this identity. So what is this answer in terms of theta? He is asking me. So what will I do first? So I need to know the sign first of all. Does that identity have any sign into it? Let's try to figure that out. So for that, what I will do, I will assume theta to be, again, this is an assumption. I will assume theta to be some acute angle. Let's say 10 degree. Anything you can assume between, let's say 5 degree, 10 degree, 30 degree, 45 degree, 60 degree, 75 degree, 89 degree, whatever you want. So 360 minus 10 degree is 350. 350 will lie in which quadrant. 350 will lie in which quadrant. Say 2 will tell me. Say 2. 350 is in which quadrant. Fast, fast. Fourth quadrant, exactly. In fourth quadrant, what is the sign of sign? They are rhyming words. What is the sign of sign? Negative. So they will external negative sign. Got it? So in this formula, which I am going to write, there will be a negative sign on the right side. External negative sign. I am not commenting about the whole value. Now, 360 degree is even multiple of 90 degree. This is even number. So sign will remain sign, exactly. So your answer will be negative sign theta. So there's no need to remember anything other than the stick which I have told you to get these identities. So any complimentary supplementary angle identity somebody gives you, you will be able to answer it in this way. So sign 360 degree minus theta is negative sign theta. Done. This identity is known to us. Yeah, it's absolutely easy to use. Okay, let's take more examples. Let's take more examples. Let's take second example. Example number two. Complete this identity and 630 degree plus theta. Complete this. You can do it very fast. You just have to practice. Because I'm explaining you things are taking a bit of time. But if you start applying it, it's very fast. Yes, I would request everybody to give the answer to this on the chat box. From there, I will come to know that how you have understood this concept. Okay, Charlie. Very good. Okay, I can see difference in the sign coming up. Why is that? So, guys, everybody should give me the same answer and of course, right answer. Yes, come on. Everybody give it a try. Right now, I can see two answers on my chat box, which is unfortunate should not happen. Why not sure? See, there are two things. Sign and expression. Okay, now sign part first of all, theta, let's say acute angle. Let's say theta is 40 degrees. Okay, let theta be 40 degrees. So 670 degree will lie in which quadrant? 670 lies in which quadrant? Fourth quadrant? Get it? I think that is clear to everybody. Anybody having said trouble with finding the quadrant? I don't think so you should be having. You already did two DPPs and all. Right? Tan in the fourth quadrant. Is it positive or negative? Tan in the fourth quadrant. Is it positive or negative? Siddharth. Siddharth Karthik. Negative sign will come. Now, a 630 degree is 7 into 90. 7 is an odd number. Odd means you have to use the complementary of the given trigonometric ratio. So what is complementary ratio of tan? Caught. So this is minus cot theta. Done. Problem is solved. Okay, so tan 360 degree plus, sorry, tan 630 degree plus theta is negative cot theta. This identity is complete now. Clear? Should I give you some questions now? Will all of you try it out? Okay, let's take at least four or five questions based on this concept. Can I go to the next slide? Can I go to the next slide? Okay, try this out. Complete the following identities. Guys and girls, I would again like to reassure you, again like to reiterate to you, these are identities. Means you can put any angle of theta in any angle for theta provided for that the particular trigonometric ratio should be defined. So it is not limited to only acute angle. Acute angle trick was only to get that external sign. That's it. Beyond that, there is no use of theta being an acute. Okay. Acute obtuse, whatever it is, it is going to work. Okay. Complete the following identities. Okay, so let me ask you cos 720 plus theta. Cosic 540 plus theta. Caught 810 degree minus theta. Seek 990 minus theta. Okay. Please put the question number while you are answering it. Okay. And try to answer them all at the same time. What is going to be the answer if it is sine to the power 4, 180 minus theta. Sine to the power 4, 180 minus theta is nothing but sine to the power 4 theta. Over. Caught it yesterday. So an even power on that particular expression basically, you know, does away with the negative signs. Right. So with respect to sign, you can by default put a positive. All you need to take care is whether it is even multiple of 90 or odd multiple of 90. That's it. Again, those who are answering, please answer it at the same go. Else what will happen? You will answer one. Somebody else will answer one. Then you will answer two. So it will get all mixed up. So press and enter only when you have solved all the four parts. Okay. Okay. Very good. Very good. Very good. Very good. Very good. Very good. Some here and there sign mistakes. I can see that is still some confusion related to locating the quadrant. Actually speaking, the quadrant that you are finding is not the actual quadrant for any angle theta. It is just a trick actually, because you don't know theta. Theta we don't know, right? Theta can be anything that the user puts. So you cannot figure out the quadrant actually without knowing the theta. So the trick which I told you is just a shortcut way. Okay. It doesn't give you the actual quadrant. Don't get me wrong. Theta is unknown to you. Theta could be like one lakh degree also. So when you locate the quadrant, it is just a trick to get the identity done. It doesn't actually give you the quadrant of that particular angle. Okay. Very good. So we'll discuss it. First of all, cost 720 plus theta. Again, in my mind, I will teach theta to be some acute angle. Okay. Let's say 15 degree. So 720 plus 15 is 735. 735 lies in the first quadrant. Okay. First quadrant cost is positive. So there will be a positive sign in my identity and 720 is nothing but eight into 90 degree. Eight is even. So cost will remain cost. So the first answer will be cost theta. Okay. You will sell for part on your back. If you have got it, correct. Very, very well done. Okay. Next one. Cosic 540 plus theta. So again, theta, as you meant to be, let's say 15 degree. So 540 plus theta will be let's say 540 plus 15 degree, which is 555. 555 will lie in the third quadrant. In the third quadrant, Cosec is negative. So it'll be negative. And since this is 6 into 90, 6 is already an even number, it will remain Cosec. So second one answer will be negative Cosec theta. Okay. Made a mistake. Manu made a mistake. Even Sharduli made a mistake. Is it fine? Any questions here with the second one? Second one is all set. Okay. Third one. 810 degree you are at this position. 810 degree means you have taken one complete, two complete and then 90 more. Right. So you are at this position minus theta. Let's say minus 10 degree. So you fall in the first quadrant. Okay. In first quadrant, everything is positive. So there will be a positive only. And this is 9 into 90 degree. 