 Hello, good morning. Welcome to the YouTube live session on Centrum Academy. Hope you guys can see my screen. So those who have joined in the session, I would request you all to type in your names in the chat box so that I know who all are attending the session. Good morning, everyone. So those who have joined in the session, please type in your names so that I know who all are attending the session. Good morning, Hamila. So let's give a minute or so for the others to join in. Good morning, Vidyota. Hi, Mac. Hope this screen is visible to all and I am clearly audible to everyone. All right, so it's exactly 9 by my watch. So let's start the session. So in this session, basically, you'll be having an online session and you can only ask your doubt by typing it in the live chat box that you can see. So hope you are using a laptop or a desktop for it so that is convenient for you to type on it. However, I don't think so. You'll have many questions. In fact, I will ask you some problems to solve. We just have to respond whether it is done or not done. So let us begin with the session. So we are going to start with the most heaviest chapter in class 11, which is trigonometry. I think it has already been started in your school. Now, this chapter is not only important for your class 11, but also for your class 12th point of view, because you will be dealing with trigonometric functions in and out. For example, calculus also you will find you will be exposed to a lot of trigonometric functions. Okay. And in class 11th, normally we spend around one and a half months just doing trigonometry. It is such a huge chapter for us. Now, trigonometry becomes slightly challenging because of the fact that there are so many identities, so there is a problem of plenty in trigonometry. There are so many identities that sometimes it is very confusing for us to apply the right identity to solve a question. Therefore, sometimes I say that you look at the end to decide the means. So look at the answer and then take the required route to get to that answer. So first of all, I'll briefly list down the subtopics that we are going to cover under trigonometry. Hi, Harin. How are you? So hope all of you had your breakfast, right? Ready for a three-hour stretch? Okay. So I'll give you an overview of this chapter first. Under this trigonometry chapter, we are going to study the following. First, I'm going to just do a recap of your trigonometric identities. Trigonometric identities that you have already done in your class 10th. Okay. It's just like a class 10th portion, right? Because I want to know how comfortable you are with your class 10 part of the trigonometry. Okay. So most of you have done high amount of, so most of you have done trigonometry. I think you would have used RS, take it all for it. And that was a very good book for trigonometric identities. So we'll do a quick test on this by solving few questions, right? The second subtopic that we'll cover under this is measurement of angles, measurement of angles. So we'll look into different types of systems for measuring angles and we'll focus more on the radian or the circular system of measurement of angles. The third subchapter under trigonometry would be trigonometric functions, would be trigonometric functions. So all the six trigonometric ratios that we know of that is sin theta, cos theta, tan theta, sec theta, cosec theta, cot theta, we'll be taking them as a function, function of x. Okay. So basically we put some input to those functions and we get an output from it. So we'll talk with respect to functions in this part of the subchapter. Next, we'll talk about trigonometric identities based on compound angles. Compound angle means more than one angle. So sin of A plus B, sin of A minus B, sin of A plus B plus C, sin of cos of A plus B plus C. So more than one angle will be involved in such identities and therefore the name is given to be compound angle identities. Okay. This would be followed by multiple and submultiple angle identities, multiple and submultiple angle identities. So as you can see right from the first chapter, we are talking about identities, identities, identities. That's why I told this chapter is full of identities. And sometimes it becomes very difficult to know which identity will help us to solve a given question. So this is a very, very practice intensive chapter. You need a lot of practice. It's a practice intensive chapter. Okay. Continuing chapter number six would be, hi Anga, hi Akshita. Next would be conditional identities. Sorry, before that we'll talk about transformation formulas. Transformation formula. Okay. This will be followed by conditional identities. Note that transformation formula are also a type of identities. So we'll talk about conditional identities. Okay. The list is not over yet. We have trigonometric equations and in equations following that. So just like we study equations in algebra, polynomial equation, equations involving exponential function, logarithmic function will also have equations involving trigonometric functions. Finally, you have a chapter called properties and solutions of triangles, properties and solution of triangles. Okay. Out of these, this entire thing would be important for your KVPY exams also. This is important for your KVPY examination also. Okay. Especially the last chapter because they include it as a part of geometry in their KVPY