 So under trigonometric ratios and identities, there is a very small thing which is left off which we call as the harmonic forms. Okay. What are harmonic forms? Let me just take a small example to illustrate this. Let us say somebody comes to you with a question. He says, hey, there is a function like this. 3 sin x plus 4 cos x. Can you, number one, sketch the graph of this? Okay. Number two, can you find the range of this function? Okay. That means can you find out the maximum or minimum value that is the max and minimum value of this function? Okay. So how will you solve this? How will you solve this particular problem? Now, sketching the graph is not that easy because here both sine curve and cos curve are involved. And please let me tell you, if you're thinking that you're going to sketch both the curves on the same xy coordinate system, that would be considered as a sum of both the graph. No, that is the wrong notion. Okay. Sketching both the function, sine function, cos function on the same coordinate axis doesn't make the graph of sum of these. Okay. So how do you sketch? That is one issue. Second thing is, how do you know the range of this function? That means how do you find the maximum or minimum value of this particular function? Okay. No issues. Okay. Just let me know that you are going for that. Okay. Okay. Now, let us understand it. Now, in this particular shape, it is difficult to do both the sketching part and finding the range of the function. So what I'm going to do here is, all of you please pay attention. I'm going to convert this. I'm going to convert this to a harmonic form. The word harmonic has come as a combination of the word harmony. Harmony means trying to make this entire function as a single trigonometric function, either a sine function or a cos function. That means I would like to convert this function either as a constant sine X plus minus some angle or you convert it as R. That is a constant cos X plus minus some angle. Here you can see that there has been a harmony established between the two different trigonometric ratios by converting it to a single trigonometric ratio. So if I'm able to convert f of X in either of the two form, in either of the two, I will be able to answer both of these questions very easily. Okay. Now, many people ask me, Sir, why do you want to convert it to a single trigonometric ratio like a sine or a cos? Because my dear students, managing a single trigonometric ratio is very easy. You can always draw the graph of R sine X plus minus alpha very easily. You can always find out the range of such a function very easily and so is true for the second one as well. Okay. So how do I do that first of all? What is my R? What is my alpha? And then how come this particular form will help me to sketch the function and range of the function? Let us try to understand through the very same example. Okay. Now let's see the process. Now you choose which form you want to convert it to. I'm leaving up to you. Which form you want to convert it to? You want to convert it to the first one or the second one? Okay. Say to all the first one to choose. So he said convert it to sine. Okay, fine. So now what I'm going to do, everybody please pay attention. Very, very important. The process I am explaining you. So this three sine X plus four cos X. Okay. This number three. This number four. Okay. Pick it up and make it as a coordinate. Okay. So what did I do? I picked up this number three and I picked up this number four and I made it as a coordinate. Okay. Sir, are we doing coordinate geometry or are we doing trigonometry? We're doing trigonometry only but a small interesting thing I want you to observe. Let's say this is a coordinate of a point P. Okay. So at this point is let's say our units away from the origin. That means this distance is capital R. Okay. And this angle is alpha. Good enough. Now tell me how can you write three and four in terms of R and alpha? You will say sir simple. You have asked a very easy question. This will become your R cos alpha. And this will become your R sine alpha. That means your R cos alpha is three and R sine alpha is a four. Yes or no? Correct. Can you find out R from here? Yes, I can easily find out R. Just square them and add them. Sir, what are you doing and why are you doing all these things? I will explain you. Give me some time. Give me some time. Okay. So here this will give me R square and sine square plus cos square will actually be a one. If you want, I can write it for the time being. So R here will become a five because R represents the distance. Isn't it? So how will you get alpha here? Alpha is very simple. Take the ratio of these two. So R sine alpha divided by R cos alpha. That will give me four upon three. And this is nothing but tan alpha is four by three. Okay. Many of you know the answer for this. This is roughly fifty threes degree, right? Roughly fifty three degree. Correct. Even if you don't know it, not going to hamper a lot of things. Right. So once you have done this, now come back to this expression. Okay. So, sir, you made a coordinate out of it. You found the distance of that coordinate from the origin. You found the angle made by that particular vector. So as to say Shalini correct with the X axis. Why are you doing this? What is the reason for this, sir? Okay. Let me explain the reason. Now this three that you have written. Can I write it as R cos alpha like