9 is an odd number. This is an odd number. So cot will become its complimentary. So it'll become tan theta. Got the point? So the third answer is tan theta. Let me check. Anybody made a sign mistake? Why a sign mistake is happening? Manu also sign mistake is happening. Guys and girls, let me tell you, these are building blocks. These are building blocks of your future concepts. If you make mistakes here, you will always make mistake in the subsequent thing. ASTC rule. ASTC rule is all trigonometric ratios are positive in the first quadrant. Only sine and cosec are positive in the second quadrant. Only tan and cot are positive in the third quadrant. Only cos and sec are positive in the fourth quadrant. This is ASTC rule. Maybe you should watch the previous lecture once again. Okay. Next one. So seek 990 minus theta. I can take theta to be let's say 30 degrees. So 960. 960 will lie in the third quadrant. 960. By the way, 990 is in this position. Okay. Correct. 990 is in this position. So minus means you are here. You are in the third quadrant. Means clockwise. Move clockwise theta from there. Okay. So this is clockwise from there. Theta. So in the third quadrant, seek is known to be negative. And this is 11 into 90. 11 is an odd number. So seek will become cosec. So last answer will be minus cosec theta. Okay. How many of you got everything correct? Very good, Vaishnav. Excellent. Excellent. Sethu also got everything correct. Awesome. Now I will slightly complicate this. Sir, why? Okay. Complete the following identities. By the way, I'm not asking you find the values. No, they are not values. Without theta, you cannot find the value. It's an identity that you are trying to complete over it. Okay. So get the question clear. So let's take slightly ugly figures. Okay. Okay. Got, let's say 117 pi by 2 minus theta. So I have changed, first of all, degrees to radians. And I have complicated this n. Okay. 117. Okay. 984 pi by 2 minus theta. I will not give you a lot. I will just give you one more. Maybe seek minus 601 pi by 2 plus theta. Okay. Meanwhile, I would request you to only do the first one here because once you do the first one, I will discuss. I will discuss with you and then we can do the subsequent ones. Compliment of seek is cosec, not cosine. Cosine is the reciprocal. Get the difference, Nikhil. Reciprocal is not the compliment. Science compliment is cost, but science reciprocal is cosec. There's a difference. Okay. First one, are you done? Just give the answer for the first one. I will discuss the first one with you. Then you can do the second and the third one because I have something interesting to tell you there. Okay. Only two different answers are sitting on my screen. Okay. Prasham, Nikhil and congratulations to NPS Kormangala. I think Pranjal got again gold medal in IMO. I hope you know he's your senior. Again, he got twice two gold medals, which is very rare, the rarest of rare. I think first attempt he got silver, then he got two golds. This fellow actually started very early. He started preparing in class six onwards. Six, he did all these, you know, geometry, tignometry, number theory, et cetera. So that helped him. So yeah, yeah, yeah. First one done. Are you caught? Also you are saying Siddharth. God, you're in Siddharth. This is an odd numbers Siddharth. It has to be a complementary of it. So it will be either plus tan theta or minus tan theta. So any of the two answers if you'd have given, I would still consider it to be, you know, having some possibility of being correct. Odd number. This is odd number. Odd number means complementary will come. So either the answer will be plus tan theta or minus tan theta. These are the only two possibilities, right? Okay. Hello, everybody will now pay attention here. Okay. Keep down your pens. Listen to me. Now, when I make this number two large, people start getting confused with respect to quadrants. Okay. When it was a multiple of pi, it was fine. Okay. In multiples of pi, which we basically applied when you were doing the allied angles, if it was odd multiple of pi used to keep your pen here, even multiple of pi used to keep your pen here, right? But now this is not a multiple of pi. It is actually a multiple of pi by two. Okay. So what change happens? Now, see here, everybody please pay attention. So how do I find my quadrant in a faster way? So that I don't have to, you know, move circle, circle, circle on my, on my question paper. Okay. Listen to this. This number, let me call it as n. Okay. If your n is a multiple of four, start with positive x-axis. Okay. Minus means clockwise by theta. Theta is the acute angle which you have assumed. Okay. If this number is a multiple of four plus one, keep your pen on this position. If your number is a multiple of four plus two, keep your pen at this position. And if your number is a multiple of four plus three, keep your pen in this position that will help you to easily figure out the quadrant without having to go circle, circle, circle, circle on your paper. Okay. Now see how easily I solve this question. 117 is nothing but four into 29 plus one. Am I right? Am I right? So it is of the type 4k plus one. So where will I keep my pen? I will keep my pen here. Then minus, minus means clockwise by some angle. Let's assume it to be 30 degrees. Okay. So clockwise I will move from this position by 30 degrees. Which quadrant did I fall in? First quadrant. First quadrant, cot is known to be positive. Now this is an odd number. So cot's complementary will come over here. So the answer to the first question is tan of theta. So let me check who all got it right. Sharduli got it right. Nikhil got it right. Prism got it right. Is it clear how it works? So this is something that new you are learning. If your expression is a multiple of pi by two, this is how you figure out the quadrants quickly. Okay. Of course, quadrant subject to your assumption of theta. In reality, we cannot know the quadrant because theta is unknown. But we assume theta to be something and then figure out the quadrant just to, just as a trick. Okay. Just as a trick. Okay. Now the second one, please do not do any mistake. See the number that you have given a Karthik. Okay. Figure out in which of the three, four category it falls. Is it a multiple of four? In fact, divided by four. See whether it leaves a remainder zero or leaves a remainder one or leaves a remainder two or leaves a remainder three. Depending upon that, you position your starting point at that particular line. For example, if it's a multiple of four, put your pen on the positive X axis. If it's the multiple of four plus one, that means it leaves a remainder of one when divided by four, then keep your pen on the positive Y axis. If it leaves the remainder of two, when you divide it by four, keep your pen on the negative X axis. If it leaves a remainder of three, when you divide it by four, keep your pen on the negative Y axis. And from there, you figure out what is the quadrant. Got it. Karthik. Don't worry. We'll take few more examples. It will be very clear to you. Okay. So I'm getting a, oh, why some people said minus, some people said plus. This is, I don't like. Why there's a mistake happening? Sir, we are sorry, sir. We are just learning here. Okay. Let me erase the previous data. See, 984 is a perfect multiple of four. If you divide it by four, by the way, how do you test whether a number is divisible by four? The shortest way is the last two numbers should be divisible by four. Okay. 84 is definitely divisible by four, right? So it will be a multiple of four. It's of the nature of four K. Of course, if you want to find K, that's a different thing, but it doesn't require me to actually get the