syllabus. Okay. Hope there is no doubt with respect to the topics involved in trigonometry. So as you can see, there are nine sub chapters. So quite a big topic it is. So normally we take around four to five classes to finish off these topics. Okay. So next one month, as you can see, you can, you have to only do trigonometry, trigonometry, trigonometry, nothing else. Okay. So let me begin with the first part of this subtopics which is recap of, recap of trigonometric identities done in class 10th. Okay. So I will give you some question to begin with because there is nothing to teach in this because you already know the three famous Pythagorean identities. What are they? So you need to just know your sine square x plus cos square x is equal to 1. By the way, why it is called an identity? Anybody can type? Why it is called an identity? What is the difference between an identity and an equation? Feel free to type on your chat box. What is the difference between an identity and an equation? So basically identity is true for any value of x, isn't it? So x can be molded in any fashion you want. That means if I want to say sine square of 5x plus cos square of 5x, it still remains 1. So this x can be molded in any fashion you want. Normally in many books you will see that they write three dashes in the equal to. That means these two are the same things. That's why it is called an identity. So identity is true for any value of the variable. Whereas an equation is true only for certain values of the variable which we call as the roots of the equation. Is that fine? So there's a difference between identity and equation and to maintain that difference in international books you would see that they use three dashes over here rather than two dashes because two dashes taken as equal to sine which means you're solving an equation kind of a thing. I'll give you another example. Let's say 2x plus 2 is equivalent to saying 2 times x plus 1. So this is not an equation that you are solving. This is an identity. This is always true no matter whatever is the value of x. Next Pythagorean identity is 1 plus tan square x is equal to cx square x. I'll also write triple dash over here just to show that we are dealing with identities. Now how do we get to the second identity? It's very simple. If you divide the first identity that means if you divide sine square x plus 4 square x by cos square x both sides you end up getting the second identity. So if you divide this by this you get a tan square. If you divide this by this you get a 1 and 1 by cos square is secant square as we all know and that's how we get the second identity. So the second identity has been derived from the mother identity which we know is sine square x plus cos square x is equal to 1. One more has been derived that is 1 plus cot square x is equal to cosecant square x. How do we get this? Simply you just divide sine square plus cos square both sides by sine square. So if you divide this by sine square if you divide this by sine square this by this will give you a 1, this by this will give you a cot square and 1 by sine square as we all know is cosecant square x. So all these three identities are basically Pythagorean identities because they have been derived from the very simple fact of Pythagorean theorem a square plus b square is equal to c square. Is that fine? Any questions so far? Yes, no. As you can feel free to chat on the type on the chat box. Okay so now time for some questions. Let's take the first question prove that great prove that under root of 1 plus sine x by 1 minus sine x is equal to c kicks plus tan x. A simple class 10th question I'm sure you would have seen this question. I'm team number of times in your class 10. So just a quick recap of what you have done in class 10 through these problems. There's nothing new that I'm going to talk about in these problems. I just want to see how comfortable you feel with your class 10. So I'm sure you all are sitting with your notebooks in front of you. So please quickly prove that these two terms are equivalent that means they are the same term 3 equal to means it's an identity. Please note such identities won't work when x is x should not be a multiple of x should not be 90 degrees here. Let me not write pi by 2 because officially I've not learned pi so let me write 90 degrees. Once you're done just type done on your chat box so that I know who all are done. At least three people should have done for me to start discussing it. Okay Manav is done. Amla is also done. Very simple question. I'm sure you would have done this in class 10. Okay Vidyota is also done. So let's quickly discuss it. So we start with the LHS side. So what do we do is within the brackets within the under root sign okay we'll multiply and divide with 1 plus sin x okay. So we'll multiply and divide with 1 plus sin x. So this gives us under root of 1 plus sin x the whole square and in the denominator we get 1 minus sin square x okay. We all know that 1 minus sin square x according to the Pythagorean identity is cos square x. So we'll replace that with a cos square x okay. Now it's like the squares have been subjected to the square root symbol so it will be as good as writing 1 plus sin x by cos x correct. Now individually divide 1 by cos x and sin x by cos x. So what I've done I have individually divided this and this right. So 1 by cos x will give me a cx and sin x by cos x will