Trigonometry Formulas and Techniques

Hello, hello, hello Ninja Warriors. Sup? How is everyone? Are you guys absolutely fine? Let me just open this chat box. Let me check whether I'm live or not. Am I live students? If I am live, then please do let me know in the chat box. Okay. So, so, so, okay, where is the live chat. Oops. Okay, okay, okay. Alright. Okay. Yes, so I can see you all over here. Hi Abhijeet, hi Taruna, hi Rohind, hi Girish, hi William. How are you all? The very first thing you have to do, you remember na? The moment you enter the session, you have to hit the like button, okay? So today's session is going to be all about multiple and sub-multiple angles, another very important topic for trigonometry and I am absolutely fine William, how are you all? How are you? Okay, so multiple and sub-multiple angles, today we are going to start a couple of formulas we are going to do today and based on that we are going to solve questions as well. So yes, That's how it is going to go. And I guess one student is asking when you are going to finish the syllabus. When are you going to finish the syllabus? See, syllabus we will be finishing approximately by the end of October or maximum by mid-November. If like one or two sessions here and there can happen, right? But estimate is when Cengage book solving? Yes, So, Cengage solving session is going to happen today at 10 o'clock. So, I don't know how many of you have noticed but since yesterday I have started a regular practice session from different different books and from different different chapters. So, can you increase volume level? My volume is at maximum Girish. Try changing your device or maybe plugging in your phones for better experience. Okay? Yes? All right, good evening, Kurva Ashwin. So yes, I was talking about the recent practice sessions. Yes, today we have any class at 10. Yes, can you, you can't see, you can't see the class at 10. Let me show it to you. I have, see I have already scheduled it. But you are not, maybe you are, it is not visible to you, kya? It is not visible. Let me just open the channel. You also open the channel, na? Here is the channel, can you see? So this is how it is. You go here and live session if you go, okay? So today what are we going to do? 10 killer problems from functions, Cengage K functions problem. 10 o'clock that is going to happen. Today is 11th July. So please hit the like button right now, okay? Alright? Yes, sequence and series also we will do. One session for sequence and series will also do. Yes, notes of the session, yes, I will provide. Okay, I will add in the description box. Okay, so shall we start today's session? Are we ready for today's session everyone? Okay, are we ready for today's session? Chalo. But before that, I just want to make sure that you Answer me this particular question. Solve this particular question for me. Ma'am, I don't know functions. But Abhijeet, that's how it is. We are going to cover each and every chapter. I can't be taking only one chapter whole year. Okay? Alright? Yes. So this question solving, question solving. So yes, announcement. Every day, almost every day. Half an hour we are going to do practice session. So mostly not every day but at least 3 to 4 days in a week we will do maths practice. Okay? Can first PU students also watch that 10 series? Yes Girish, you can watch that, okay. It's just that if there are core class 12th chapters, that session, that will be trouble for you, suppose you have not studied it. But yeah, mostly quadratic, sequence, these chapters, basics of function also is there in first PU, right? So you can attend it. Chalo, let's start this question. Tan 14 degree, tan 31 degree plus tan 14, tan 31. Okay. So, these kind of questions I think I guess we have done in the last class. Tan 14 degree, tan 31, tan 14 into tan 31. So, 14 and 31, can anybody tell me? 14 and 31 has a relation. So, 45 ko. What is happening? Mosquito everywhere. Mosquitoes. So many mosquitoes today. I don't know from where they are coming. Tan 45 degree is equals to what? Tan. 14 plus 31. Can I write? Now tan 45 degree is 1 and tan a plus b. Tan a plus b is what? Tan a plus tan b whole upon 1 minus tan a tan b. Okay? Getting it? Understanding it? Everyone? So now if you just cross multiply, you will realize tan 14 plus tan 31 is equals to 1 minus tan 14 tan 31. So, actually the correct answer is just cross multiply and rearrange option A is absolutely right. I am use good night liquid. Yeah, good night liquid all out all these devices my mosquitoes are sitting on that device. They don't care about what do you do. Okay. Yes. Now, let's talk about multiple and sub multiple angle. okay multiple Multiple and sub-multiple angle. Okay? Very important session. Trust me. Very, very important session. Okay? Important not only just for trigonometry but we are going to use these formulas in calculus also. So, this session make sure that you attend today's session with utmost dedication and attention. Okay? Yes, clear? Chalo. So the very first formula is sin 2 theta. Sin 2 theta is actually 2 sin theta cos theta. Can anybody tell me how to prove it? Can anybody tell me how to prove it? Sin 2 theta is 2 sin theta cos theta. Tell me how to prove this thing. So ma'am sin 2 theta is 2 sin theta cos theta. It is actually sin theta plus theta. sin a plus b Right? sin is now sin plus cos . Okay? So sin became what? Tell me. sin cos theta plus cos theta. So this is 2sin theta. Yes or no? Getting it? Understanding this formula? Okay. Now one more thing, one more formula for sin we can do this. What I'm going to do is 2 sin theta by cos theta, I'm dividing it by 1. Can I do that? Sin 2 theta is 2 sin theta cos theta divide by 1. And this 1 will be replaced by sin square theta plus cos square theta. Okay, sin square theta plus cos square theta. Whole upon, can you see? Now, step by step, same thing we are going to do multiple times. Now divide 2 sin theta cos theta by cos theta and here also divide by cos theta. What will you get 2 sin theta cos theta? So cos theta cos theta cancel. So 2 sin theta by cos theta is tan theta and you know what? See actually you need to divide by cos square theta not cos theta cos square theta. So what will happen 2 sin theta cos theta by cos square theta right? So, 1 cos and 1 cos gone. 1 cos, 1 cos gone. Sin by cos is tan. Why 1 changed into that ma'am? Because we are creating different formulas. Sin 2 theta in terms of theta. Sin 2 theta in terms of tan. Manipulation, right? This is what we call manipulations. Sin square theta by cos square theta is tan square theta. So, can you see if you divide by cos square theta, what will you get? Cos square by cos square is 1. Sin square by cos square is tan square. So, this is another formula. So, this is the very first formula. You can see the second formula is, ok, sorry. Second formula, let me write it over here, which is right, yes. 2 tan theta upon 1 plus tan square theta, ok. Can you see? So, this is you have to remember. Okay, will you be able to remember it? Okay, ready? Ready for the sine 2 theta is what? Now tell me, remember. Ma'am, sine 2 theta is 2 sine theta cos theta and sine 2 theta is 2 tan theta upon 1 plus tan square theta. Now one thing, understand what are these formulas. See, if sin 2 theta is 2 sin theta cos theta, what is happening over here? 2 theta is becoming theta. That means whatever is present over here, we are going to divide it by 2. So here if the formula, if the question is sin theta, then what will you do? Ma'am, sin theta is 2 sin theta by 2 cos theta by 2. Okay? So remember it that way. 2 theta. on the right side becomes what? Theta by 2. Okay? Alright? Yes? Now cos theta. Cos 2 theta. Now cos 2 theta has so many formulas. Let me tell you. Cos 2 theta, the very first formula is cos square theta minus sine square theta. Okay? Again, can you prove it for me? Can you prove it? Chalo ma'am, let's prove na? Let's prove. Why not? Ma'am, same thing. Cos 2 theta is cos theta plus theta. Cos a plus b is cos a cos b minus sin a sin b. Cos a cos b minus sin a sin b which is nothing but cos square theta minus sin square theta. Now, one more thing, one more thing. Convert cos square. Okay. Convert sin square theta in terms of cos square theta. Do that convert sin square theta in terms of cos square theta. So, cos square theta minus 1 minus cos square theta. So, cos square plus cos square is 2 cos square theta minus 1. So, one more formula 2 cos square theta minus 1 and now change this cos square theta. into sine. So, 1 minus sine square theta minus 1. This will be 2 minus 1, 1 minus 2 sine square theta. So, one more formula. 1 minus 2 sine square theta. Can you see 1, 2, 3? Okay. So many formulas for cos 2 theta. 1, 2, 3. Fourth formula is already coming your way. Same thing we have done. Remember, see cos 2 theta Ma'am, you told us cos square minus sin square. Okay? And now ma'am, what will you do is divide it by 1. Divide by 1. Abhijeet, are you here? Now, one more time. 1 will be replaced by cos square theta plus sin square theta. 1 will be replaced by cos square theta plus sin square theta. And eventually, Now what you have to do is divide the numerator and denominator by cos square theta. Divide the numerator and denominator by cos square theta. So what will happen? Cos square theta upon cos square theta is 1. Sin square by cos square is tan square. Now cos square by cos square is 1. Sin square by cos square is So, one more formula. Okay? So, now let me write one more formula after this which is yes. 1 minus tan square theta upon 1 plus tan square. You don't have to remember the proof proof but yes. What you need to do is you need to remember white color. No. Green utilized. Orange. Hmm. Okay, don't worry. I am here. We'll remember formulas together. Right now, just get the glimpse of it. So, there are two things. Ma'am, we saw sin 2 theta which is 2 sin theta cos theta and that is 2 tan theta upon 1 plus tan square theta. Okay, I told you know 50 plus formulas in trigonometry. Then, cos 2 theta is cos square theta minus sin square theta. 2 cos square theta minus 1, 1 minus 2 sin square theta and 1 minus tan square upon 1 plus tan square. Now, I will tell you, I will tell you how to remember it ma'am. I don't know, please tell me. First of all, please remember sin is bad guy. Sin is a bad guy. Because sin, sin, sin is what? Do you know Google? S-I-N, sin. Sin is what? Negative. Not a positive thing. Always remember, sin is negative. Sin is negative. Minus sin in front of sin. Now, obviously these two formulas are similar. One is 2 cos square theta minus 1. And other one is 1 minus 2 sin square theta. So, I think this is very much similar. It's just that sin square theta has a negative sign. Because of the sin. Negativity in sin. And you know what? This formula. Okay, now this formula. Especially this cos 2 theta is. Now sin 2 theta and cos 2 theta understand. Both are same formulas actually. Sin 2 theta is also having 1 plus tan square theta. Cos 2 theta is also having 1 plus tan square theta. Just the numerator part is different. Here it is 2 tan theta. Here it is 1 minus tan square theta. Now 1 by 1. 