K. By the way, if you want to know it, it's going to be 246. Am I right? Okay. So it's a perfect multiple of four. So you place your, you start, you keep your pen here. Okay. Now, minus, minus means clockwise theta. Assume some theta in your mind. That's a 60 degree. So clockwise move by 60 degree. Which quadrant did you fall in? Which quadrant did you fall in? Fourth quadrant. Okay. Fourth quadrant, Cossack is positive or negative? Cossack is positive or negative in fourth quadrant. Tell me. Tell me. Tell me. Tell me. Arun Dutty. Tell me. Negative, no? Arun Dutty. So why did you write that? Okay. And 984 is an even number. So Cossack remains Cossack. So answer is minus Cossack theta. Got it. So those who got minus Cossack theta, give yourself a pat on your back. Absolutely right. Which axis to start from? I've already written, no? If you have a multiple of four, start with positive x-axis. If you have multiple of four plus one, that means remainder is one. Start with positive y-axis. If you have multiple of four plus two, start with negative x-axis. If you have multiple of four plus three, start with the negative y-axis. I've written it on the screen. What is the confusion? Okay. Last question. If everybody gets this right, I will not give you any more. If you get it wrong, I'll give you two more. Sir, we want two more. We will make mistakes. Any trick you use here, I want the answer, Manu. You want to use graph, you use graph. You want to use any trick, use it. Okay, Shardali. Very good. It cannot be seek, because 601 is an odd number. Come on. So it has to be, go see. Oh, sorry. Manu, Manu, not Setu. Sorry, sorry, Setu. I hope you didn't feel bad. Oh, sir. Today I will cry, sir. You took my name for all reasons. Oh, well, an ideal cry, no? Okay, Siddharth. Very good. Guys, this is something which many people make mistakes. And I can see that mistake happening. What is this mistake? The mistake is people considering 601 as actually 4k plus 1. This is the biggest mistake. Right? It is not 4k plus 1 type. It is 4k plus 3 type. See here, minus 601 is 4 into minus 151 plus 3. What do you people or what many people do here is that they ignore this negative sign and this day 601 is 4 into 150 plus 1. As a result, they start thinking it as 4k plus 1 type, which is wrong. It is actually minus, it is actually 4k plus 3 type. Okay. So you are here. Okay. Plus means anticlockwise by theta means you land up in the fourth quadrant. In fourth quadrant, seek is already positive. Correct? And 601 is an odd number. So it will be complementary of that. So the answer to this question is positive cos theta, which I think only few of you said, most of you went for minus cos theta. Okay. So be very, very careful. You may be tricked like this. Okay. One another method is you could actually make it as seek 601 pi by 2 minus theta also, because as we have already discussed, seek and cos, they do not care about the negativity of the angle. So why not we make this angle, multiply this angle with a minus sign? In this case, your 601 will actually be 4k plus 1 type. That means you are at this location. Minus means clockwise by theta. So you land up in the first quadrant that still makes it positive. That still makes it positive. Positivity is not going to change because of that. Right. And still your answer is going to be positive. So no matter whatever approach you apply, result will remain the same. It is not going to change. Is it fine? Any questions, any concerns? Okay. Good. Now that you understood this, we are now going to move towards, and let's say a few questions. Maybe one or two questions you will take on this. Okay. A few questions. Yeah. Maybe this one. Simple one. Find the value of this expression. I'm so sorry. Can I go back to the previous slide? I think Karthik and Seethu have some questions. See, what trick I use here? You're asking about this guy. See, do you remember this thing that I told you? This and this are same things? Okay. No. So whether you write this or you take a negative of this, it doesn't make any difference to the answer. No. Which one? The one before you give, which one? This whole trick you're talking about. Karthik, I didn't get you what you want. The one during complementary and supplementary angles. Okay. Okay. One second. You want me to take you to the theory part of it. After 180 series, I just told you that you can start using this trick. What is there you didn't get? And why are you asking it so late? You did eight questions and you're now asking, I didn't get this. All right. Take it down here. Mine, we take it down. This is a materialistic taking it down. Take it intangibly. Tangibly, you will always get it after the class. Okay. Let me know once you're done. Everybody's waiting for you. Quick question. I think now you'll be able to answer these questions like this. Isn't it? Tell me the answer. Hey you, so much time. I thought you'll be bombarding with me with the answer. Yes, Nikhil, my dear, what is that equal to? Oh, you're getting C square. I read it as sine square. Okay. Okay. Just any charlotte correct. Just have a relook at it, Nikhil, because costs, you're still treating cost complimentary as seek. You have confusion between reciprocal and complimentary. Nikhil, you still have confusion between reciprocal and complimentary. Reciprocal and complimentary and not same things other than in tan it happens. In tan, cot is the reciprocal also and complimentary also. Very good. So costs, first of all, costs 270 plus theta. Okay. Now, in my mind, I'll treat theta to be 10 degree. 270 plus 10 takes me to the fourth quadrant. Fourth quadrant, it is positive. This is odd multiple of 90. So it will become sine theta. So costs 270 plus theta is sine theta. Okay. Cost 90 minus theta. Now you already know it. Okay. So this whole guy, this whole guy is sine square theta. Okay. Next, sine 270 minus theta 270 take theta as acute angle 270 minus theta will land you up in the third quadrant and third quadrant sign is negative 270 is an odd multiple of 90. So it will become complimentary of sign, which is cost. So sine 270 minus theta is minus cost theta or negative cost theta and cost theta is anyways cost theta. So this will become negative of negative cost theta, cost theta, which is actually sine square theta plus cost square theta, which is one two. Answer is one. Is it fine? Any questions, any concerns? Why are you doing mistake in such small questions? Can you go to the next one? Next one, next one, next one, next one, next one. Okay. Okay. Now these type of questions also many people are asking me how to solve it. I think these were there in your DPPs as well. Okay. Sine square theta is equal to this. Then x must be. Okay. Now how to solve this question? Basically, you have to remember your domain range concept. We know sine theta actually lies between minus one to one. So sine square theta will actually lie between zero to one. Are you getting my one? So this quantity that you have, this should actually be greater than zero and less than one. Okay. So solve this as two different inequalities. One is x square plus y square plus one by two x is greater than equal to zero. And the other inequality is x square plus y square plus one by two x is less than equal to one. Correct? Now here, the only thing that will come in your mind is x should be positive. That's it. Okay. Let's look into this. It should be greater than equal to