give me a tan x and this is nothing but your right hand side and hence proved. Is that fine? Very simple question to begin with. Let me take another one if tan x plus cx is 3 by 2 okay. If tan x plus cx is 3 by 2 find the values of number 1 tan x number 2 cx number 3 sin x. Once you find those values please type it on the chat box. Find the values of find the value of the following tan x cx and sin x. Once you're done just write tan x is this cx is this sin x is this on the chat box so that I know whether your answer is right or wrong and eat mubarak to all of you. Any idea anyone not getting the answer don't worry Mehak we'll be solving it by the way anybody else who thinks who has got close to getting the answer okay not a issue not a issue let's discuss let's discuss this. So guys we'll the moment we look at this the identity which comes to my mind is this identity secant square x is 1 plus tan square x right okay so this implies that secant square x minus tan square x is equal to 1 yes or no right now here we can you make use of the formula a square minus b square right so we can factorize this by using the formula a square minus b square where I am treating c kicks as a and tan x is b so when you break it up as a minus b times a plus b this is what we get right does this ring a bell I'm sure by now you would have understood the approach for it okay now we already know the value of c kicks plus tan x that is given to us in the question as 3 by 2 so I'll replace that with 3 over 2 right which clearly implies that c kicks minus tan x is going to be 2 by 3 okay let me call this equation as equation number 1 and let me call this equation as equation number 2 now a simple way to get the values of tan x and c kicks is let's add these two equations let's add 1 and 2 so if you add these two equations you can see that tan x will get cancelled off and you will get two c kicks on your left hand side on the right hand side you will get this is that fine oh no no no with the earth we have to give the value the the word of the meaning of the word value means it'll give you a numeric value for it not an expression right yeah so from here c kicks could be written as half of 3 by 2 plus 2 by 3 I think this value is going to be 9 plus 4 13 13 by 6 into 2 that's going to be 13 by 12 so this is the answer that we are looking for so this value would be 13 by 12 is that fine right now there's no surprise how we'll find the value of tan x we can subtract 1 and 2 so if you subtract 1 and 2 equations okay so c kicks will get cancelled and tan x will double up so you'll have 2 tan x is equal to 3 by 2 minus 2 by 3 which is nothing but 5 by 6 right so tan x value would be 5 over 12 so this answer is going to be 5 over 12 getting it everybody okay now what is sin x sin x can be easily obtained by using this formula tan x by c kicks isn't it why why does tan x by c kicks give us sin x like because we all know that tan x is sin x by cos x okay and if I divide by c kicks okay this and this will become one and will be left just with a sin x okay so once we have found out tan x and c kicks we just need to divide these two results so my answer is going to be 5 over 13 is that fine with everyone now looking at the solution you found it easy so one last question I'll give you on this and then we'll move on to measurement of angles prove that sin x minus cos x plus 1 by sin x plus cos x minus 1 is equal to or it is equivalent to saying c kicks plus tan x okay so prove this identity that sin x minus cos x plus 1 over sin x plus cos x minus 1 is equivalent to c kicks plus tan x I'm sure this question you would have seen in class 10 once you're done just type it on the chat box that you're done and need at least three people to have solved this before I start solving this awesome amla good how about others akshita vidyota harain manav anugha mahek awesome good oh wow you guys are too fast very good hope you're solving it just not typing done simply yeah chavi done okay great so we'll start with the left hand side so what we'll do in the left hand side we'll divide this expression by cos x throughout okay so we'll divide both the numerator and the denominator by cos of x okay so overall I'm not disturbing the expression because both the numerator and denominator have been divided by cos x okay so when you divide sin x by cos x we get a tan x okay so this becomes tan x minus 1 plus a c kicks right and here also we get tan x plus 1 minus a c kicks okay now what I'm going to do next is something which most of you would have experienced in class 10th this one that you see on the numerator this one that you see on the numerator this one I'm going to write that as secant square x or you can say minus 1 itself you can write it as tan square x minus secant square x I'm sure you would have done this a lot of times while solving class 10th questions okay so the moment we do that I can write that minus 1 as tan square x minus secant square x okay denominator I'm not going to disturb it denominator I'm not going to disturb it okay now we all know that this can be factorized this part can be factorized as tan x plus c kicks and tan x minus c kicks so let me do that so this is tan x minus c kicks times tan x plus c kicks whole divided by tan x minus c kicks plus 1. Now let me pull out tan x plus c kicks common from both the terms on the numerator so when I do that within brackets I will get 1 plus