1 by 1 we will do it. We will not do it everything at once. One formula at a time. Alright, so today's session is going to be all about remembering these formulas only. So just understand like these sin2θ, cos2θ and tan2θ. What is happening? Math file is saying idea about her. Okay. Now tan 2 theta, tan 2 theta. Ma'am tan 2 theta is 2 tan theta upon 1 minus tan square theta. What is this? Let's understand. What is this? So ma'am tan 2 theta is 2 tan theta upon 1 minus tan square theta. So it is tan a plus b. Remember tan 2 theta is tan a plus b. Danny plus tan b. So tan theta plus tan theta upon 1 minus tan a tan b. Can you see you get we will get this only. So this is the formula of tan 2 theta. Yes Mr. Perfect. Now one more important observation which is which can be a point of confusion but I will clear your confusion. You know what? Sine 2 theta formula. And tan2 theta formula, both formulas are very similar. I am talking here about sin2 theta in terms of tan. Can you see? Sin2 theta is 2 tan theta upon 1 plus tan square theta. Tan2 theta is 2 tan theta upon 1 minus tan square theta. So just one sin is there. Just one sign difference. In tan, there is a negative sign. But in sine to theta, there is a positive sign. But you know what? Keep everything in mind. Gradually, I am going to tell you everything. Now, I think it's the time to solve a couple of questions. Let's solve. Let's solve. I am going to help you with the formulas. So, they are talking about tan 2a. Tan a is given. Which formula are you going to use? Ma'am, turn. 2 theta is 2 tan theta upon 1 minus tan square theta. This is what we are going to use. Use it, tell me quickly. Abhijeet, go and watch, re-watch it. I have explained it very nicely. Divide by 1, write it sin square plus cos square and eventually now the expression whole, numerator and denominator gets divided by cos square theta. Okay, so I think it's an easy question ma'am. tan 2a is equals to 2 tan a upon 1 minus tan square. Direct formula based na. 2 into 1 by 3, 1 minus 1 by 3 whole square. So, 2 by 3, 2 by 3 and 1 minus 1 by 9 is 8 by 9. 3 ones are 3, 3 threes are 9. So it is 3 2's are 6 by 8 which is equals to 3 by 4. Easy question? Yes? Very direct formula based question. Initially I will make sure that you only remember the formulas. I am not doing anything which is very difficult. Okay? Now some prove that type of questions so that you get hang of how to solve a question. Some prove that. Okay? By the way before prove that, before prove that Hold on, hold on, hold on. Before prove that, two gold formulas, two gold formulas. Before prove that, we have to talk about two gold formulas, okay? Two gold formulas. See, do you remember cos 2 theta? So, I am cos 2 theta is equals to 2 cos square theta minus 1 and 1 minus 2 sin square theta. These are not the gold formulas. Okay? These are the reference formulas. These are cos 2 theta. Now, these are the reference formulas. Two gold formulas. Number 1. Number 1. Tell me the value of 2 cos square theta. Good evening Sanjay Shweta. Come. Right? timing. 2 gold formula. Gold. Why gold? Gold means very good. Like very expensive. Okay. So, cos 2 cos. Can you tell me 2 cos square theta value and 2 sin square theta value. Quickly. So, ma'am 2 cos square theta is 1 plus cos 2 theta. And 2 sin square theta will be 1 minus cos 2 theta. Can you see over here what is happening? What is happening students? What is happening? What is happening? Sign has a negative sign. Sin, sin, sin, sin, negative. S-I-N, sin is equals to sin. Sin means what? That sin. Sin means in Hindi we call it as paap, something very evil. Sin is evil. That is why it is negative. That is why you will see negativity, okay, in sin. That's how you remember. When you have two similar formulas, please remember, it is but obvious, it is so human psychology. If at all you are remembering two similar things. At one time. It is so obvious that you are going to get confused that which one is this? 2 cos square theta is 1 plus or 1 minus. Ma'am, I got confused. But now you have to remember sign negative. Now it will be done. Okay. Chalo, let's solve. Now I am going to give you 3 questions back to back. Let's see how many of you are able to solve it. Okay, try. So, 1 minus cos 2 theta ma'am is actually a. Left hand side ma'am, 2 sin square theta. Whole upon sin 2 theta ma'am, 2 sin theta cos theta. So, clearly what we can do is ma'am, 2, 2 cancel, sin, sin cancel. So, this is tan theta. Done, done ma'am. Easy question, easy question. Easy question, yes. Same 1 minus cos 4 theta now. So now 1 minus cos 4 theta will become what? 2 sin square 2 theta ma'am. And sin 4 theta will become what? 2 sin 2 theta cos 2 theta. Now again you need to cancel. 2 2 cancel, sin sin cancel. So tan 2 theta. So ma'am this is also easy, not very difficult. Now cos 2 theta upon 1 plus sine 2 theta. This is something which is slightly on a trickier side. Cos 2 theta ma'am. Now I have so many formulas for cos 2 theta. What to do? So let's try. Let's try the very first formula which is a balanced formula. Cos 2 theta is 2 cos square theta minus sine square theta. Now 1 plus sine 2 theta. 1 plus sine 2 theta. So, can you see sin 2 theta is actually 2 sin theta cos theta and 1, 1. Now, 1 will be replaced by Abhijit again. Abhijit is so traumatized that ma'am randomly how come you are writing sin square theta plus cos square theta instead of 1. That is the whole point. That is the whole thing. Which one will be replaced by sin square theta plus cos square theta? That person. If you are able to think that, that person is like something else. So now this one will be replaced by sin square theta plus cos square theta. Why I did that? The answer lies in the next step. Sin square theta plus cos square theta plus 2 sin theta cos theta. Can you see now why did I do that? Because this is cos theta plus sin theta whole square. Can you see? This is actually cos theta plus sin theta whole square. Cos square theta minus sin square theta. a square minus b square is a plus b, a minus b. Some manipulation. Keep on solving. You will get your answer. Don't worry. So clearly cos theta plus sin theta, cos theta plus sin theta gone. Okay? cos theta minus sine theta upon cos theta plus sine theta. Okay, now you have to divide everything by cos theta. Like numerator divided by, remember we have done this, come on. Cos minus sine, cos plus sine, previous lecture only. Divide numerator and denominator both by cos theta. So, if you divide numerator by cos theta, it will be 1 minus sine by cos is tan and 1 plus tan theta. Which is actually equal to, do you remember the previous class formula? 1 minus tan, 1 plus tan is tan, 45 minus theta. Getting it? Done, done, done. What do you think about these three problems? Good? Ready for the next? Move on from the x. Ready for the next? Okay? This. something trending Instagram thing okay chalo are we ready for a question now you have to give me an answer ready for like you have to give me the answer okay chalo Chalo, ready ready? 1 minus 2 sine square pi by 4 plus theta. Okay? I don't know formulas ma'am. Which formulas? Which formulas? I have just taught you Abhijit. What are you saying? Please don't say that. I just taught you. How can you say something which I just taught you? See, remember? Okay, which formula will be applicable? 1 minus sine square something. Suppose 1 minus sine square theta is what? Ma'am, 1 minus sine square theta is? cos 2 theta. 1 minus sine square theta is what? cos 2 theta. So 1 minus sine square anything is equals to what? cos 2 times that same thing. Yes or no? Now cos 2 times same thing. So cos 2 times pi by 4. So pi by 2 plus 2 theta. Cos pi by 2 plus 2 theta. is actually sin cos pi by 2 plus pi by 2 ASTC, cos pi by 2 plus theta, second quadrant, cos is negative. So, minus sin 2 theta. So, option D is absolutely right. Okay, one thing I was like thinking about one thing and it was for class 9th and 10th students. basically. I was thinking of bringing some sessions for 9th and 10th class and that is not going to be your usual school sessions like 9th and 10th NCRT I am not going to solve. Something which is Jai Tau we are doing it and I think 9th and 10th students some students like Abhijit are also attending it but I was thinking about something specific to 9th and 10th. Can it be like Olympiad. can it be ntsc nsc js like something like that okay so let me think about it but if at all you want something you want to give suggestions so feel free to give suggestions in the comment session section okay all right yes abhijeet okay Ready, ready? This question good, good, good? Okay, chalo. Now prove that. sin theta upon 1 plus cos theta. See, now sin theta can be written down as 2 sin theta by 2 cos theta by 2. And it is so obvious, so intuitive, the right hand side is all about theta by 2, right? So, 1 plus cos theta. Now, 1 plus cos theta is 2, see, plus sin. So, 2 cos square theta by 2. Now, start cancelling. 2, 2 gone. Cos theta, cos theta gone. Tan theta by 2. No ma'am, only focus on second PUJ, it's my opinion. Girish, I'm not going to like, I'm going to take, JE sessions will be there, I'm not disturbing that, okay? Just apart from that, like one more thing I want to start for 10th. Don't worry, I will not disturb you people. Chalo, one question quickly, Answer me right away. 1 plus cos x, ma'am, 2 cos square x by 2. Upon sin x will be 2 sin x by 2 cos x by 2. Okay? Ma'am, 2, 2 cancel. Cos, cos cancel. Cos x by 2, sin x by 2. Cot x by 2. Yes? Everyone? Getting it? Understanding now? Are you able to remember all the formulas? Tell me if you are able to remember the formulas. Okay, let's go. Now it's time to talk about some special angles. Remember 15 degree. Now similar way 22 and a half degree is also special angle. 