zero. No, not equal to zero. It should be positive. Okay. If you look at this, it gives you x square plus y square plus one. Okay. Less than equal to two x. Now how did I take two x on the top? How did I multiply here two x? Because I knew my x is positive. So two x will also be positive. And in an inequality, you may have also done in school that if you are multiplying both sides by a positive quantity, the inequality doesn't change. Correct? Now this is as good as saying x square minus two x plus one plus y square is less than equal to zero. Okay. This is as good as saying x minus one, the whole square plus y square is less than equal to zero. Now the sum of two squares cannot be less than zero. It could be max to max equal to zero. So from here, only one conclusion comes that this could be equal to zero. And if it is equal to zero, my dear friend, then x has to be one. That means both of these quantities have to be zero individually. That means x has to be one and y has to be zero. That's the only possibility. So this question was asking you, what is x? x has to be one here. Okay. So these type of questions you will see quite a lot in your DPPs as well. I think some of you are asking me also how to solve these kind of questions. So similar type of questions in terms of seek and coseek is also there. So apply this methodology. Okay. Asetu, you have some doubt in the previous solution. Let me know what is your doubt? Only after copying this, where did square go here? You want me to go to the previous slide? Anybody wants me to go to the previous slide? Any explanation you want? Any question? Yes, yes, yes. Why not? Unmute yourself. Previous to previous slide. Yes, Setu, tell me. So can you hear me? Yeah, I can hear you. Sir, in this trick here in the second last step, in that where you wrote sec minus theta is sec theta. So here sec 601 pi by 2 here. When we use the trick, 601 is 4 into 150 plus 1. So we start from 4k plus 1 and then we go and then we go clockwise. So we get it in first quadrant. Sine is there, right? Sine is possibly there. Oh, for sine. See, quadrant, quadrant, so nobody knows. I am saying that we don't know the quadrant till we know our theta. Nobody can guess the quadrant. This is a trick only. And with this trick, you will manage to get the sine. Now, whether you are getting it by doing this trick or that trick, sine will be positive there. That's it. That's what we are trying to say. Okay, don't judge me by the quadrant here because quadrant, nobody knows. How will you come to know the quadrant? It is just a trick. At the point, Setu, let's take few more questions based on this. I think I didn't pick up many questions based on it. Okay, we can take this one, by the way, maybe a simple one for you to solve. Prove that this is equal to negative one. Prove that this is equal to negative one. After this, we'll take a break. On the other side of the break, we'll be discussing compound angle identities that is very important for us. Just say done once you're done. No need to give me any explanation. This is all theta, by the way. This is theta, theta, theta. The print is not that great actually, so I wrote theta. Excellent. Very good, Nikhil. Dan Prasim, very good. Okay, numerator, let us write it. Cost 90 plus theta is minus sine theta. Seek minus theta is just seek theta. Tan 180 minus theta is minus tan theta. Denominator. Seek 360 degree minus theta is seek theta. Sine 180 plus theta is minus sine theta. Cost 90 minus theta is tan theta. Now see what will get cancelled? Sine will get cancelled. In fact, this whole thing will get cancelled with this whole thing. Seek will get cancelled with this. This will get cancelled with this, leaving you with a minus one, which is your RHS. Okay, so we will now take a break because already 6.15. We'll take a break till 6.30 and on the other side of the break, we will discuss our compound angle identities that is long waiting for us. See you on the other side of the break. Enjoy your break. So dear students, the next concept that we are going to do is even very, very more important than whatever we have done so far because that is the heart and soul of your class 11 trigonometry and that is called compound angle identities. Compound angle identities. This chapter is full of identities, identities, identities everywhere. Now we are going to talk about compound angle identities. Now the identities which you learned earlier, whatever identities you learned, let's say complementary, supplementary, negative, Pythagorean, whatever identity you learned, there was only one single variable involved, theta or X, whatever, isn't it? Now we are going to talk about such identities where there will be two unknowns or two angles involved. So what's a compound angle? The word compound here is basically a representative of, let's say if you're adding to angle or you're subtracting to angle or it could be more than two angles also. For example, if you're doing this or let's say you're doing a mix and match of these operations. So all such angles are basically called compound angles. Now word compound is already familiar to you in chemistry. Basically it's a pure substance made up of elements, it's a compound, isn't it? So in case of trigonometry also we are going to learn how to find sine, cos, tan, cot, etc. for these compound angles. I'm going to start with two angles A and B as of now and we'll start with our cos A plus B identity. So we will derive this identity first of all and then using this identity we'll be deriving other compound angle identities. So let us derive what is cos A plus B. I know everybody knows the result. Cos A cos B minus sin A sin B. So we will officially derive it in the session now. But before that I would like you to answer one small question of mine. So I have a very small question to ask. Let's say this is a unit circle. This is a unit circle. Unit circle means a circle whose radius is 1. Now I have a question for you. If at an angle theta let's say I basically found out a point over here. Let's say I call this point as capital X. So we know that this is a unit circle. So this is 1 and let's say this angle is theta and I reach a point X on this circle. I have written that point as a capital X point. Can you tell me in terms of theta what is the coordinate of X? What is the X coordinate and what is the Y coordinate of X in terms of theta? Who will tell me? So people are telling me it is cos theta comma sin theta. Now my question here is let's say if this theta were here. Let's say this was your full theta and I reach a point here Y. What will be the coordinates of Y? Will your answer change? Will your answer change? If yes how? Okay. First of all they just didn't have this question. So how is it cos theta and sin theta? First of all tell me this one. Forget about Y point C. They just didn't X coordinate is nothing but this length. This is your X coordinate and this is your Y coordinate. So in a right angle triangle if this is 1 and this is theta what is the base? Base is cos theta, height is sin theta. Right? So this is your X coordinate. This is your Y coordinate. Isn't it? Okay. Is that clear to you Tereshwini now? Okay. Now some of you are saying the answer is not going to change even if I take my theta in the second coordinate. But some of you are saying my answer will become minus cos theta comma sin theta. Now which one is correct and why? Okay. To surprise of