tan x minus c kicks right why this one because this you can treat it as this into 1 okay so this one term will be left off and this term would be left off so these double tick mark terms would be left off in the brackets whole divided by tan x minus c kicks plus 1 okay so these two would be cancelled off isn't it because they are the same terms yes or no leaving me back with tan x plus c kicks and that's what we wanted to prove so this is our RHS right hence proved right now I'm not claiming that this is the only way to do it there are multiple ways to do it okay I'll show you another way to do it guys that's the problem in dignometry you have so many ways to solve it that many a times we get stuck which way to follow which way not to follow my suggestion would be follow that way which is time saving for you which is time saving for you so I'll show you another method method number two second method okay the same question uh sine x minus cos x plus 1 by sine x plus cos 6 minus 1 okay so what I'll do here is that I will write the numerator term as something like this sine x minus off I'll take minus sign common from the last two terms so from minus cos 6 and plus 1 I have taken a minus common okay all of you please listen to this very very carefully and in denominator also I'll put this bracket simply just to segregate between these two terms right now what I'm going to do is I'm going to multiply I'm going to multiply sine x plus cos x minus 1 both in the numerator and in the denominator okay so as you can see on the numerator it is following the formula of a minus b and a plus b so it'll become a square minus b square that is sine square minus cos x minus 1 the whole square and in the denominator it would become sine x plus cos x minus 1 the whole square right so let me expand so in the numerator if you expand you get a sine square x you get a minus cos square x you get a plus 2 cos x and you get a minus 1 okay denominator you can apply the formula of a plus b plus c whole square as you can see this is your a b and c you cannot remove the brackets from here if you want okay we all know the formula for that which is a square plus b square plus c square plus 2 a b plus 2 b c plus 2 c a so this can be written as sine square x cos square x plus 1 2 a b would be 2 sine x cos x 2 b c would be minus 2 sine x and minus 2 cos x finally okay now we all know that sine square plus cos square is going to be 1 so I can erase this and put a 1 instead so this can be erased and we can put a 1 in place of it right and we also know from the numerator sine square minus 1 please note that please note that since sine square plus cos square is equal to 1 I can say sine square x minus 1 is negative cos square x so instead of sine square minus 1 I can put a negative cos square x and again I have a negative cos square x plus 2 cos x okay in the denominator I will have 2 plus 2 sine x cos x minus 2 sine x minus 2 cos x okay so I can club these two terms and make a single minus 2 cos square in fact I can take a minus cos x common also minus 2 cos x common it will become a cos x minus 1 term right now denominator also if you see you can write it as 2 times 1 minus sine x minus cos x plus sine x cos x okay if you see closely this term is actually 1 minus sine x into 1 minus cos x try expanding it you will realize that it comes out to be the same as what has been written in the denominator so you can say minus 2 cos x cos x minus 1 and in the denominator you have 2 1 minus sine x into 1 minus cos x okay so I can cancel 2 I can cancel a minus cos x minus 1 with this leaving me with this result cos x by 1 minus sine x okay now cos x into 1 minus sine x you can again multiply and divide with 1 plus sine x okay this will give you cos x times 1 plus sine x by cos square x you know 1 minus sine x 1 plus sine x will be 1 minus sine square which is nothing but cos square 1 cos 1 cos will go off so it will be 1 plus sine x over cos x and now you can individually divide 1 by cos x plus sine x by cos x which is nothing but c kx plus tan x which is nothing but your right hand side hence proved but here you realize this method is very very long yes or no so if I've been if I'm given an option I will choose the first method over this because time is a very big factor time is a very big factor while solving these questions so if your approach is right you will get the answer anyhow but the idea is to choose that approach where the time consumption is the least yes or no one final question before we move on find the value of find the value of sine square 5 degree sine square 10 degree sine square 15 degree all the way till you reach sine square 90 degree value means you'll get a number from it let's see who is able to solve this first so basically you're adding sine square 5 sine square 10 sine square 15 sine square 20 sine square 25 all the way till you reach sine square 90 just type in the response on the chat box once you have found it 17 plus 1 by root 2 no manav that is not also correct 9 is wrong but you're very close to the answer manav with dhuta no that is also wrong 10 is not right akshita mek chavi 9.5 absolutely correct absolutely correct that's correct manav okay all of you please pay attention over here now last term will definitely be one undoubtedly right we know that the value of sine 90 from our junior classes that sine 90 is 1 so sine square 90 will also be 1 okay now