22 and a half degree is actually 45 degree by 2 is equals to 22 and a half. 45 is pi by 4. So pi by 4 into 2 pi by 8. Okay, so that is nothing but 22 and a half. So remember it first of all the degree and. Something has motivated ma'am. Ma'am has taken many initiatives just in one week. Practice session, ninja formulas, classes for 9th and 10th. No, it's not like that. See, now it depends also. See, it's already July. So, I mean, it's there. I don't know if you have seen me when there was CBSE classes and then when there was JEE main, I was taking 8-8 hours classes. So yes, exam time is there. So now if I will not work hard, you will not understand the gravity of like J main. If I am teaching for half an hour daily and I will be like, do it. Then you will be like, okay, okay, I maybe maybe J main you will be like, like this only. So understand the gravity of it. So for that, we have to work hard, right? Please take for 9th. 9th, 10th combined. Okay, don't. 9th, 10th. I am not making a difference. Because see Olympiad, NSCJS is all about like 9 students can also give Olympiad 10th also. Same paper. Same syllabus. For Olympiad there is no discrimination. Right? Okay. Hi Anushka. Good evening. Chalo. Tell me now how to do it. Tell me how to do it. Sign 22 and a half degree. Sign. 22 and a half degree. Chalo. Gold formula. Remember, use gold formula. 2 sine square theta is 1 minus cos 2 theta. Okay? Komal is also here. Are you guys attending my session for the very first time? This is Shiva Kumari. Shiva Kumari from 2025 batch. Sin square theta is 1 minus. So theta, if theta is 22 and a half degree, 2 theta will be 45 degree. Yes. Okay. So now put the value 2 sin square 22 and a half degree is equals to 1 minus cos 45 degree. Why are we doing this? Because 22 and a half degree I want to calculate. With the help of 45 only. So I will need the help of 2 theta. So this is 1 minus 1 by root 2. Whole upon 2 is sine square 22 and a half degree. Now tell me one thing. Tell me one thing. One thing. Just tell me sine 22 and a half degree. Will the value be positive or negative? Sine 22 and a half degree whether the value will be positive or negative. Okay. So this is root 2 minus 1 upon 2 root 2. It will be a positive value. So now square you will remove. You will get square root and plus minus sine. But we know sine 22 and a half degree is positive. I will only talk about the positive value. Okay? So, sin 22 and a half degree is nothing but square root of root 2 minus 1 upon 2 root 2. Yes? Can you see? This is sin pi by 8. Now, sin pi by 8 or sin 22 and a half degree will be cos 67 and a half degree. Okay, 67 and a half degree. See, 67 and a half will be what? 3 pi by 8. 3 pi by 8 will be, so this is cos 3 pi by 8. Sine pi by 8 cos 3 pi by 8 will be equal, yes or no? Alright, I have came back just to see your teaching again. Kya baat hai? Are you feeling nostalgia, Shiva Kumari? Chalo! Chalo, one quiz, quiz time, quiz time. Let's see. Let's see, one quiz time. Cos 22 and a half degree. Now, 2 cos square theta is 1 plus cos 2 theta. Okay? And you know what? cos 22.5 degree is also a positive value right so this is 2 and this is square root cos theta is square root 1 plus cos 2 theta by 2 ma'am these are okay but what about pi by 12 values huh Rohit did you forget we did pi by 12 value where in the compound angles lecture How can you forget pi by 12 is 15 degree? Sine 15 degree, cos 15 degree. Last class we have done that already. So I don't think so you have to like do it, do it. Don't do it. Think about it. Same thing. Angle is same. Can you see the similar thing is going to happen? It's just that 1 plus cos 45 by 2 will happen. So 1 plus cos 45 that means there is a plus sign. Over here, so option, which option is correct? Tell me. So, option A is correct. Okay, this is cos. So, cos pi by 8 is equals to sin 3 pi by 8 is equals to what? Square root of root 2 plus 1 by 2 root. Yes? Okay? Clear? everyone okay Chal next quiz on your screen next quiz on your screen ma'am not like that just like science seven and a half degree big calculations that question hmm ma'am follow of chapters please follow na hi flow you are asking flow of chapters you're asking flow of chapters will be like this only now I have given you the plan also it will be similar only flow of chapters I have shared one video go watch that video let me share it to you one more time do you want me to do it flow of the chapters will be similar right bounce back plan Ma'am, am I only the one who is feeling hard on all question only in maths? Hmm, Abhijeet, maths is a little bit tough, huh? Maths is tough. Okay, so let's do this question. First of all, no, this question is not tough. Only you are feeling. Sorry, you are feeling only. Can you not, Abhijeet, chal tell me. Step by step and you tell me. Which step is difficult in this question? You see this question, you'll be like, sign it, sign it. cos a cos a. So can you take common 2 sin a cos a? 2 sin a cos a can be taken common or not? Yes Abhijeet, are you with me? 2 taken common sin a gone, cos a gone, so cos square a. Minus here, cos a gone, sin a gone, 2 gone, so sin square a. 2 sin a cos a is sin 2 a and cos square minus sin square is cos 2a. Abhijit, which step is difficult? Tell me. Now, multiply 2 and divide by 2 because we know that this is the formula. 2 sin 2a cos 2a. 2 sin 2a cos 2a. So, ma'am, 2 sin 2a cos 2a is sin 4a upon 2. Sin 4a upon 2 option B. Now, tell me Abhijit, is that difficult? Which step is difficult? Hmm? But yes, you are in class 9th. It will be difficult for you. I know. Okay? So, don't trouble yourself so much. Now, one thing, prove the following and please remember it also. This thing. Hi, Suji. How are you? Good evening. Good evening. Now, I got somewhat clear. Abhiji, don't get disheartened. See, I will tell you one thumb rule in mathematics. If you are not able to understand in the class, like if you are not able to at all understand in class, that's no big deal. But if 10, 20, 30% also you are understanding, just re-watch that lecture. By re-watching, you will be like now your understanding will go 60%. And make sure, not watching, watching. Sit with pen. Start solving with me. Do same thing twice, doing same thing twice actually is, is, will actually make you better in anything. Yes or no? Doing same thing twice. Suppose Abhijit or any other student, you have to give a presentation on, about life in your school. You'll be like, okay ma'am, this is my first presentation, etc, Now you'll be like, okay, you have written the pointers and then you went there and then five minutes speech you have given. But then you fumbled a lot. You got like so many issues also. You came back, you were not satisfied. You'll be like, okay, let's do one thing now. If now I will give you another opportunity. Now you know the loopholes. Now you know yourself also that where exactly I do not do good. So you'll be like no writing pointers was a bad idea. I was not fully prepared. I was thinking I was prepared but now I know more preparation is required. Where exactly I have to work. So you'll be like now I'll write full script for five minutes and the next speech obviously will be better. So doing second time will also make you better in that thing. Okay. It's just that you have to make more effort. Pure efforts nothing else. Now prove this following. cos theta plus sin theta whole square and remember it also see cos theta plus sin theta whole square remember it 1 plus sin 2 theta I am not helping you just open it cos square plus sin square is 1 and 2 cos theta sin theta is what sin 2 theta okay now cos theta minus sin theta will be 1 minus Okay, so do you want me to do this A part? A part is already done. Okay, talk about this. Talk about this. cot theta minus tan theta. So cot theta is cos theta by sin theta. tan theta is sin theta by cos theta. So in trigonometry, none of the things strike you. None of the things strike you. Just convert tan and cot in terms, just convert everything in terms of sin and cos. Okay? Majority of the time it works. Now sin cos. So this is sin theta cos theta. cos square minus sin square. Okay, now I clearly think that 2 is required in the denominator specially. So, 2 cos square theta minus sin square theta is 2 cos 2 theta. Whole upon 2 sin theta cos theta is sin 2 theta. So, clearly this is 2 cot 2 theta. Can you see step by step? Easily you will get it. Easily you will get it. Okay? Alright, done, done, done. See, are you now understanding multiple and submultiple angles, everyone? Okay, let's solve this question. Let's see how many of you are able to solve this question. Yes, Jainith, I'll do that. Thendral, thank you, thank you for writing good things over here. That we also need motivation. Okay. Root 3 cosec 20 degree minus sec 20 degree. Are you solving it? So, now I told you, whenever you get stuck, root 3 upon sine 20 minus 1 upon cos 20. Ok, now take LCM. I told you when you get stuck, convert everything in terms of sine and cos. Now, sine 20 cos 20 will become LCM. Root 3 cos 20 minus sine 20. Now, this is slightly different question actually. Slightly different question. Some different kind of manipulation will come into the picture. You will see me doing something for the very first time over here. Let me tell you what is this. Okay, let me tell you what is this. First of all, tell me that this is root 3 cos 20 minus sin 20. So, what I am going to do is, I am going to multiply 2 and divide by 2. Can I do that here? Multiply 2 and divide by 2. Why? Because root 3 by 2 and 1 by 2 are two values which I know. Root 3 by 2 is what? Root 3 by 2 is ma'am cos 30. And sin 60. 1 by 2 is sin 30 and cos 60. So, these two values are very well known to us. Okay? Alright. Now, one more thing. I am going to do one more thing. Now, sin 20 cos 20. So, 1, 2 and 1, 2 over here. Because 2 sin 20 cos 20 is also a formula. Now, do it. Apply it. So, two 2's are first of all 4. 2 sin 20 cos 20 is sin 40. And now, here. inside the bracket, let me tell you. Root 3 by 2 cos 20. Okay? So root 3 by 2 can be written down as sin 60 and this is cos 60. Yes? So sin 60 cos 20. Sin 60 cos 20 minus cos 60 sin 20. Cos 60 sin 20. Yes? Okay, getting it? So now this is 4 upon sine 40 degree. Sine A cos B minus cos A sine B. Sine A cos B minus cos A sine B. Sine A cos B minus cos A sine B is sine A minus B. So now can you see sine 40 and sine 40 is gone. So 4 is absolutely right. option C. CCC. CC. Okay, all right everyone. Yes. Now some more formulas we are going to talk about. Okay. Triple angle formulas. Now is the time we do triple angle formulas. Three more formulas. That's it. I'm not going to stretch it. Three more formulas. I think I am getting lagged chat. Is it? Or you know, you guys are not active on chat now. Chalo, let's do triple angle formula. Tell me, sin 3 theta, cos 3 theta and tan 3 theta. Three formulas we are going to learn. Sin 3 theta. Sin 3 theta is 3 sin theta minus 4 sin cube theta. Cos 3 theta is 4 cos... cube theta minus 3 cos theta and you know what tan theta welcome to the live session cosmic a welcome to the brand new session okay for you tan 3 theta tell me tell me you are not going to tell me i know i have to tell you but tan 3 theta listen to me very carefully do not get panic because slightly difficult formula but ninja mam is there to tell you the trick also Okay, tan 3 theta is 1 minus 3 tan square theta. Sorry, sorry, sorry. Tan 3 theta is 3 tan theta minus tan cube theta. Whole upon 1 minus 3 tan square theta. Now, you will be like, ma'am, how to remember? Hold on. So, trick to remember is, okay, first of all, First of all, number one, both have negative sign in numerator and negative in denominator. Negative in numerator, negative in denominator. Second, diagonally there is 3. Diagonally 3 is there. Diagonally 3, 3. Can you see? 3 and 3. Okay? Yes, Tarun Raja. Welcome to the session. Welcome. Are you enjoying it? Were you there since the very beginning? Rama Raju is to late today. Okay. Alright. Diagonally, what we are going to do is, can you see 3, 3 and 1, 1. Third thing is, can you see one more important point to notice. Numerator has odd power. 