this? Okay. Then sine X, I'll copy it as it is. This four that I have written. Can I write it as R sine alpha? Okay. Into cos X. Okay. If you see this term very, very carefully, it will remind you of your compound angle identity. You will say, are you, sir? This is nothing but R sine X plus alpha. Okay. And this is what actually I wanted. See here. This is what I wanted. Isn't it? I have achieved it. So let's put the nitty gritties in this particular expression. So your R would actually be, or your R was actually five and alpha was actually 53 degree. Okay. Let's say I'm approximately taking as 53. Okay. Now, once I obtained this, is it easy for us to answer these two questions? Yes. Of course it will be easy for you to answer this question. So let us start with sketching part first. Okay. So how do you sketch the graph or five signs? So let's say you have now this function in front of you. How do you sketch this graph? Okay. So very simple. You'll say, sir, just draw the graph of sine X first. Okay. I'm just making it in dotted way, because this is not my final graph. So I'm just making a dotted graph. Okay. So this is your sine X graph. Okay. Of course your top end will be one minus one in the bottom. Okay. Passing through the origin. Okay. Now let's go step by step. This is the graph of sine X. Okay. Let me call this as Y equal to sine X graph. Now, if I want five sign X, what do I do? Let's say next step is five sign X. You'll say sir simple increase the amplitude of the graph by five times. Right. So this, this whole graph will actually become extended along the Y axis, something like this. I mean, I maybe not by scale. I'm not making it. So it gets extended five times like this. Okay. So this is the graph of five sign X. The top end will come at five. Bottom end will come at minus five. In fact, this on UG also don't worry about it. And now finally tell me if I have to make the graph of five sign X. This packet is not required. Five sign X plus 53 degree. What do I do to this graph? What do I do to this graph? Tell me people who are there in the bridge course transformation of graph chapter. You'll say sir shift this by 53 degree left. Absolutely. So when I shift this 53 degree left, your final graph will look like this. Pardon me. If I have done some, you know, graphing error. Okay. This is how the graph is going to look like. Okay. Let me show this exactly. How does it look on? From there we'll be able to appreciate in a much better way. Okay. So graphing became possible. Okay. So from this graph, I'll be able to tell its range also. Isn't it? Won't I be able to tell the range? You'll say yes sir. It has gone to a maximum of five. Okay. In fact, without the graph also you can tell the range. So if I ask you what is the maximum value this function can take, you'll say this maximum value is going to be five. Why? Because sign of any angle, this can become a max of one. So this whole thing will become a max of this whole thing will become a max of five. Isn't it? Because sign of any angle can maximum become one. Okay. Five sign x is just the extension of the sign x graph five times. Its amplitude became five times. Think like that. Right. So it was oscillatory graph. Right. It was basically a sinusoidal graph. Its amplitude became five times. Think as if somebody took it and stretched it along the y axis. Okay. Similarly, what will be the minimum value of this function? You'll say sir, it will be minus five because this fellow can take a minimum value of minus one. That means overall this function will have a minimum value of minus five. So this graph will oscillate from a high value of five to a low value of minus five. That would be the range of the graph. Okay. Now, you may get this question in your mind. Sir, can I use calculus for this if I know it? Of course, if calculus is known to you, then you don't have to rely on these methods. Okay. But sometimes let me tell you for these simple questions, you should not go for calculus. Calculus is like a, you can say it's a heavy weapon. Okay. Don't use to kill. Don't use it to kill insects. Okay. So don't use calculus, you know, for simpler questions where you can manage without it also. Now, time to show you this on your, uh, let's go to use you and see whether have I got a correct graph for this function. And does my graph actually have a range of minus five to five. But before that, before I change the screen, anybody has any questions. Okay. Now, if I wanted, I could have converted things in terms of cost also, just like I converted things in terms of sign, I would have converted things in cost also. How other way around, I would have taken three as our sign alpha and four as our cost alpha that that we would come into costs. Okay. But converting it to costs or converting it to sign will not have any implication with respect to the graph. There will not be any change in the graph. There will not be any change in the range. Don't worry about it. So you are free to choose whichever form you want to convert it to. So converting it to cost, you can try it out as your homework. How did you find alpha? Okay. So Noel, what did I do? I divided the, let's say I call this as the first equation. And I call this as a second equation. I did two divided by one. And I got this. RR got canceled and I got tan alpha, tan alpha is four by three. So alpha value, normally four by three is a very well known angle. I don't know