most of you the answer is not going to change. It is just going to be cos theta comma sin theta. In fact it is not going to change even if my theta were in the third coordinate. Let's say I reached a point Z. It will still be cos theta comma sin theta. Okay. Now why it is not going to change? Because please note that the sign that you are trying to put externally that is already accounted for in cos theta value. So if your cos theta is, if your theta is in the second quadrant, cos theta is already negative. Correct? So why you are putting an external negative sign to it? If you are putting an external negative sign to it, in fact you are making it positive, which will be wrong in that case. Okay. So this is very important for you to know that no matter whichever quadrant you are referring to, sin and cos signs will change as per the requirement of the coordinates there. You don't have to put an external sign to it. Okay. So the sign is already taken into account. For example, let's say if you are in the third quadrant, let's say Z is in the third quadrant. In third quadrant sin and cos, sorry, the x and the y coordinates should both be negative, which is coming now because cos theta and sin theta, if theta is in the third quadrant are both negative. So it is already accounted for. Why I have to put an external negative sign? I don't have to put an external negative sign. In fact, if I do, I'll make it wrong. Okay. So this is something very important because based on this, we are going to now figure out a few things. Let us say I do another unit circle, okay, slightly bigger one this time. And this is my y-axis. This is my x-axis. Okay. Now, this is again a unit circle. In this circle, at an angle of A, I take a point, let's say p. Okay. And of course, let's say this point is q. Okay. And let's say at another angle, b from here, I obtain a point q. And at a negative angle minus b from here, I obtain a point R. By the way, this negative just shows the direction because I've taken it in a clockwise manner. Okay. So this is my, this is my four points on your screen right now. Okay. Now I would request everybody to give me the coordinates of all these four points which you see on your screen right now. Okay. Write your answer like this. q coordinate, p coordinate, r coordinate. By the way, I have my mistake named two coordinates as q. So p, q, r, let me call this as s. Sorry, two coordinate, two points I named it by the same alphabet. Yeah. So give me a response as what is the coordinates of q, p, r, and s or p, q, r, and s. Let's see. Let's see if you have taken the learnings from the previous small exercise that we had. Okay. Do you all agree that q has to be one comma zero? Do you all agree q has to be one comma zero? Okay. p has to be cos a comma sine s has to be cos a plus b comma sine a plus b and r has to be cos minus b sine minus b. Correct. Okay. Now cos minus b sine minus b is as we were saying cos b comma minus sine b. Okay. Now all of you check whether you have got similar results or not. Okay. Now how am I going to use these? All of you please pay attention. I want you all to focus on these two triangles in this particular diagram. One is the triangle, p, o, r. Okay. And another is the triangle q, o, s. Okay. So all of you please pay attention to these two points that you see on your, these two triangles that you see in the diagram one is q, o, s. And the other triangle is p, o, r. What can you say about these two triangles? What can you say about these two triangles? Are they related to each other? You'll say, yes, sir, they are congruent to each other. Right? By, by which rule of congruency? Sus. Okay. s, a, s. Because the sandwich angle is same a plus b in both of them. By the way, don't get carried with the negative b. Negative just shows the direction. Magnitude wise, it's b only. Okay. So when you talk about geometry, we don't care about the direction of the angle. We just take the magnitude. Okay. So these two triangles are congruent. If these two triangles are congruent, can I say q, s, that is the green side length is equal to p, r, which is your blue side length. Correct? Now, in your coordinate geometry, you have learned that when you know the coordinates of ends of two points, or you know the coordinates of two points, what is the line segment joining them? You already know the formula, isn't it? So what is the length of a line segment joining x1, y1 and x2, y2? You'll say, sir, under root of x1 minus x2 the whole square. So in this case, x1 minus x2 will be, by the way, this has got hidden somewhere, x1 minus x2 will be cos a plus b minus 1, the whole square, plus y1 minus y2, which is nothing but sin a plus b minus 0, the whole square. Correct? This is your qs, isn't it? What will be p, r similarly? p, r similarly will be, correct me if I'm wrong, cos a minus cos b whole square and sin a minus minus sin b, minus minus means plus. Correct? Do you all agree with this? Do you all agree with this relation that I have written on the screen right now? Okay, square both sides just to get rid of the under root sign and expand. So if you square this side and expand it, you will end up getting cos square a plus b minus 2 cos a plus b plus 1 and this will be nothing but sin square a plus b. On the right side, you'll end up getting cos square a cos square b minus 2 cos a cos b and sin square a sin square b plus 2 sin a sin b. Okay? Now, let's simplify even further. Cos square a plus b sin square a plus b will become a 1. There wasn't only one sitting over here, so let me write that down as well and minus 2 cos a plus b, let's copy it as it is. Okay? On the right hand side, you will see this will become a 1, this will become a 1, so 1 plus 1 and this will become negative 2, you can take common, you have cos a cos b minus sin a sin b. Till this step, everybody's fine? Any questions here? Cancel off 2 and 2 from this side. Okay? So you'll end up getting minus 2 cos a plus b as minus 2 cos a cos b minus sin a sin b. Okay? Cancel off these minus 2 as well. So ultimately, what do you see? Ultimately, you see the formula of cos a plus b. So the formula of cos a plus b becomes cos a cos b minus sin a sin b. Now everybody, I would request you to demarcate some pages in your notebook for writing down all the identities which we are going to, I know, write today or maybe no subsequent class as well, so that those identities are readily accessible by you. You don't have to turn 50 pages to go, sir. Where is that identity gone? Okay? So demarcate some page. Maybe you can leave five, six sites. Okay? For writing down whatever formula we are deriving right now. Okay? So the first formula that we derived in compound angle is cos a plus b formula and cos a plus b formula. Now I'm completing it. It's cos a cos b minus sin a sin b. I will box it up also so that when you're referring to your notes, you will have an i on it. Okay? So this is your number one formula for the day. Is it fine? Any questions? Now, thanks to this identity that now we will be able to find cos of several angles which otherwise we would not have been able to find out. For example, I could find out cos 75 degree from it. Isn't it? Can we find cos 75 from it? We'll say, sir, yes. Take as 30, bs 45. So right hand side, you know what is cos 30, cos 45, sin 30, sin 45. We can also find cos 105 degrees from it. Okay? So some unknown trigonometric ratios can now also be figured out