1, 3. Denominator has even power. 0, 2. So, it is 0, 1, 2, 3. So, Ninja ma'am, do you remember Ninja ma'am? N. So, this is 0, 1 and then 2, then 3. 0, 1, 2, 3. 0, 1, 2, 3. That means denominator will have 0 and 2. So, 1 is what? Degree 0. And square theta. Can you see tan square theta? And 1, 3. So, tan power 1 and tan cube. This trick toh I have not given it to your seniors. 2025 aspirants will be like, ma'am you toh didn't give me that trick in the last year. This is a new trick for specially for 2026 students, okay? So yes, how will you remember tan 3 theta is? Now see, both numerator, denominator, negative sign, 3, 3 diagonally is there and you know what? You know what? This is 1, this is also 1. So, 3, 1. And now, let's talk about degree. Degree, how will it start? So, how do you write n? Ninja ma'am, right? n. So, like this. Now, 0, 1, 2, 3. So, denominator is having 0 and 2. Numerator is 1 and 3. Right? Easy. Okay, yes Rohan what happened? What happened Rohan? Now quickly tell me the answer of this question Okay, cos 3x. Ma'am, you told us cos 3x is 4 cos cube x minus 3 cos x is the formula for cos 3x. Plus 3 cos x is kept as it is. 1 plus cos 2x. Gold formula. Gold formula. Do you remember 1 plus cos 2 theta is? 2 cos square theta. So, this is 2 cos square x. Now, cancel. 3 cos x, 3 cos x. Gone. So, now 4 by 2 will be 2. Cos cube by cos square is cos x. Can you see option C is absolutely right? Easy or not? Easy or difficult? Okay. Now, What are we going to do is we are going to talk about one special angle. One special angle. Okay? One special angle. Now you have to tell me actually. I am not going to tell you. We have to calculate the value of sin 18 degree. How do you, what do you think? Like how can we calculate 18 degree? And you know what? See. Whenever we talk about special angles like 15 degree, 22 and a half degree. Or any sort of these kind of things, right? We always relate. See, 15 degree tell me how did we calculate? Ma'am, 15 degree is 45 minus 30. So, we always take help of standard angles. To calculate special angle value, what do we do? We think about, we relate that 18 degree or 15 degree or 22.5 degree to special angle, to standard angle. Yes or no? Yes, so 15 degree was 45 minus 30. Both 45, 30 were standard angles. Then we did that pi by 8, that 22 and a half degree. So, 22 and a half degree is half of 45. So, we thought about 45 is the immediate relatability in terms of 22 and a half. Now, think about 18 degree. 18 degree, think about it. 18, what can we do with 18? So, that we will get a Standard angle. What can we do to 18? See 18 degree is theta. Now see 5 Rama Raju is saying but Rama you are not telling me the right thing. Okay. Ma'am will the degree come in decimal? Yes Tarun Raja we have just done this. 22.5 degree in this session. Just go back you will see. Go back you will see. Okay, so 18 degree. Now, multiply 5. 5 into 18 is 90 degree is equals to 5 theta. Multiply by 5. Multiplication with 5 will give you what? Multiplication with 5 will give you the standard angle. Okay, alright. So, this is sine 18 degree. So, sine 18 degree. So, sine See, now start with sin 90 degree is equals to sin 5 theta. Okay? So, 1 is equal to sin 5 theta. Yes or no? Ma'am, we know 60 value. Yes, but 60 and 18. Okay, you are talking about 18 into 3. But that is going to be complicated. Okay? That is going to be complicated. See, now understand. 90 degree, see do like this. 5 theta is equals to pi by 2. 5 theta is equals to pi by 2. Can we do this? 5 theta being pi by 2? Yes ma'am. So now 5 theta is 2 theta plus 3 theta is equals to pi by 2. Yes? Are you getting it? Understanding it? Okay. So now what you're going to do is 2 theta is equals to pi by 2 minus 3 theta. 2 theta can be written down as pi by 2 minus 3 theta. Take sine both the sides. So sine 2 theta is equals to sine of pi by 2 minus 3 theta. So can you see sine 2 theta is equals to what? Sine 90 minus 3 theta is cos 3 theta. So this is what we have got. We just need to solve it. Sin 2 theta is equals to cos 3 theta. We just have to solve it. What do we want? What do we want? Tell me. From here, this is equal to sin theta. So we are actually, from this equation, what is the agenda? Agenda is to calculate the value of sin theta. Sin 2 theta is? Sin 2 theta? is equals to cos 3 theta. Please do not forget eventually we want the value of sin theta. So, this is 2 sin theta cos theta and cos 3 theta anyone. cos 3 theta. cos 3 theta is what? 3 cos theta minus 4 cos cube theta. Now can you see this cos theta is coming out? cos theta is coming out. Yes, so it become what? 2 sin theta is equals to 3 minus 4 cos square theta. See, now what do you want? Eventually you want sin theta. Cos square theta, can we do not write sin square theta? Think about writing everything in terms of sine theta. Ok, so 2 sine theta, 3 minus 4 is minus 1 plus 4 sine square theta, 4 sine square theta minus 2 sine theta minus 1 equals to 0. Ok. Now quadratic equation in sine theta, you just want sine theta, that's it. Sine theta is minus b, so 2 plus minus b square, so 4 minus, so plus 4, 4 is 16 by 2 4 is 8. Okay, so now I'm going to do something, please do not panic. 2 1 is 2, 2 4 is 8. Okay, so like I am just dividing whole numerator denominator by 2. 4 and this is going to become 4. So, this is sine theta is actually becoming 1 plus minus. Now, understand 1 plus minus. Okay. 1 plus root 5 by 4 or sine theta can be 1 minus root 5 by 4. Just tell me one thing. Sine 18 degree will be positive or negative. Obviously, sin 18 degree will be positive because of Being in first quadrant, 1 minus root 5. See root 5 value is 2 point something. So 1 minus Tarunakirti 4. Have I done anything wrong over here? 4 cos square 4 cos cube minus 3. I have done something else. Oops! I have done something else. 4 cos cube, right? 4 cos cube theta minus 3 sin theta. So that means I will have to erase everything. Okay? Sorry, minus 3 cos theta. Okay, so cos theta, this is cos theta, cos theta, this is square. So, 2 sin theta is equals to 4 cos square theta. 4 cos square theta is this, 4 minus 4 sin square theta. Yes, minus 3. Okay, so this is 2 sin theta is equals to 1 minus 4 sin square theta. Call everything in the left hand side. 4 sin square theta plus 2 sin theta. And minus 1 is equals to 0. Yes. Okay. So now sine theta will become what? Minus 2 plus minus square root of b square plus 4 4's are 16 upon 2. So sine theta will become what? Minus 1 plus minus. See if you divide by 2. Root 20 will be what? 2 root 5. So root 5 by 2 into 4. So eventually it will be. 4. Okay? Sin theta is minus 1 plus minus root 5 by 4. Now understand, either you will take plus sin or minus sin. Minus 1 plus root 5 is a positive thing. Minus 1 minus root 5 is a negative thing. So, sin 18 degree will obviously positive. So, root 5 minus 1 by 4. This is the value of sin 18 degree. By the way, sin 18 degree is nothing but sin pi by 10. So, sin pi by 10 value please keep remember, please do remember it is root 5 minus 1 by 4. Okay everybody? Yes? Getting it? Understanding it? Hmm. So, just remember this sin pi by 10 is equals to cos 2 pi by 5. Okay, sin 18 and cos 72 are equal which is root 5 minus 1 by 4. Okay? Alright? Cool? Chalo, now let me talk about something over here. Yes, so we are done with today's session, multiple and sub-multiple. Okay, oops, oops, I cannot drag it. I cannot drag it. Let me, yes, this is how it is. Can you see that this is, this was the video I was talking about. So, I have made a full-fledged three, like four months plan. Can you see four months plan I have? created for you I have uploaded this video okay so now if you can see over here I will proceed and see first of all I have told you that how to customize my given plan as per your need obviously I am giving you a planner that doesn't mean that everybody will follow the same planner you have to little bit here and there modify it as per your schedule as per your timetable Based on that weekly timetable, I want all of you to make daily timetable. I have given you 4 months plan. I have given you weekly plan as well. And then I want everyone of you to convert it into daily targets. Okay? What is the plan? Where is the plan? Let me just move forward and you will be like, this plan I have made for you. 4 months planner by Ninja Ma'am. Week 1, it starts with July. And it ends with, if you just go over here, so I have told you that this planner and week by week, what are the important chapters, right? Okay, so just check over here. Okay, so can you see week 16? It is still 27th October. Physics and chemistry, yes. What you can do is next time when Deeksha ma'am will come live. Just ask them to share similar timetable. They will share. Okay. Yes. So make sure that you are following it. This is for you all to make sure that you guys are on track. Okay. Because sometimes we feel we have time. We have time. But you see there are so many chapters. And this is just for mathematics. Okay. This is just for mathematics. For 2027 a similar plan I will share. Don't worry Sanjay Shweta. Okay. Yes, but right now this is for 2026. So before October, at least 70% syllabus, at least I am saying. At least 70% syllabus should be done. Okay, otherwise you will lag. That is what I have told in this video. Make sure that you watch this video and make sure that you hit the like button also. Like, like, like, make sure that you like. Okay, so yes, this was all about today's session. and ma'am max attempts what are you talking about max attempts mean means what what do you want okay so now i think i am going to meet you one more time huh 10 o'clock are you ready for that 10 functions questions 2026 aspirants and 2027 also because functions is common to both your j2026 and 2027