we also know simultaneously that sine square 85 is as good as sine square 90 degree minus 5 correct let me write this square as the overall square on this so that we are able to identify that this is nothing but cos of 5 degree so it's as good as cos 5 degree whole square that's nothing but cos square 5 right okay so if you see sine square 5 and sine square 85 would be as good as saying when you join these two terms the one in the front and the one in the second last position they would be as good as saying sine square 5 plus cos square 5 that's going to be 1 similarly sine square 10 and sine square 80 degree will also give you 1 similarly sine square 15 and sine square 75 will also give you 1 if you continue doing that you would realize that sine square 40 degree and sine square 50 degree will also give you 1 correct now one term would be left off in middle which is sine square 45 but i think we know the value of sine 45 which is 1 by root 2 so 1 by root 2 square will be half okay now count how many ones are there so from sine square 5 to sine square 40 there would be eight ones there would be eight ones coming up right there's a one over here also and there's a half over here also so altogether your answer would be eight that is these eights one half here and the last term one so your answer would be nine and a half right absolutely correct mano well done good shallow now we'll talk about the next part of our subtopic which is called the measurement of angles measurement of angles oh okay okay okay so with this i i'm sure you would have identified that mistake okay so let's talk about measurement of angles now what is an angle first of all what's an angle how do you define an angle let's try to understand this what's an angle so angle is basically a parameter that defines how much a ray let's say there's a ray which is originating at o right how much has this ray rotated about the initial point okay any parameter which defines how much is this rotation will be actually called an angle okay where we call this as the initial side and here we call this as the terminal side okay so something let's say if there's a rotation from initial to the terminal side in an anticlockwise fashion like this now this is an anticlockwise fashion and this rotation is 30 degree we call this as angle is 30 degree we call this angle as 30 degree okay but remember in case of angles we actually have a sign convention the sign convention is if you have an angle which has been obtained by rotating the ray in an anticlockwise direction that means your initial and terminal as such that terminal has been obtained by rotating this ray anticlockwise direction then this angle would be a positive angle so anticlockwise is a positive angle if your rotation is like this that means from initial side if you go in this direction that means in a clockwise sense like this let's say you went by 30 degree then you'll call this as a minus 30 degree so clockwise sense rotation is treated as negative in case of angles okay so going forward even angles would have signs so a positive direction of rotation will be considered as an anticlockwise direction a negative direction of angle a negative value of the angle would be considered when you are rotating the ray in a clockwise direction okay so for example let's say if I rotate this from here to let's say this position like this in this direction okay we'll call this angle as plus 270 degrees plus 270 degrees but if you rotate this from this position to this position in this direction we'll call the very same angle as minus 90 degrees understood right so naming convention says that these two are different angles however both these angles are coterminal angles now what is coterminal angles this term will come across a lot in tignometry okay coterminal angle means angles which have angles which have angles which have the same terminal position or the same terminal side okay for example 30 degree and 390 degree would be coterminal this would be coterminal okay can you give me some angles which are coterminal please type it on your chat box can I say 0 degree and 360 degree are coterminal yes or no any other example you can give me just now we discussed 90 degree and 450 degrees will be coterminal no no no 60 and 300 are not coterminal they are allied angles there's something called allied angles we'll call we'll talk about that no no no no you're not understood see coterminal means both these angles will have the same terminal position for example let's say 10 degree okay okay this is a terminal position if I take a complete rotation and come back at the same point that means 370 degree okay that 370 and 10 degree will be coterminal if I take one more revolution and come back to the same position that is 730 degrees this will also be coterminal are you getting it so 10 370 730 these are all coterminal angles are you getting this point read the definition they should have the same terminal side are you getting this point okay so if you read this word coterminal you should understand that the angles differ from each other the difference between two angles if theta and phi are coterminal this is a symbol for difference this is a symbol for difference the difference should be a multiple of 360 degrees where n is a integer okay then there would be coterminal angles so everything regarding angle is clear what is an angle when is an angle a positive