whether Deere said has spoken about it ever. So four by three basically corresponds to 53 degree. Okay. So Noel, is it fine? Any questions? Ajat, today I was, I was starting my session and I got a message that some meeting is already going on, on this particular ID. And I realized that one of you, I don't want to name the person had joined some trigonometry session. Hmm. And that person is there in this meeting. Please do not click any old link of zoom send to you. Even my mistake. Because as a message, I will get that there is some other meeting going on. You cannot join the meeting. Okay. So one of you had joined trigonometry meeting of the last class. Please do not do that. Okay. Is it fine? Any questions? Okay. Because I get a mail that so and so has started that meeting. Because I am the license owner of this zoom. So I'll get a mail directly that so and so student has logged in and is trying to have a meeting there. Okay. All right. Anyway, let's go to the, let's go to the, I'll just minimize it. Yeah. So my function was three sign X plus four Cossack's. So three sign X plus four Cossack's. Okay. Please have a look at it. This is the graph that we had actually drawn. Three sign X plus four Cossack's here. So the top end of the graph is a plus five. The bottom end of this graph is at minus five. Okay. And how did I draw the graph also? I will tell you what did I use to make the graph. Just a quick revision of graph and is transformation topic that we had done in our bridge course. out the transformation from there on. I'll just mute this for time being a special focus to say to you are wondering know what will happen when you multiply with 5. So see here. So this was the sign next. Now if you multiply this with 5, see what happens on the x axis, it doesn't displace anywhere. It just becomes higher oscillations. So five times the oscillation okay. And now finally we displace this by, we displace it by some angle. So sign x plus. Let me write 53. I don't know whether it's going to take degrees over here. 53 degrees. Wonderful. Okay. I think it has taken 53 degrees. All right. So this is this is the graph that you will finally get. I think this matches with our original graph. No, which how much you displace it here more than 53 degrees, I believe. Okay. Anyways, let me write it as tan inverse of delete. Yeah. Let me write it as plus plus tan inverse of our tan four by three. Okay. This graph and this graph, as you can see, they both match. I'm switching both of them on together. Yeah. You see that? Right. So both the graphs are same graphs. Okay. So this is how you can actually sketch the graph. And of course, as I told you, you can easily find out the domain and the range. What happens if your x becomes 5x? Okay. Now say to saying, if I make x as 5x, what will happen to the graph? You tell me what will happen to the graph? The graph is going to shrink by five times. You want to see? You want to see what will happen? Okay. This is a sine x graph. Let me make a five over here. You see that? It got shrunk by five times. Okay. Anyways, so here is a generic theory with respect to whatever we have discussed over here. So basically when you have been, I'm just write a general form. Okay. Basically, if you have been given a function like a sine x plus b cos x, okay, you can always convert your function in terms of r sine x plus minus alpha or r cos x plus minus alpha. Okay. Let me not write alpha here. Let me write a beta because there's a difference between alpha and beta values, whatever. And the basis of this, you can sketch the graph. By the way, r value, I can directly tell you r value will always be under root of a square plus b square. Okay. Alpha will depend upon which form you are actually choosing. So as per the form you are trying to convert it to. So any of the two forms you can choose as per your whims and fancies. Okay. And you can get your r will be same for both of them. It will not change. Alpha will vary depending upon which, you know, this angle is going to change depending upon which form you are trying to convert it to. So one important thing comes out from here. Since we already know my r, I can always comment that the maximum value of this function will always be r only. Isn't it? Just like in this case, our max value came out to be 5 and 5 was your r. Isn't it? And your min value will always come out to be negative of under root a square plus b square, which is actually negative r. That is what we got here also as a negative 5. Okay. This is a very interesting theory that you should all note down because directly maximum minimum questions are asked on this particular concept. Okay. Now a couple of things to be noted over here. Many people ask this question, sir. Can I apply this to, you know, any scenario? That means let's say if it is a sin x and b cos 2x, can I apply it? Or let's say a sin 2x and b cos 5x, can I apply it? No. This formula or this direct result that you are using for maximum and minimum can only be applied when these two angles are same. If these two angles are not same, you cannot apply the direct formula. Then you have to break it up and bring it to terms which are having the same angle. We will see it through some questions. Okay. One or two questions we will take up and we'll see how it works. Okay. So please note down whatever I have written on the screen right now because I'm going to switch my screen to the next