by using this formula. Okay? So please note this down. And now we are going to use this result to find the subsequent results. Now I will not derive this formula from scratch. So I will, I will not reinvent the wheel. I'll be using this result to get the subsequent formulas. Can I go to the next slide? Anything that you would like to copy here or ask here, please do let me know. By the way, many people asked me, sir, will this derivation be asked? No. This derivation will not be asked. Then everybody. Okay. Should I go to the next side? Okay. Now let's use this result only, find out the formula for cos a minus b. So how will I find cos a minus b from this result? What should I do? What should I do? Any, any hint? Right? So you will say that, sir, this is an identity. This is an identity. You can play with your A and B. You can change their signs. Anything you want to do, you can do with them. You can interchange them. You can say A is equal to B only. You can change B with a minus B also. Okay. So what I'm going to do, I'm going to precisely do this activity. I'm going to write this as cos a plus negative B. Okay. Now in your formula, just treat your B as a negative B. Okay. So in this formula, in this formula, replace B with a negative B. So when you do that, you end up getting cos a plus negative B as cos a cos negative B minus sin a sin negative B. Now, cos is anyways, unperturbed by the negativity of the angle. So cos minus B is as good as cos B cos, cos has no, I mean, effect on the negativity of the angle because it's an even function. We have already seen it. And sin minus B will become minus sin B. So minus minus will become a plus. And there you go. This basically gives you the formula of cos a minus B. So this is the second formula in our kitty. Okay. So please make a note of this. Any questions? Any questions? Any concerns? Okay. Now you are going to tell me, how do I get the formula of sin a plus B by using the result that we have already derived? I don't want to reinvent the wheel. I don't want to use any other way. I already know the identities to identities, which I've derived using it. Can I get sin a plus B? If yes, how? Tell me what should I replace with what? That is what I want to know. And in which formula? What should I replace with what and in which formula to get my answer? Just tell me that. Okay. So Prism, what you're trying to say is that in cos a minus B formula, should I replace a with 90 minus a? Is that what you're trying to say? Exactly. Very good. So we have cos a minus B formula, which is cos a cos B plus sin a sin B. In this formula, what I'm going to do is I'm going to replace, I'm going to replace a with 90 degree minus a. Guys, again, it's an identity. I can play with it. Right. I can play with the identity, but whatever I do on one side, I have to do the same on the other side as well. Now, this basically becomes 90 minus a plus B. Okay. By the way, this becomes sin a cos B cos a sin B. What is cos 90 minus any angle? Sin sin of that angle. So it doesn't it become sin a plus B? So there you go. The formula is in front of us. Sin a plus B is sin a cos B plus cos a sin B. Okay. So note this down. This is our third formula for the day. Okay. So now tell me how do I get sin a minus B formula? Quickly. You can use any of the three results which we have derived so far. See, I'm just making your memorization as less as possible so that you don't have to remember too many stuffs because I know this chapter is full of identities. So I want you to know one of them and start linking the other to it so that somehow if you forget also, but my experience is that nobody forgets these formulas unless and they'll you have never practiced in the military. Right. Shardili. Right. Say to replace your B with a minus B. So in the previous formula, if I replace my B with a minus B, I will end up getting, I will end up getting sin a cos minus B plus cos a sin minus B, which is as good as saying, which is as good as saying sin a cos B minus cos a sin B. So please make a note of this. I'll write it again on top over here minus cos a sin B. So this is the fourth formula derived from the very same result. So whatever I'm putting in box, you keep writing them down in that sheet that you have left aside for noting down these identities. So cos series, sin series is done. Let's do the same for tan as well quickly. Can I go to the next page? Can I go to the next slide? Anything that you would like to copy from here? Have you already done it? Let's go to the next. Now what about tan a plus B formula? Now you will say it's simple. We already know that tan a plus B is sin a plus B upon cos a plus B and we already know the formula for both of them. So this is sin a cos B plus cos a sin B and this is cos a cos B minus sin a sin B. Now fine, this formula is good enough but normally we intend to get this formula in terms of, in terms of tan, in terms of tan only. So what do we do to convert everything in tan? So you'll say it's simple. Divide both the numerator and denominator by cos a cos B. Okay. So when you divide all the terms on the numerator and denominator by cos a cos B, this is what you will see, cos sin a cos B by cos a cos B will give you a tan a. Cos a sin B divided by cos a cos B will give you a tan B. Cos a cos B divided by cos a cos B will give you a 1. Sin a sin B divided by cos a cos B will give you a tan a tan B. Okay. So there you go. This is the formula that you will be getting. I'll write it down over here. This is tan a plus tan B by 1 minus tan a tan B. Okay. Now please note that this is an identity. This is an identity. I agree with you this is an identity but you cannot put any value of a, b or a plus b which is actually a odd multiple of pi by 2. So it should not belong to odd multiple of pi by 2. Please don't down. Neither a nor b nor a plus b by any case should be coming out to be an odd multiple of 90 degree else this condition or all these else this identity is not going to work because the constituent functions themselves will be not defined. Are you getting my point? Now many people will say sir right side, left side both will be undefined so they can still be equal. See undefined quantities are cannot be equated. They are not defined only. Forget about they are not existent. Forget about comparing them. Okay. So I know you would be thinking left will be undefined, right also will be undefined. So they can be still be equal. So this identity works there also. No. Never say to undefined things are equal. Okay. Is it fine? So please note down this is your fifth formula in the list. Is it fine? Now can I get the formula for tan a minus b? Very easy. Let's do that. So for tan a minus b, all you need to do is replace your replace your b with a minus b. So I will not write it again. I'll just use this result and wherever there is a b, I will replace with a minus b. Remember tan minus b is minus tan b because it's an odd function. Here also you'll have one minus minus which is plus tan a tan b. So please make a note of this. This is your identity number six. Here also your a b or a minus b should not be a odd multiple of 5 by 2 or 90 degree. Is it fine? Any questions? Any questions? Okay. Now I have a simple exercise for all of you over here. Please note this down first. Then I'll give you that exercise. Very small exercise. So give me the formula for cot a plus b in terms of cot. Everybody, please try this