angle when is an angle negative angle what is the meaning of coterminal angles okay tell me one thing how would you generate minus 765 degrees how would you generate a minus 765 degrees so if this is your initial side how do you generate minus 765 degrees so you'll say you will go clockwise rotation one complete is minus 360 one more complete is minus 720 and then 45 more so basically this would be the position of 700 minus 765 degrees minus 80 and 280 can be said that is correct yeah manu 20 and minus 340 can be said that would be coterminus yes you're correct okay now let us talk about what are the systems for measurement of angles so let's talk about systems of systems of measurement of angles just like for that there is a SI unit there's an MKS unit so there's also systems for measurement of angles so the first system that we are going to talk about is the sex adjustable system this system of measurement of angles was given to the world by the Britishers when these Britishers were ruling the world okay after the industrial revolution they became very powerful they started dominating the entire world they started making colonies then they gave this system the British system where they said that one right angle one right angle would be treated as 90 degrees okay one degree is going to be 60 minutes so please pronounce it as minutes okay and one minute is going to be 60 seconds pronounce this as seconds not the second that we have in our time clock not the minutes that we have on our time clock it's just different minutes and different seconds okay so if I say 34.256 degrees can you convert this into degrees minutes and seconds just a quick should not take you more than one minute to do it yeah type in your screen what is 34.256 degrees in terms of degrees minutes and seconds done is it too difficult so first of all this is nothing but 34 complete degrees okay and it is 2.56 degrees left off so I have to convert this in terms of minutes and seconds so what I'll do is in order to convert 0.256 degrees into minutes I'll convert multiply this with 60 okay because you know one degree is 60 minutes so 0.256 degrees would be nothing but 0.256 into 60 okay that would give you approximately if I'm not wrong so 0.256 into 60 will give you 15.36 so these many minutes would be there which is equivalent to saying 15 complete minutes and 0.36 minutes now this 0.36 I can convert it to seconds by again multiplying with 60 right so 0.36 if you multiply with 60 that will approximately give you 21.6 okay so you can round this up to 22 seconds and therefore this value will be 34 degrees 15 minutes 22 seconds simple yeah absolutely absolutely correct Manav the next system that we are going to talk about very briefly not just for your information I'll tell you this system this system is called the centi symbol system now as you all know that even French were ruling us for a considerable amount of time right French Dutch Portuguese Britishers they all ruled India right so they this is their system the French system okay which is very less known because the French were overpowered by Britishers and chased away from their colonies so in this system we say one right angle is 100 grades they had a grade system not degree system 100 grades one grade was equivalent to 100 minutes now this minute is different from the British minute okay again this is also minute but it is not the same as the British minute and one minute was again 100 seconds again this is different from the seconds of the Britishers now since everywhere 100 100 100 comes the word centi has come over here right just like centum academy 100 so that's how these naming system was done okay so we are not going to study too much about it is just for your information we are going to directly jump upon the most important of all the systems which is called the circular system which is the modern day system that we follow in circular system angles are measured in terms of radiance so angles are measured in terms of radiance okay now how do we define one radian first of all one radian is written as one with a super script of c this c stands for circular okay just like one degree we write a knot on the top for case of for case of radiance we write a c on the top so how do we define one radian if you take a circle let's say we have a circle okay and we have the radius of the circle as r and there's an arc length let's say pq whose length is also r that means this length is also same as the radius of the circle then the angle subtended at the center of the circle would be said to be one radian right so whenever we have a new unit we learn its definition isn't it for example how do you define one newton they'll say one newton is the amount of force which will generate an acceleration of one meter per second square in a body of mass one kg isn't it that's a similar way i'm defining a radian one radian is the angle subtended by an arc length this is your arc length l whose length is same as the radius of the circle so if you have a one centimeter circle and if this is one centimeter then the angle subtended would be one radian if there's a two centimeter radius circle and this r cleanse is also two centimeter then again the angle subtended would be one radian if you have a one kilometer circle radius and the the arc length is also 1 kilometer, then the angle subtended by the center will be the same as 1 radian ok. In