page. I'll take hardly 10 more minutes and then I'll start with straight lines. Okay. Actually not straight lines. We are going to start with the prerequisites to learning straight lines. Are you, sir? That important a chapter that we need prerequisites? All right. Yes. Of course. Okay. Time to take questions. Let's take questions. Tignometry is here. Let's take this question. I know it's a proof that question, but assume that these values are not given to you. Let's say this value is hidden from you. This also value is hidden for you. Just find out what is the range of this function? What is the range of this function? Find the range. You will automatically get the range as between minus 4 to 10. Okay. Let's do this and if you're done, just write are done on the chat box. Don't get confused. In the earlier slide, I had taken angle as X. Here they've taken angle as theta. It's just the change of the name of the variable involved. Concept remains the same idea. Concept is not going to change. Okay. Okay. Could I write my answer as maximum value is 5 square plus 3 square under root? Can I say this is the max value? Can I say that? Will it be right to say this? Will it be right to say this? And can I say min value is negative of this? Will it be right to say this? You say of course no, else it would have matched with your minus 4 and 10. So this is not going to work. Why is it not going to work? Because as I already told you, this formula works when these two angles are same, but they're not the same. So you cannot apply this as the max and the min value. At the question being just 5 cos theta plus 3 sin theta, then of course I can say maximum value is root 34. Minimum value is minus root 34. But in this case, I cannot say so because these two angles are different. Yes or no? So what do I have to do then? Simple. You need to sit and expand this at least the second term you need to expand and write it as cos theta cos pi by 3 minus sin theta sin pi by 3. So let's expand it even further. So it's 5 cos theta. This will be 3 by 2 cos theta and this will be minus 3 root 3 by 2 sin theta. Am I right? Now club the cos theta terms together. So this is going to give you 13 by 2 cos theta minus 3 root 3 root 3 by 2 sin theta. Now you have this expression. Can I now say the maximum value of this expression? The max value of this expression, I can easily say will be under root of 13 by 2 whole square and 3 root 3 by 2 whole square, which is nothing but 1 by 2. 13 square is 169 and this is 27, which is I think, I think is root 196. So which 1 minus 6 is 14. So max value is 3. But by the way, sorry, max value 7. By the way, there was also a 3 sitting over here. So I should write that 3 also everywhere. And this 3 is going to remain 3. It is not going to be affected. So your max value will be this plus 3, this plus 3, this plus 3, okay. This plus 3, this plus 3, which makes it 10. In other words, your f theta will always be less than equal to 10 because it is the max value of that particular expression. So ideally it will be lesser than equal to that value always. Is it fine? How did it work out in this case? Any questions, any concerns? Anybody? Clear? Or any doubt is there? So can you also now figure out what will be the minimum value of this function theta? You say, sir, now you have already done all the hard work. We just have to change the sign here. So they just let me know when you once you're done so that I can go down and done. Okay, great. So what is going to be the minimum value of this expression? You'll say, sir, simple. All you need to do is take this as a negative of this quantity. And you don't have to redo the whole thing because we have already seen that this value comes out to be 7. So it will be minus 7 plus 3. So minus 7 plus 3 will be a negative 4. So your f of theta will be always be more than negative 4. Okay, so let us combine these two. Let's call this as 1 and 2. So if you combine these two, we can claim that the f of theta lies between your f of theta lies between minus 4 to 10. This is what we wanted to do. Okay. Do you think the maths is asking neat exams? Shalini? No, maths is not in need. You will not be tested on maths if you are a neat aspirant. But if you're asking for CET very much, this question is very much asked in CB. Okay. In need only PCB is tested. PCMB is not tested. Okay. Is it fine? Any questions, any concerns? Please note this down and we'll take one last question and then we will close this particular topic for the time being, not permanently, for the time being. So we'll wait when your school will start again with trigonometry because we have trigonometric equations left to be completed. And one of the most important topics for your KVPY also which is called properties of triangles that will also be taken up. Okay. Can we go to the next page? Done everybody? Okay. One last question we'll take up. Find the maximum and the minimum value of this expression. Okay. Many times you will ask me, sir, maximum value when you're asking, why can't we do this thing? We'll make sine x also one, cos x also one, everything has one. We'll make it maximum. See in the previous example, in the first case which I gave you, if you wanted to make a maximum value out of it, right? Why can't I say, okay, the maximum value of this is three, the maximum value of this is four. So maximum value of this is seven. Why can't I say this? Why did my answer not match