Hello, hello, hello Ninja Warriors. Sup? How is everyone?

Are you guys absolutely fine? Let me just open this chat box. Let me check whether I'm live or not. Am I live students? If I am live, then please do let me know in the chat box.

Okay. So, so, so, okay, where is the live chat. Oops. Okay, okay, okay.

Alright. Okay. Yes, so I can see you all over here.

Hi Abhijeet, hi Taruna, hi Rohind, hi Girish, hi William. How are you all? The very first thing you have to do, you remember na? The moment you enter the session, you have to hit the like button, okay? So today's session is going to be all about multiple and sub-multiple angles, another very important topic for trigonometry and I am absolutely fine William, how are you all?

How are you? Okay, so multiple and sub-multiple angles, today we are going to start a couple of formulas we are going to do today and based on that we are going to solve questions as well. So yes, That's how it is going to go. And I guess one student is asking when you are going to finish the syllabus.

When are you going to finish the syllabus? See, syllabus we will be finishing approximately by the end of October or maximum by mid-November. If like one or two sessions here and there can happen, right? But estimate is when Cengage book solving? Yes, So, Cengage solving session is going to happen today at 10 o'clock.

So, I don't know how many of you have noticed but since yesterday I have started a regular practice session from different different books and from different different chapters. So, can you increase volume level? My volume is at maximum Girish. Try changing your device or maybe plugging in your phones for better experience.

Okay? Yes? All right, good evening, Kurva Ashwin. So yes, I was talking about the recent practice sessions.

Yes, today we have any class at 10. Yes, can you, you can't see, you can't see the class at 10. Let me show it to you. I have, see I have already scheduled it. But you are not, maybe you are, it is not visible to you, kya? It is not visible. Let me just open the channel.

You also open the channel, na? Here is the channel, can you see? So this is how it is.

You go here and live session if you go, okay? So today what are we going to do? 10 killer problems from functions, Cengage K functions problem. 10 o'clock that is going to happen.

Today is 11th July. So please hit the like button right now, okay? Alright?

Yes, sequence and series also we will do. One session for sequence and series will also do. Yes, notes of the session, yes, I will provide. Okay, I will add in the description box. Okay, so shall we start today's session?

Are we ready for today's session everyone? Okay, are we ready for today's session? Chalo. But before that, I just want to make sure that you Answer me this particular question. Solve this particular question for me.

Ma'am, I don't know functions. But Abhijeet, that's how it is. We are going to cover each and every chapter. I can't be taking only one chapter whole year. Okay?

Alright? Yes. So this question solving, question solving. So yes, announcement.

Every day, almost every day. Half an hour we are going to do practice session. So mostly not every day but at least 3 to 4 days in a week we will do maths practice. Okay? Can first PU students also watch that 10 series?

Yes Girish, you can watch that, okay. It's just that if there are core class 12th chapters, that session, that will be trouble for you, suppose you have not studied it. But yeah, mostly quadratic, sequence, these chapters, basics of function also is there in first PU, right?

So you can attend it. Chalo, let's start this question. Tan 14 degree, tan 31 degree plus tan 14, tan 31. Okay. So, these kind of questions I think I guess we have done in the last class. Tan 14 degree, tan 31, tan 14 into tan 31. So, 14 and 31, can anybody tell me?

14 and 31 has a relation. So, 45 ko. What is happening? Mosquito everywhere. Mosquitoes.

So many mosquitoes today. I don't know from where they are coming. Tan 45 degree is equals to what? Tan.

14 plus 31. Can I write? Now tan 45 degree is 1 and tan a plus b. Tan a plus b is what?

Tan a plus tan b whole upon 1 minus tan a tan b. Okay? Getting it? Understanding it? Everyone?

So now if you just cross multiply, you will realize tan 14 plus tan 31 is equals to 1 minus tan 14 tan 31. So, actually the correct answer is just cross multiply and rearrange option A is absolutely right. I am use good night liquid. Yeah, good night liquid all out all these devices my mosquitoes are sitting on that device.

They don't care about what do you do. Okay. Yes.

Now, let's talk about multiple and sub multiple angle. okay multiple Multiple and sub-multiple angle. Okay? Very important session.

Trust me. Very, very important session. Okay?

Important not only just for trigonometry but we are going to use these formulas in calculus also. So, this session make sure that you attend today's session with utmost dedication and attention. Okay?

Yes, clear? Chalo. So the very first formula is sin 2 theta.

Sin 2 theta is actually 2 sin theta cos theta. Can anybody tell me how to prove it? Can anybody tell me how to prove it?

Sin 2 theta is 2 sin theta cos theta. Tell me how to prove this thing. So ma'am sin 2 theta is 2 sin theta cos theta.

It is actually sin theta plus theta. sin a plus b Right? sin is now sin plus cos .

Okay? So sin became what? Tell me. sin cos theta plus cos theta. So this is 2sin theta. Yes or no?

Getting it? Understanding this formula? Okay.

Now one more thing, one more formula for sin we can do this. What I'm going to do is 2 sin theta by cos theta, I'm dividing it by 1. Can I do that? Sin 2 theta is 2 sin theta cos theta divide by 1. And this 1 will be replaced by sin square theta plus cos square theta.

Okay, sin square theta plus cos square theta. Whole upon, can you see? Now, step by step, same thing we are going to do multiple times. Now divide 2 sin theta cos theta by cos theta and here also divide by cos theta.

What will you get 2 sin theta cos theta? So cos theta cos theta cancel. So 2 sin theta by cos theta is tan theta and you know what? See actually you need to divide by cos square theta not cos theta cos square theta.

So what will happen 2 sin theta cos theta by cos square theta right? So, 1 cos and 1 cos gone. 1 cos, 1 cos gone. Sin by cos is tan. Why 1 changed into that ma'am?

Because we are creating different formulas. Sin 2 theta in terms of theta. Sin 2 theta in terms of tan.

Manipulation, right? This is what we call manipulations. Sin square theta by cos square theta is tan square theta. So, can you see if you divide by cos square theta, what will you get? Cos square by cos square is 1. Sin square by cos square is tan square.

So, this is another formula. So, this is the very first formula. You can see the second formula is, ok, sorry.

Second formula, let me write it over here, which is right, yes. 2 tan theta upon 1 plus tan square theta, ok. Can you see? So, this is you have to remember. Okay, will you be able to remember it?

Okay, ready? Ready for the sine 2 theta is what? Now tell me, remember. Ma'am, sine 2 theta is 2 sine theta cos theta and sine 2 theta is 2 tan theta upon 1 plus tan square theta. Now one thing, understand what are these formulas.

See, if sin 2 theta is 2 sin theta cos theta, what is happening over here? 2 theta is becoming theta. That means whatever is present over here, we are going to divide it by 2. So here if the formula, if the question is sin theta, then what will you do? Ma'am, sin theta is 2 sin theta by 2 cos theta by 2. Okay?

So remember it that way. 2 theta. on the right side becomes what?

Theta by 2. Okay? Alright? Yes? Now cos theta. Cos 2 theta.

Now cos 2 theta has so many formulas. Let me tell you. Cos 2 theta, the very first formula is cos square theta minus sine square theta. Okay?

Again, can you prove it for me? Can you prove it? Chalo ma'am, let's prove na?

Let's prove. Why not? Ma'am, same thing.

Cos 2 theta is cos theta plus theta. Cos a plus b is cos a cos b minus sin a sin b. Cos a cos b minus sin a sin b which is nothing but cos square theta minus sin square theta.

Now, one more thing, one more thing. Convert cos square. Okay.

Convert sin square theta in terms of cos square theta. Do that convert sin square theta in terms of cos square theta. So, cos square theta minus 1 minus cos square theta. So, cos square plus cos square is 2 cos square theta minus 1. So, one more formula 2 cos square theta minus 1 and now change this cos square theta. into sine.

So, 1 minus sine square theta minus 1. This will be 2 minus 1, 1 minus 2 sine square theta. So, one more formula. 1 minus 2 sine square theta. Can you see 1, 2, 3?

Okay. So many formulas for cos 2 theta. 1, 2, 3. Fourth formula is already coming your way.

Same thing we have done. Remember, see cos 2 theta Ma'am, you told us cos square minus sin square. Okay?

And now ma'am, what will you do is divide it by 1. Divide by 1. Abhijeet, are you here? Now, one more time. 1 will be replaced by cos square theta plus sin square theta. 1 will be replaced by cos square theta plus sin square theta. And eventually, Now what you have to do is divide the numerator and denominator by cos square theta.

Divide the numerator and denominator by cos square theta. So what will happen? Cos square theta upon cos square theta is 1. Sin square by cos square is tan square. Now cos square by cos square is 1. Sin square by cos square is So, one more formula.

Okay? So, now let me write one more formula after this which is yes. 1 minus tan square theta upon 1 plus tan square.

You don't have to remember the proof proof but yes. What you need to do is you need to remember white color. No. Green utilized. Orange.

Hmm. Okay, don't worry. I am here.

We'll remember formulas together. Right now, just get the glimpse of it. So, there are two things.

Ma'am, we saw sin 2 theta which is 2 sin theta cos theta and that is 2 tan theta upon 1 plus tan square theta. Okay, I told you know 50 plus formulas in trigonometry. Then, cos 2 theta is cos square theta minus sin square theta. 2 cos square theta minus 1, 1 minus 2 sin square theta and 1 minus tan square upon 1 plus tan square.

Now, I will tell you, I will tell you how to remember it ma'am. I don't know, please tell me. First of all, please remember sin is bad guy. Sin is a bad guy. Because sin, sin, sin is what?

Do you know Google? S-I-N, sin. Sin is what?

Negative. Not a positive thing. Always remember, sin is negative. Sin is negative.

Minus sin in front of sin. Now, obviously these two formulas are similar. One is 2 cos square theta minus 1. And other one is 1 minus 2 sin square theta.

So, I think this is very much similar. It's just that sin square theta has a negative sign. Because of the sin.

Negativity in sin. And you know what? This formula. Okay, now this formula.

Especially this cos 2 theta is. Now sin 2 theta and cos 2 theta understand. Both are same formulas actually.

Sin 2 theta is also having 1 plus tan square theta. Cos 2 theta is also having 1 plus tan square theta. Just the numerator part is different.

Here it is 2 tan theta. Here it is 1 minus tan square theta. Now 1 by 1. 1 by 1 we will do it. We will not do it everything at once.

One formula at a time. Alright, so today's session is going to be all about remembering these formulas only. So just understand like these sin2θ, cos2θ and tan2θ. What is happening? Math file is saying idea about her.