out. Give me the formula of cot a plus b in terms of cot. And just say I'm done in the chat box once you're done. Done. Done. Okay. Great. Great. Okay. So you'll see it's simple. cot a plus b is reciprocal of tan a plus b and you've already taught us what is tan a plus b. So it is nothing but 1 by tan a plus tan b upon 1 minus tan a tan b. Now we know that the denominator's denominator will actually become a numerator. Okay. But unfortunately this answer is in terms of tan. But I want in terms of cot. Okay. So what I'm going to do is I'm going to multiply both numerator and denominator with cot a cot b. On doing that, I will end up getting cot a cot b minus 1 by now remember tan a multiplies with cot a cot b will give you a cot b and tan b multiplied with cot a cot b will give you a cot a. So there you go. This is it. This is it. Let me write it down here. It becomes cot a cot b minus 1 by cot b plus cot a. Many people ask me, sir, is this important to remember this since we already know tan a plus b? Yes, there are some questions we should directly ask you to evaluate cot off compound angles. So you should be knowing this result also. Now here, please note down neither a nor b nor a plus b should belong to a multiple of pi. Else, else things will become undefined. Things will become undefined. Is it fine? Okay. Now it is not a surprise to get cot a minus b now. Now all of you please pay attention for cot a minus b. I'll just change my b with a minus b in the previous result. So I will treat it like this. Okay. And I would use this result that I have derived over here right now. So I will get cot a cot minus b minus 1 by cot minus b plus cot a. Okay. So we know that cot minus b is actually minus cot b because cot is also an odd function just like tan is. But when you look at this result, there is too many negativity involved. So what do we normally do? We multiply both numerator and denominator with a minus 1. Okay. So that gives us cot a cot b plus 1 upon cot b minus cot a. Okay. So let me list it down over here. So it becomes cot a cot b minus 1 by cot b minus cot a. Now please be very, very careful. There are some students who think they know the formula and in that overconfidence they write cot a cot b minus 1 by cot a minus cot b that will be wrong. That will give you a negative of the answer. Okay. How do we remember so many formulas? Did I produce this formula from memory? No, right? I gave you a trick to remember all of them. The same way you will remember it yesterday. And second thing that yesterday is when you practice it, when you apply it, these formulas will already sit in the back of your mind. Right? Do you remember how to walk? Do you remember how to eat? Do you remember how to, do you remember how to speak in English or maybe your mother like me? No. Because you're doing it every day and day in and day out. So you end up, this is a part of your readily accessible memory. Okay. What we call as random access memory for in our computers. Okay. So that will be easily remembered to you once you start applying it. Okay. Yes. One important thing I would like to, sorry, one important thing I would like to bring to your notice which I already told you. This is cot b minus cot a. Don't do cot a minus cot b. It will be wrong. Okay. Here also a, b or a minus b, none of them should belong to multiples of pi. So six, no, not six. I think eight formulas you did so far. Okay. I forgot how to, you'll get ample time to recall that. Don't worry. Okay. So should we take questions because I think we have derived too many formulas. Let's concretize our understanding through some problems, time for some problems. Can I go to the next slide if you have all copied from here? All right. So let's begin with some problems. Okay. Let's start with this question. Prove that tan 70 degree is tan 20 plus two tan 50. Just let me know by writing done if you're done with it. Prove that tan 70 degree is tan 20 plus two tan 50. Done. Okay. See guys, it's very simple. If you see tan 70, tan 70 can be written as tan 20 plus 50, isn't it? So let's use our identity which we learned now. Tan 20 plus 70 is like tan a plus b, right? So it's tan a, tan b by one minus tan a tan b. Can I know? Okay. Let's take this to the other side and multiply. So when you multiply, you get tan 70 minus tan 70, tan 20, tan 50 is equal to tan 20 plus tan 50. Okay. Now, a very small thing that most of you would have observed over here is that tan 20 is cot 70. Okay. Cot 70 is one by tan 70. So can I say tan 20 into tan 70 is actually a one? So this guy is actually a one. Okay. So this whole result becomes tan 70 minus tan 50 equal to tan 20 plus tan 50. Okay. And there you go. Make tan 70 the subject of the formula. It gives you tan 20 plus two tan 50 on the other side. Done. Got it. Now the trick was you knowing this fact, this guy is going to be a one because one is cot 70, other is tan 70. They're reciprocal of each other. So they'll cancel each other out giving you a one. Is this fine? Any questions? Any concerns? Any questions? Any concerns? Can you take the next one? If you're done with this. Okay. Let's take this one. If a plus b is 45 degrees, then show that one plus tan a times one plus tan b is actually two guys. There is a twin question to this as well, which I will write it down here. There's a twin question to this which says cot a minus one into cot b minus one is also two. If your a plus b is equal to 45 degrees. Okay. Let's prove both of them won't take much of your time. So let's do it. And just write done on your chat box if you're done with it. Done. Excellent. Both the facts, both the proofs position or only one of them just the first. Okay. First one done. Very good. Okay. Okay. So this is quite a simple for the first one. You just have to take tan on both the sides, right? Take tan on both the sides. We already know that tan a plus b is tan a plus tan b upon one minus tan a tan b and tan 45 degrees of one. Okay. Let's take the denominator to the right side. You'll get tan a plus tan b as one minus tan a tan b. Okay. Now let's bring this term to the left side. So tan a tan b plus tan a plus tan b is equal to one. Okay. Now I'm going to do a small manipulation over here. I'm going to add a one to both the sides. Hey, very good question. Very good. Okay. I'm going to add one to both the sides. Now see why did I do that? Because now it will become factorizable. So if you take tan a common from the first two terms and let's say one common from the second two terms, this is what we see. And this is precisely what I wanted to prove. This actually is tan a plus one times tan b plus one and that's equal to two hence proved. Okay. Excellent. Okay. So some of you have already done with the second one as well. Very good. Second one. Those who are done with the first one or those who have copied the first one, you can start working on the second one. Any questions, any concerns, please do let me know. Okay. So now try the second one out. This was the first one. Quate minus one times called b minus one is equal to two. Let me know done when you're done with it. Okay. So right now I'm seeing three people having done the second one. Okay. They're just funny. Okay. Very well done. Siddharth Karthik also done. These two are potential questions in school exam as well. Okay. So let's do the same process what we did with tan. Okay. So take caught on both the sides. Caught a plus b formula