with this? My answer came out to be five actually. Do you remember the very first question which we took as an example? Why not seven? Can anybody explain me that? Correct Nikhil? Absolutely. Because sine and cos cannot simultaneously become a one. Right. That is the main reason why we had to do all these drama, right? If sine x and cos x become simultaneously one at some value of x, I would happily choose that to be my, you know, position of maximum value or if they become simultaneously minus one, I could happily make that as my minimum value. But unfortunately, there's a lag between sine and cos. They cannot become simultaneously one or they cannot become a simultaneously minus one. Absolutely right. Correct. All right. So please attempt this question. This is the last question for trigonometric ratios and identities and give me a response in the chat box. Okay, Shalini, anybody else? Very good. Now here you will realize that the expression has a mix of x and two x, but thankfully by knowing my double angle identity, I could convert this term as three into two sine x cos x. Correct. Okay. And two sine x cos x is actually sine two x. Now, just recall what I had said in the previous discussion in like, I think last to last slide. When you have the same angle sitting over here, you can use directly the formula for maximum and the minimum values. So the max value of this is simply going to be under root of three square plus four square. That's a five. Absolutely right. Those who give that answer and minimum value will be minus under root three square plus four square, which is minus five. Is this fine? Any questions? Any questions with respect to this? Okay, Prishan, did you realize a mistake? Clear now? Okay. Now I'll give you, we will talk about that later on when we do calculus. As of now, let's not go into calculus part of it. Well, last question, I would just take this up. Find the maximum and minimum value values of the following function. Let's take two sine square x plus four sine two x plus three. Now, see here, there is a sine square x, there's a sine two x. Okay. Let's see. Again, please don't use calculus even if you know it. So whatever we have done so far is good enough to solve this question. If you're done, you can write it done also. There's no need to actually sit and type out if the figures are ugly. Okay. Don't take, don't bother yourself for typing it out. Just say done. Nikhil is done. Very good. How about this? Nikhil, you're from Kormangala or HSR? Or neither? Okay, Kormangala. Let's wait for one more minute. If anybody is trying hard, please finish it off in the next one minute. Then we'll discuss it out. Okay, Prishan is done. Shardul thinks she's done. Good, good, good, good. How about others? Noel is done. Very good, Noel. Okay, okay, take your time. Okay, let's discuss it without much ado. See the problem here is there is a, you know, sine square sitting over here. Earlier we had us just, you know, without any power, right? And second problem is that there is an x sitting here and there's a two x sitting over here. So what should I do then? Okay. So basically both these problems will cancel out each other. How? Let's see. Can I convert sine square x in terms of cos 2x? If yes, you'll say you'll have to recall your double angle formulas, right? So please recall. I'll just write it down over here. Recall that we had done cos 2a as 1 minus 2 sine square a, isn't it? So I can say 2 sine square a is 1 minus cos 2a, right? Put your a as x and your problem is solved actually because this will become 1 minus cos 2x. Okay. So overall, this expression simplifies out to a familiar one, which we can actually handle out. Okay, minus cos 2x plus or 4. Don't be scared by this minus sign over here. It doesn't make any impact to the result that we had taken out. So the same a sine x b cos x format will work here also. Okay. It's just that your b is now or a whatever you want to call it is a negative quantity, but that doesn't change our maximum minimum results formula. So if you look at this expression, the max value of this expression will be 4 square minus 1 square under root. Okay. So this will be your maximum value of the given function and that comes out to be if I'm not mistaken, 4 plus root 17. Okay. The minimum value of this expression will come out to be negative of this and plus 4. See this 4 is not going to change. This 4 is a constant, right? It is not dependent on any x. So 4 will remain 4 for the whole of its life. Okay. Only this guy is going to change depending upon the x values. So the maximum that can take is under root of 4 square plus minus 1 square under root, which is root 17. The minimum it can take is negative root 17. So your answer to the minimum value will be this. Okay. So if somebody asks you for the range of the given function, you can say with confidence that it will be 4 minus root 17 to 4 plus root 17. Okay. So those who said done, did you get the same result or did you get something else? Same result. Excellent. Very good. Awesome. So for the time being, my dears, friends, I'm going to give this chapter a pause. Okay. I'm going to come back again to it with some more concepts. In fact, we have done almost only around 60% of the chapter trigonometry. 40% is still left. Okay. But we'll come back to it when you do it in the second semester. All right. All set. Everybody's fine. Happy. Okay. So let's start the new chapter.