Okay. Now tan 2 theta, tan 2 theta. Ma'am tan 2 theta is 2 tan theta upon 1 minus tan square theta. What is this? Let's understand.

What is this? So ma'am tan 2 theta is 2 tan theta upon 1 minus tan square theta. So it is tan a plus b.

Remember tan 2 theta is tan a plus b. Danny plus tan b. So tan theta plus tan theta upon 1 minus tan a tan b. Can you see you get we will get this only. So this is the formula of tan 2 theta.

Yes Mr. Perfect. Now one more important observation which is which can be a point of confusion but I will clear your confusion. You know what? Sine 2 theta formula. And tan2 theta formula, both formulas are very similar.

I am talking here about sin2 theta in terms of tan. Can you see? Sin2 theta is 2 tan theta upon 1 plus tan square theta.

Tan2 theta is 2 tan theta upon 1 minus tan square theta. So just one sin is there. Just one sign difference. In tan, there is a negative sign. But in sine to theta, there is a positive sign.

But you know what? Keep everything in mind. Gradually, I am going to tell you everything. Now, I think it's the time to solve a couple of questions. Let's solve.

Let's solve. I am going to help you with the formulas. So, they are talking about tan 2a. Tan a is given. Which formula are you going to use?

Ma'am, turn. 2 theta is 2 tan theta upon 1 minus tan square theta. This is what we are going to use. Use it, tell me quickly.

Abhijeet, go and watch, re-watch it. I have explained it very nicely. Divide by 1, write it sin square plus cos square and eventually now the expression whole, numerator and denominator gets divided by cos square theta.

Okay, so I think it's an easy question ma'am. tan 2a is equals to 2 tan a upon 1 minus tan square. Direct formula based na. 2 into 1 by 3, 1 minus 1 by 3 whole square.

So, 2 by 3, 2 by 3 and 1 minus 1 by 9 is 8 by 9. 3 ones are 3, 3 threes are 9. So it is 3 2's are 6 by 8 which is equals to 3 by 4. Easy question? Yes? Very direct formula based question. Initially I will make sure that you only remember the formulas.

I am not doing anything which is very difficult. Okay? Now some prove that type of questions so that you get hang of how to solve a question. Some prove that. Okay?

By the way before prove that, before prove that Hold on, hold on, hold on. Before prove that, two gold formulas, two gold formulas. Before prove that, we have to talk about two gold formulas, okay?

Two gold formulas. See, do you remember cos 2 theta? So, I am cos 2 theta is equals to 2 cos square theta minus 1 and 1 minus 2 sin square theta.

These are not the gold formulas. Okay? These are the reference formulas.

These are cos 2 theta. Now, these are the reference formulas. Two gold formulas. Number 1. Number 1. Tell me the value of 2 cos square theta. Good evening Sanjay Shweta.

Come. Right? timing. 2 gold formula.

Gold. Why gold? Gold means very good. Like very expensive. Okay.

So, cos 2 cos. Can you tell me 2 cos square theta value and 2 sin square theta value. Quickly. So, ma'am 2 cos square theta is 1 plus cos 2 theta.

And 2 sin square theta will be 1 minus cos 2 theta. Can you see over here what is happening? What is happening students? What is happening? What is happening?

Sign has a negative sign. Sin, sin, sin, sin, negative. S-I-N, sin is equals to sin.

Sin means what? That sin. Sin means in Hindi we call it as paap, something very evil. Sin is evil.

That is why it is negative. That is why you will see negativity, okay, in sin. That's how you remember. When you have two similar formulas, please remember, it is but obvious, it is so human psychology.

If at all you are remembering two similar things. At one time. It is so obvious that you are going to get confused that which one is this? 2 cos square theta is 1 plus or 1 minus. Ma'am, I got confused.

But now you have to remember sign negative. Now it will be done. Okay. Chalo, let's solve.

Now I am going to give you 3 questions back to back. Let's see how many of you are able to solve it. Okay, try.

So, 1 minus cos 2 theta ma'am is actually a. Left hand side ma'am, 2 sin square theta. Whole upon sin 2 theta ma'am, 2 sin theta cos theta. So, clearly what we can do is ma'am, 2, 2 cancel, sin, sin cancel. So, this is tan theta.

Done, done ma'am. Easy question, easy question. Easy question, yes.

Same 1 minus cos 4 theta now. So now 1 minus cos 4 theta will become what? 2 sin square 2 theta ma'am. And sin 4 theta will become what?

2 sin 2 theta cos 2 theta. Now again you need to cancel. 2 2 cancel, sin sin cancel. So tan 2 theta. So ma'am this is also easy, not very difficult.

Now cos 2 theta upon 1 plus sine 2 theta. This is something which is slightly on a trickier side. Cos 2 theta ma'am. Now I have so many formulas for cos 2 theta.

What to do? So let's try. Let's try the very first formula which is a balanced formula. Cos 2 theta is 2 cos square theta minus sine square theta.

Now 1 plus sine 2 theta. 1 plus sine 2 theta. So, can you see sin 2 theta is actually 2 sin theta cos theta and 1, 1. Now, 1 will be replaced by Abhijit again. Abhijit is so traumatized that ma'am randomly how come you are writing sin square theta plus cos square theta instead of 1. That is the whole point. That is the whole thing.

Which one will be replaced by sin square theta plus cos square theta? That person. If you are able to think that, that person is like something else. So now this one will be replaced by sin square theta plus cos square theta. Why I did that?

The answer lies in the next step. Sin square theta plus cos square theta plus 2 sin theta cos theta. Can you see now why did I do that?

Because this is cos theta plus sin theta whole square. Can you see? This is actually cos theta plus sin theta whole square.

Cos square theta minus sin square theta. a square minus b square is a plus b, a minus b. Some manipulation. Keep on solving. You will get your answer.

Don't worry. So clearly cos theta plus sin theta, cos theta plus sin theta gone. Okay?

cos theta minus sine theta upon cos theta plus sine theta. Okay, now you have to divide everything by cos theta. Like numerator divided by, remember we have done this, come on. Cos minus sine, cos plus sine, previous lecture only.

Divide numerator and denominator both by cos theta. So, if you divide numerator by cos theta, it will be 1 minus sine by cos is tan and 1 plus tan theta. Which is actually equal to, do you remember the previous class formula?

1 minus tan, 1 plus tan is tan, 45 minus theta. Getting it? Done, done, done. What do you think about these three problems? Good?

Ready for the next? Move on from the x. Ready for the next? Okay? This.

something trending Instagram thing okay chalo are we ready for a question now you have to give me an answer ready for like you have to give me the answer okay chalo Chalo, ready ready? 1 minus 2 sine square pi by 4 plus theta. Okay?

I don't know formulas ma'am. Which formulas? Which formulas? I have just taught you Abhijit. What are you saying?

Please don't say that. I just taught you. How can you say something which I just taught you?

See, remember? Okay, which formula will be applicable? 1 minus sine square something. Suppose 1 minus sine square theta is what? Ma'am, 1 minus sine square theta is?

cos 2 theta. 1 minus sine square theta is what? cos 2 theta.

So 1 minus sine square anything is equals to what? cos 2 times that same thing. Yes or no?

Now cos 2 times same thing. So cos 2 times pi by 4. So pi by 2 plus 2 theta. Cos pi by 2 plus 2 theta. is actually sin cos pi by 2 plus pi by 2 ASTC, cos pi by 2 plus theta, second quadrant, cos is negative. So, minus sin 2 theta.

So, option D is absolutely right. Okay, one thing I was like thinking about one thing and it was for class 9th and 10th students. basically.

I was thinking of bringing some sessions for 9th and 10th class and that is not going to be your usual school sessions like 9th and 10th NCRT I am not going to solve. Something which is Jai Tau we are doing it and I think 9th and 10th students some students like Abhijit are also attending it but I was thinking about something specific to 9th and 10th. Can it be like Olympiad. can it be ntsc nsc js like something like that okay so let me think about it but if at all you want something you want to give suggestions so feel free to give suggestions in the comment session section okay all right yes abhijeet okay Ready, ready?

This question good, good, good? Okay, chalo. Now prove that. sin theta upon 1 plus cos theta. See, now sin theta can be written down as 2 sin theta by 2 cos theta by 2. And it is so obvious, so intuitive, the right hand side is all about theta by 2, right?

So, 1 plus cos theta. Now, 1 plus cos theta is 2, see, plus sin. So, 2 cos square theta by 2. Now, start cancelling. 2, 2 gone.

Cos theta, cos theta gone. Tan theta by 2. No ma'am, only focus on second PUJ, it's my opinion. Girish, I'm not going to like, I'm going to take, JE sessions will be there, I'm not disturbing that, okay? Just apart from that, like one more thing I want to start for 10th. Don't worry, I will not disturb you people.

Chalo, one question quickly, Answer me right away. 1 plus cos x, ma'am, 2 cos square x by 2. Upon sin x will be 2 sin x by 2 cos x by 2. Okay? Ma'am, 2, 2 cancel.

Cos, cos cancel. Cos x by 2, sin x by 2. Cot x by 2. Yes? Everyone? Getting it?

Understanding now? Are you able to remember all the formulas? Tell me if you are able to remember the formulas. Okay, let's go. Now it's time to talk about some special angles.

Remember 15 degree. Now similar way 22 and a half degree is also special angle. 22 and a half degree is actually 45 degree by 2 is equals to 22 and a half. 45 is pi by 4. So pi by 4 into 2 pi by 8. Okay, so that is nothing but 22 and a half.

So remember it first of all the degree and. Something has motivated ma'am. Ma'am has taken many initiatives just in one week.

Practice session, ninja formulas, classes for 9th and 10th. No, it's not like that. See, now it depends also. See, it's already July. So, I mean, it's there.

I don't know if you have seen me when there was CBSE classes and then when there was JEE main, I was taking 8-8 hours classes. So yes, exam time is there. So now if I will not work hard, you will not understand the gravity of like J main.

If I am teaching for half an hour daily and I will be like, do it. Then you will be like, okay, okay, I maybe maybe J main you will be like, like this only. So understand the gravity of it. So for that, we have to work hard, right? Please take for 9th.

9th, 10th combined. Okay, don't. 9th, 10th.

I am not making a difference. Because see Olympiad, NSCJS is all about like 9 students can also give Olympiad 10th also. Same paper.