we just now learned is caught a caught b minus one by caught b plus caught a. Okay. And caught 45 degrees one. So let's cross multiply. So caught a caught b minus one is caught b plus caught a send these two guys to the left side and send this one to the other side. So it ended up getting caught a caught b minus caught b minus caught a equal to one. Now add a one to both the sides. Let me write it in different colors so that you can identify it. So from these two terms take caught b common and from these two take terms take minus one common. So this is anyways caught b minus one times caught a minus one. Let me write it ahead of it is equal to two. This is what we wanted to prove. Okay. So any of these two questions or any of these two twins can be asked in your school exams as well. Okay. By the way, based on the first result, there is a very interesting question that probably you will see in your DPPs as well. Let's take that up. Meanwhile, anything that you would like to copy or anything that you would like to ask, do let me know. Okay. Should we now go to the next question? This question says prove that the product of one plus tan one degree, one plus tan two degree, one plus tan three degree, one plus tan four degree, one plus tan five degree, till you reach one plus 45 degree is equal to two to the power 23. So I will write it in single line. Actually, this is what the problem says. I'll write some terms before the last term also. We have to prove that this fellow is equal to two to the power of 23. It's quite a big number, by the way, two to the power 23. Any idea how to do this problem? Okay. Anybody got a clue how to do it? That's the darts. You're absolutely right. Okay. So see the last term is definitely a one plus one, which is two. No doubt about it, isn't it? And this recall, recall in the previous page, we have done this that if A plus B is 45 degrees, then one plus tan A into one plus tan B is actually a two, isn't it? Can I say this criteria will be satisfied by these two gentlemen? Let's say this is A and this is B. Are they adding up to 45 degree? Yes. So one plus tan A into one plus tan, we see everything is multiplied. So let's say I group these two terms together. Can I say they will multiply to give you a two? Can I say the same will be true for these two terms also? This will also give you a two. Can I say the same will be true for these two terms also? This will also give you a two. Like that, if you continue doing this, you will realize that you will end up getting 22 times because there are 44 such terms. And if you start pairing every two, two, two, two, two, there will be 22 such pairs created. And each one will give you a two, two, two, two. So basically, you'll have two into two into two, 22 times, and there is one two in yellow, which I'll write separately. So altogether, altogether, this will be nothing but 223 times. That is your RHS. That is your RHS. Is it clear? Is it fine how it works? So if you know the previous result, you can easily get the answer to this. Okay, of course, if this question was given to you in isolation, maybe you have to prove this result. Maybe you have to prove this result in order to use it. So as a single question, as a standalone question, this should have been a slightly bigger question to solve because it would incorporate you to prove this result as well. Since we had already proved it in the previous page, so we did not do it again over there. Is it fine? Any questions, any concerns do let me know. Okay, can we move on to the next one? Let's take another one. Cos beta minus gamma plus cos gamma minus alpha plus cos alpha minus beta is equal to negative 3 by 2. Prove that these two terms, that is cos alpha plus cos beta plus cos gamma and sin alpha plus sin beta plus sin gamma both are equal to zero. Both are equal to zero. Next class, when we meet, we are going to talk about transformation formulae. We're going to talk about multiple and some multiple angle identities. We're also going to talk about conditional identities. So that would be the, I can say, the concluding part of trigonometric ratios and identities. After that, we'll have one class on trigonometric ratio equations and one class on properties and solution of triangles. So three more classes would be required to finish off trigonometric. And then we can start with inequalities, straight lines, complex numbers, whatever is going on. Anybody who could prove this? Okay, let's do this question. This is a very interesting question actually. Let's say I call this term as P and I call this term as a Q. Okay, so let's say P is your cos alpha cos beta cos gamma and Q is your sin alpha plus sin beta plus sin gamma. Okay, now I would request you all to do this operation for me. P square plus Q square. Okay, that means I'm asking you to find the squares, of course, simplified as well of these two terms. Oh, sorry. Okay, so A plus B plus C whole square formula, everybody knows that's going to be A square, B square, C square, plus two AB, plus two BC, plus two CA, or two AC, whatever. Okay, similarly, sine square alphas, sine alpha, sine beta, sine gamma square, I will write it below this because I want to pair something up. Maybe by looking at it, you will understand what do I intend to do. Now, when you add this, when you add this, now see here, this adds up to give you, what does this add up to give you? One, this adds up to give you, one, this adds up to give you, one. Okay, now look here, this adds up to give you to give what? You will say, sir, this adds up to give you two cos alpha minus beta, right? Yes or no? Similarly, these two will add up to give you what? Two cos beta minus gamma, and these two will add up to give you what? Two cos alpha minus gamma. Yes or no? So it says, sir, so what? How does it help us to solve the question? All of you, please again pay attention. This is going to be three and this is going to be twice of cos alpha minus beta, cos beta minus gamma, cos gamma minus alpha, and this result is actually provided to us. This result is already given to us. See here, this result is already given to us as negative 3 by 2. Isn't it? See cos alpha minus beta, cos beta minus gamma, and cos gamma minus alpha, or you do alpha minus gamma, it doesn't make any difference to cos, because cos is an even function. So even if you want to write, you can write it as cos gamma minus alpha. So this guy is already given to us as a negative 3 by 2. So as a result, your answer becomes a 0. Now answer a small question of mine. If some of the squares of two real numbers add up to give you a 0, then what is your conclusion about those two real numbers? What is your conclusion about those two real numbers? They must individually be 00 each, which means cos alpha plus cos beta plus cos gamma has to be 0, and same will be true with sin alpha, sin beta, sin gamma. And this is what we wanted to prove. Good question, no? So when these questions come in school exams or competitive exams, people are wondering how to do it. By the way, there's one more method to solve it by using complex numbers, but since you're not aware of it, we will talk about it when we learn complex numbers. So I will conclude my today's session with this. Next class, stay tuned for more identities. So those five sites that you have left off, they are going to be filled in the next class. Okay. All right. Now coming to doubts, some of you had doubts.