Same syllabus. For Olympiad there is no discrimination. Right?

Okay. Hi Anushka. Good evening. Chalo. Tell me now how to do it.

Tell me how to do it. Sign 22 and a half degree. Sign.

22 and a half degree. Chalo. Gold formula. Remember, use gold formula.

2 sine square theta is 1 minus cos 2 theta. Okay? Komal is also here.

Are you guys attending my session for the very first time? This is Shiva Kumari. Shiva Kumari from 2025 batch. Sin square theta is 1 minus.

So theta, if theta is 22 and a half degree, 2 theta will be 45 degree. Yes. Okay. So now put the value 2 sin square 22 and a half degree is equals to 1 minus cos 45 degree.

Why are we doing this? Because 22 and a half degree I want to calculate. With the help of 45 only.

So I will need the help of 2 theta. So this is 1 minus 1 by root 2. Whole upon 2 is sine square 22 and a half degree. Now tell me one thing.

Tell me one thing. One thing. Just tell me sine 22 and a half degree.

Will the value be positive or negative? Sine 22 and a half degree whether the value will be positive or negative. Okay. So this is root 2 minus 1 upon 2 root 2. It will be a positive value. So now square you will remove.

You will get square root and plus minus sine. But we know sine 22 and a half degree is positive. I will only talk about the positive value. Okay? So, sin 22 and a half degree is nothing but square root of root 2 minus 1 upon 2 root 2. Yes?

Can you see? This is sin pi by 8. Now, sin pi by 8 or sin 22 and a half degree will be cos 67 and a half degree. Okay, 67 and a half degree.

See, 67 and a half will be what? 3 pi by 8. 3 pi by 8 will be, so this is cos 3 pi by 8. Sine pi by 8 cos 3 pi by 8 will be equal, yes or no? Alright, I have came back just to see your teaching again. Kya baat hai?

Are you feeling nostalgia, Shiva Kumari? Chalo! Chalo, one quiz, quiz time, quiz time.

Let's see. Let's see, one quiz time. Cos 22 and a half degree.

Now, 2 cos square theta is 1 plus cos 2 theta. Okay? And you know what?

cos 22.5 degree is also a positive value right so this is 2 and this is square root cos theta is square root 1 plus cos 2 theta by 2 ma'am these are okay but what about pi by 12 values huh Rohit did you forget we did pi by 12 value where in the compound angles lecture How can you forget pi by 12 is 15 degree? Sine 15 degree, cos 15 degree. Last class we have done that already. So I don't think so you have to like do it, do it. Don't do it.

Think about it. Same thing. Angle is same.

Can you see the similar thing is going to happen? It's just that 1 plus cos 45 by 2 will happen. So 1 plus cos 45 that means there is a plus sign. Over here, so option, which option is correct?

Tell me. So, option A is correct. Okay, this is cos.

So, cos pi by 8 is equals to sin 3 pi by 8 is equals to what? Square root of root 2 plus 1 by 2 root. Yes?

Okay? Clear? everyone okay Chal next quiz on your screen next quiz on your screen ma'am not like that just like science seven and a half degree big calculations that question hmm ma'am follow of chapters please follow na hi flow you are asking flow of chapters you're asking flow of chapters will be like this only now I have given you the plan also it will be similar only flow of chapters I have shared one video go watch that video let me share it to you one more time do you want me to do it flow of the chapters will be similar right bounce back plan Ma'am, am I only the one who is feeling hard on all question only in maths?

Hmm, Abhijeet, maths is a little bit tough, huh? Maths is tough. Okay, so let's do this question.

First of all, no, this question is not tough. Only you are feeling. Sorry, you are feeling only.

Can you not, Abhijeet, chal tell me. Step by step and you tell me. Which step is difficult in this question? You see this question, you'll be like, sign it, sign it. cos a cos a.

So can you take common 2 sin a cos a? 2 sin a cos a can be taken common or not? Yes Abhijeet, are you with me?

2 taken common sin a gone, cos a gone, so cos square a. Minus here, cos a gone, sin a gone, 2 gone, so sin square a. 2 sin a cos a is sin 2 a and cos square minus sin square is cos 2a.

Abhijit, which step is difficult? Tell me. Now, multiply 2 and divide by 2 because we know that this is the formula. 2 sin 2a cos 2a.

2 sin 2a cos 2a. So, ma'am, 2 sin 2a cos 2a is sin 4a upon 2. Sin 4a upon 2 option B. Now, tell me Abhijit, is that difficult?

Which step is difficult? Hmm? But yes, you are in class 9th.

It will be difficult for you. I know. Okay? So, don't trouble yourself so much.

Now, one thing, prove the following and please remember it also. This thing. Hi, Suji. How are you?

Good evening. Good evening. Now, I got somewhat clear. Abhiji, don't get disheartened. See, I will tell you one thumb rule in mathematics.

If you are not able to understand in the class, like if you are not able to at all understand in class, that's no big deal. But if 10, 20, 30% also you are understanding, just re-watch that lecture. By re-watching, you will be like now your understanding will go 60%. And make sure, not watching, watching. Sit with pen.

Start solving with me. Do same thing twice, doing same thing twice actually is, is, will actually make you better in anything. Yes or no? Doing same thing twice. Suppose Abhijit or any other student, you have to give a presentation on, about life in your school.

You'll be like, okay ma'am, this is my first presentation, etc, Now you'll be like, okay, you have written the pointers and then you went there and then five minutes speech you have given. But then you fumbled a lot. You got like so many issues also.

You came back, you were not satisfied. You'll be like, okay, let's do one thing now. If now I will give you another opportunity. Now you know the loopholes.

Now you know yourself also that where exactly I do not do good. So you'll be like no writing pointers was a bad idea. I was not fully prepared. I was thinking I was prepared but now I know more preparation is required.

Where exactly I have to work. So you'll be like now I'll write full script for five minutes and the next speech obviously will be better. So doing second time will also make you better in that thing. Okay. It's just that you have to make more effort.

Pure efforts nothing else. Now prove this following. cos theta plus sin theta whole square and remember it also see cos theta plus sin theta whole square remember it 1 plus sin 2 theta I am not helping you just open it cos square plus sin square is 1 and 2 cos theta sin theta is what sin 2 theta okay now cos theta minus sin theta will be 1 minus Okay, so do you want me to do this A part?

A part is already done. Okay, talk about this. Talk about this.

cot theta minus tan theta. So cot theta is cos theta by sin theta. tan theta is sin theta by cos theta. So in trigonometry, none of the things strike you.

None of the things strike you. Just convert tan and cot in terms, just convert everything in terms of sin and cos. Okay? Majority of the time it works.

Now sin cos. So this is sin theta cos theta. cos square minus sin square.

Okay, now I clearly think that 2 is required in the denominator specially. So, 2 cos square theta minus sin square theta is 2 cos 2 theta. Whole upon 2 sin theta cos theta is sin 2 theta. So, clearly this is 2 cot 2 theta. Can you see step by step?

Easily you will get it. Easily you will get it. Okay? Alright, done, done, done. See, are you now understanding multiple and submultiple angles, everyone?

Okay, let's solve this question. Let's see how many of you are able to solve this question. Yes, Jainith, I'll do that. Thendral, thank you, thank you for writing good things over here.

That we also need motivation. Okay. Root 3 cosec 20 degree minus sec 20 degree.

Are you solving it? So, now I told you, whenever you get stuck, root 3 upon sine 20 minus 1 upon cos 20. Ok, now take LCM. I told you when you get stuck, convert everything in terms of sine and cos. Now, sine 20 cos 20 will become LCM.

Root 3 cos 20 minus sine 20. Now, this is slightly different question actually. Slightly different question. Some different kind of manipulation will come into the picture.

You will see me doing something for the very first time over here. Let me tell you what is this. Okay, let me tell you what is this. First of all, tell me that this is root 3 cos 20 minus sin 20. So, what I am going to do is, I am going to multiply 2 and divide by 2. Can I do that here?

Multiply 2 and divide by 2. Why? Because root 3 by 2 and 1 by 2 are two values which I know. Root 3 by 2 is what?

Root 3 by 2 is ma'am cos 30. And sin 60. 1 by 2 is sin 30 and cos 60. So, these two values are very well known to us. Okay? Alright.

Now, one more thing. I am going to do one more thing. Now, sin 20 cos 20. So, 1, 2 and 1, 2 over here. Because 2 sin 20 cos 20 is also a formula. Now, do it.

Apply it. So, two 2's are first of all 4. 2 sin 20 cos 20 is sin 40. And now, here. inside the bracket, let me tell you. Root 3 by 2 cos 20. Okay? So root 3 by 2 can be written down as sin 60 and this is cos 60. Yes?

So sin 60 cos 20. Sin 60 cos 20 minus cos 60 sin 20. Cos 60 sin 20. Yes? Okay, getting it? So now this is 4 upon sine 40 degree. Sine A cos B minus cos A sine B.

Sine A cos B minus cos A sine B. Sine A cos B minus cos A sine B is sine A minus B. So now can you see sine 40 and sine 40 is gone.

So 4 is absolutely right. option C. CCC. CC.

Okay, all right everyone. Yes. Now some more formulas we are going to talk about.

Okay. Triple angle formulas. Now is the time we do triple angle formulas. Three more formulas. That's it.

I'm not going to stretch it. Three more formulas. I think I am getting lagged chat.

Is it? Or you know, you guys are not active on chat now. Chalo, let's do triple angle formula. Tell me, sin 3 theta, cos 3 theta and tan 3 theta. Three formulas we are going to learn.

Sin 3 theta. Sin 3 theta is 3 sin theta minus 4 sin cube theta. Cos 3 theta is 4 cos... cube theta minus 3 cos theta and you know what tan theta welcome to the live session cosmic a welcome to the brand new session okay for you tan 3 theta tell me tell me you are not going to tell me i know i have to tell you but tan 3 theta listen to me very carefully do not get panic because slightly difficult formula but ninja mam is there to tell you the trick also Okay, tan 3 theta is 1 minus 3 tan square theta.

Sorry, sorry, sorry. Tan 3 theta is 3 tan theta minus tan cube theta. Whole upon 1 minus 3 tan square theta. Now, you will be like, ma'am, how to remember? Hold on.

So, trick to remember is, okay, first of all, First of all, number one, both have negative sign in numerator and negative in denominator. Negative in numerator, negative in denominator. Second, diagonally there is 3. Diagonally 3 is there. Diagonally 3, 3. Can you see? 3 and 3. Okay?

Yes, Tarun Raja. Welcome to the session. Welcome.

Are you enjoying it? Were you there since the very beginning? Rama Raju is to late today.

Okay. Alright. Diagonally, what we are going to do is, can you see 3, 3 and 1, 1. Third thing is, can you see one more important point to notice. Numerator has odd power. 1, 3. Denominator has even power.

0, 2. So, it is 0, 1, 2, 3. So, Ninja ma'am, do you remember Ninja ma'am? N. So, this is 0, 1 and then 2, then 3. 0, 1, 2, 3. 0, 1, 2, 3. That means denominator will have 0 and 2. So, 1 is what? Degree 0. And square theta.

Can you see tan square theta? And 1, 3. So, tan power 1 and tan cube. This trick toh I have not given it to your seniors.

2025 aspirants will be like, ma'am you toh didn't give me that trick in the last year. This is a new trick for specially for 2026 students, okay? So yes, how will you remember tan 3 theta is?

Now see, both numerator, denominator, negative sign, 3, 3 diagonally is there and you know what? You know what? This is 1, this is also 1. So, 3, 1. And now, let's talk about degree.

Degree, how will it start? So, how do you write n? Ninja ma'am, right? n. So, like this.

Now, 0, 1, 2, 3. So, denominator is having 0 and 2. Numerator is 1 and 3. Right? Easy. Okay, yes Rohan what happened? What happened Rohan?

Now quickly tell me the answer of this question Okay, cos 3x. Ma'am, you told us cos 3x is 4 cos cube x minus 3 cos x is the formula for cos 3x. Plus 3 cos x is kept as it is. 1 plus cos 2x. Gold formula.

Gold formula. Do you remember 1 plus cos 2 theta is? 2 cos square theta. So, this is 2 cos square x. Now, cancel.

3 cos x, 3 cos x. Gone. So, now 4 by 2 will be 2. Cos cube by cos square is cos x.

Can you see option C is absolutely right? Easy or not? Easy or difficult?

Okay. Now, What are we going to do is we are going to talk about one special angle. One special angle.

Okay? One special angle. Now you have to tell me actually.

I am not going to tell you. We have to calculate the value of sin 18 degree. How do you, what do you think? Like how can we calculate 18 degree?

And you know what? See. Whenever we talk about special angles like 15 degree, 22 and a half degree.

Or any sort of these kind of things, right? We always relate. See, 15 degree tell me how did we calculate? Ma'am, 15 degree is 45 minus 30. So, we always take help of standard angles.

To calculate special angle value, what do we do? We think about, we relate that 18 degree or 15 degree or 22.5 degree to special angle, to standard angle. Yes or no? Yes, so 15 degree was 45 minus 30. Both 45, 30 were standard angles. Then we did that pi by 8, that 22 and a half degree.

So, 22 and a half degree is half of 45. So, we thought about 45 is the immediate relatability in terms of 22 and a half. Now, think about 18 degree. 18 degree, think about it.

18, what can we do with 18? So, that we will get a Standard angle. What can we do to 18?

See 18 degree is theta. Now see 5 Rama Raju is saying but Rama you are not telling me the right thing. Okay.

Ma'am will the degree come in decimal? Yes Tarun Raja we have just done this. 22.5 degree in this session.

Just go back you will see. Go back you will see. Okay, so 18 degree.

Now, multiply 5. 5 into 18 is 90 degree is equals to 5 theta. Multiply by 5. Multiplication with 5 will give you what? Multiplication with 5 will give you the standard angle. Okay, alright.

So, this is sine 18 degree. So, sine 18 degree. So, sine See, now start with sin 90 degree is equals to sin 5 theta.

Okay? So, 1 is equal to sin 5 theta. Yes or no? Ma'am, we know 60 value.

Yes, but 60 and 18. Okay, you are talking about 18 into 3. But that is going to be complicated. Okay? That is going to be complicated.

See, now understand. 90 degree, see do like this. 5 theta is equals to pi by 2. 5 theta is equals to pi by 2. Can we do this?

5 theta being pi by 2? Yes ma'am. So now 5 theta is 2 theta plus 3 theta is equals to pi by 2. Yes?

Are you getting it? Understanding it? Okay. So now what you're going to do is 2 theta is equals to pi by 2 minus 3 theta.

2 theta can be written down as pi by 2 minus 3 theta. Take sine both the sides. So sine 2 theta is equals to sine of pi by 2 minus 3 theta. So can you see sine 2 theta is equals to what?

Sine 90 minus 3 theta is cos 3 theta. So this is what we have got. We just need to solve it.

Sin 2 theta is equals to cos 3 theta. We just have to solve it. What do we want? What do we want?

Tell me. From here, this is equal to sin theta. So we are actually, from this equation, what is the agenda?

Agenda is to calculate the value of sin theta. Sin 2 theta is? Sin 2 theta? is equals to cos 3 theta. Please do not forget eventually we want the value of sin theta.

So, this is 2 sin theta cos theta and cos 3 theta anyone. cos 3 theta. cos 3 theta is what? 3 cos theta minus 4 cos cube theta. Now can you see this cos theta is coming out?

cos theta is coming out. Yes, so it become what? 2 sin theta is equals to 3 minus 4 cos square theta.

See, now what do you want? Eventually you want sin theta. Cos square theta, can we do not write sin square theta? Think about writing everything in terms of sine theta. Ok, so 2 sine theta, 3 minus 4 is minus 1 plus 4 sine square theta, 4 sine square theta minus 2 sine theta minus 1 equals to 0. Ok.

Now quadratic equation in sine theta, you just want sine theta, that's it. Sine theta is minus b, so 2 plus minus b square, so 4 minus, so plus 4, 4 is 16 by 2 4 is 8. Okay, so now I'm going to do something, please do not panic. 2 1 is 2, 2 4 is 8. Okay, so like I am just dividing whole numerator denominator by 2. 4 and this is going to become 4. So, this is sine theta is actually becoming 1 plus minus.

Now, understand 1 plus minus. Okay. 1 plus root 5 by 4 or sine theta can be 1 minus root 5 by 4. Just tell me one thing. Sine 18 degree will be positive or negative.

Obviously, sin 18 degree will be positive because of Being in first quadrant, 1 minus root 5. See root 5 value is 2 point something. So 1 minus Tarunakirti 4. Have I done anything wrong over here? 4 cos square 4 cos cube minus 3. I have done something else.

Oops! I have done something else. 4 cos cube, right? 4 cos cube theta minus 3 sin theta. So that means I will have to erase everything.

Okay? Sorry, minus 3 cos theta. Okay, so cos theta, this is cos theta, cos theta, this is square. So, 2 sin theta is equals to 4 cos square theta.

4 cos square theta is this, 4 minus 4 sin square theta. Yes, minus 3. Okay, so this is 2 sin theta is equals to 1 minus 4 sin square theta. Call everything in the left hand side. 4 sin square theta plus 2 sin theta. And minus 1 is equals to 0. Yes.

Okay. So now sine theta will become what? Minus 2 plus minus square root of b square plus 4 4's are 16 upon 2. So sine theta will become what?

Minus 1 plus minus. See if you divide by 2. Root 20 will be what? 2 root 5. So root 5 by 2 into 4. So eventually it will be. 4. Okay?

Sin theta is minus 1 plus minus root 5 by 4. Now understand, either you will take plus sin or minus sin. Minus 1 plus root 5 is a positive thing. Minus 1 minus root 5 is a negative thing.

So, sin 18 degree will obviously positive. So, root 5 minus 1 by 4. This is the value of sin 18 degree. By the way, sin 18 degree is nothing but sin pi by 10. So, sin pi by 10 value please keep remember, please do remember it is root 5 minus 1 by 4. Okay everybody?

Yes? Getting it? Understanding it? Hmm. So, just remember this sin pi by 10 is equals to cos 2 pi by 5. Okay, sin 18 and cos 72 are equal which is root 5 minus 1 by 4. Okay?

Alright? Cool? Chalo, now let me talk about something over here.

Yes, so we are done with today's session, multiple and sub-multiple. Okay, oops, oops, I cannot drag it. I cannot drag it. Let me, yes, this is how it is.

Can you see that this is, this was the video I was talking about. So, I have made a full-fledged three, like four months plan. Can you see four months plan I have?

created for you I have uploaded this video okay so now if you can see over here I will proceed and see first of all I have told you that how to customize my given plan as per your need obviously I am giving you a planner that doesn't mean that everybody will follow the same planner you have to little bit here and there modify it as per your schedule as per your timetable Based on that weekly timetable, I want all of you to make daily timetable. I have given you 4 months plan. I have given you weekly plan as well. And then I want everyone of you to convert it into daily targets.

Okay? What is the plan? Where is the plan? Let me just move forward and you will be like, this plan I have made for you. 4 months planner by Ninja Ma'am.

Week 1, it starts with July. And it ends with, if you just go over here, so I have told you that this planner and week by week, what are the important chapters, right? Okay, so just check over here.

Okay, so can you see week 16? It is still 27th October. Physics and chemistry, yes. What you can do is next time when Deeksha ma'am will come live.

Just ask them to share similar timetable. They will share. Okay.

Yes. So make sure that you are following it. This is for you all to make sure that you guys are on track.

Okay. Because sometimes we feel we have time. We have time. But you see there are so many chapters.

And this is just for mathematics. Okay. This is just for mathematics. For 2027 a similar plan I will share.

Don't worry Sanjay Shweta. Okay. Yes, but right now this is for 2026. So before October, at least 70% syllabus, at least I am saying.

At least 70% syllabus should be done. Okay, otherwise you will lag. That is what I have told in this video.

Make sure that you watch this video and make sure that you hit the like button also. Like, like, like, make sure that you like. Okay, so yes, this was all about today's session. and ma'am max attempts what are you talking about max attempts mean means what what do you want okay so now i think i am going to meet you one more time huh 10 o'clock are you ready for that 10 functions questions 2026 aspirants and 2027 also because functions is common to both your j2026 and 2027

Transcript for:Trigonometry Formulas and Techniques

Transcript for:
Trigonometry Formulas and Techniques