general, in general, in general theta radian let us say theta radian is defined as the arc length divided by the radius. For example, if I say there is a circle, there is a circle whose radius is r and I take an arc length of 3r, let's say this is my arc length of 3r. Then what is the angle subtended by this arc length at the center of the circle? What is this theta? So you will say the value of this theta would be 3r by r which is nothing but 3 radians. Is that fine? So this formula is something that we need to use while solving many problems on this particular concept of measurement of angles ok. So then theta radians is obtained by dividing the arc length by the radius. Now a very simple question to all of you, if I take a circle, if I take a circle and my arc length is the entire circumference of the circle, let's say this is my entire arc length ok. So my arc length is 2 pi r ok. So what is the angle subtended by this at the center? In radians. So you will say the angle subtended by this at the center theta would be 2 pi r by r which is nothing but 2 pi radians ok. Now we also know that the angle subtended by the circumference on the center of the circle is 360 degrees. So there is now a relationship between degrees and radians. So this is equivalent to 360 degrees. Yes absolutely Shavi. So we say that 2 pi radians is 360 degrees. So from here I can conclude that pi radians will be equivalent to 180 degrees. So from here now we can know how much is 1 radian actually in terms of degrees. So all of you please tell me by using your calculator or whatever how much is 1 radian in terms of degrees, minutes and seconds. Let's see who does it first. So pi radians is 180 degrees. So how much is 1 radian in terms of degrees? So you should have an idea about how big is this radian 1 radian in terms of degrees. Exactly in degrees minutes and seconds. Ok so Manav says 57 degrees 17 minutes and 45 seconds. Anybody else? Ok so mostly all of you are getting the same. So basically it's very simple calculation. So 1 radian would be 180 degree divided by pi. Ok this comes out to be 57.2958 degrees. So it's obvious that there would be 57 complete degrees. Now 0.2958 you multiply it with 60. Ok when you do that let's see what you get. So 0.2958 multiplied with 60 will give you approximately 17.75. Ok so you will have 17 minutes over here. And then remaining 0.75 if you multiply with 60 you will get approximately 45. So this is going to be 45 seconds. So this gives you a rough estimation how big 1 radian is. 1 radian is very close to 57 degrees. So 57 degrees 17 minutes 45 seconds to be precise. Ok how much is 1 degree in terms of radians? How much is 1 degree in terms of radians? It's approximately 0.0175 radians. Ok so radians is a much bigger unit as compared to degrees. That means 3.14 is sufficient to cover 180 degrees. And therefore it finds it's use in plotting of graphs. We normally refer the angles in radians when we are plotting a graph of sin x and cos x. Because it is much convenient the scale is much manageable as compared to the bigger scales of degrees and grades. Ok that's why this system has been taken up for the modern day maths because we cannot draw 360 degrees on the x axis. It's much convenient to draw 2 pi or a pi or a pi by 2 etc. Now one very important thing that we need to know here is that we need to think in terms of radians. Think in terms of radians. We are so used to degrees, degrees, degrees that we take a lot of time to convert angles in terms of radians. Ok but going forward now say goodbye to degrees and welcome radians any time when you are mentioning angles talk in terms of radians. Ok so you are all aware of these basic angles that you did in class 10th. 0, 30, 45, 60, 90. Just try to understand how much are these angles in terms of radians. Then it becomes very easy to find out the angles associated with these angles which we know as the allied angles. Ok so before I move on to the radian there is something called allied angles. Allied angle means the angles which are associated with angles which are associated with 0 degree, 30 degree, 45 degree, 60 degree, 90 degree they are called allied angles. For example 135 degree it's associated with 45, it's an allied angle. 120 degree it is associated with 60 degree so that's an allied angle. So we will talk about this allied angle later on in this course today. Meanwhile let me give you the idea of these angles in radians. So 0 degree is 0 radians, 30 degree is 5 by 6 radians, 45 degree is 5 by 4 radians. Ok 60 degree is 5 by 3 radians. This is 5 by 2 radians. Ok so if somebody says how much is 300 degrees in radians. You don't have to break your head this is 5 into 60 correct. So 60 is already known to you as 5 by 3 so you can say it's 5 5 by 3 radians. Now guys you would see here that I have dropped the C term. I used to write C for radians right. So I have stopped writing it. So in your books you will not see those C terms written. Is that fine? So if I just say 5 by 2 what does it mean 90 degrees. I would not say 5 by 2 with a superscript of C. Ok if I say 450 degrees how much radians is this? You know it's 5 into 90 that's going to be 5 5 by 2. If I say 330 degrees you know it's 11 into 30. So 30 is already known as 5 by 6. So you will quickly say 11 5 by 6. If I say 150 degrees you know it's 5 into 30. So you will quickly say 5 5 by 6 like that. Ok so try to think in terms of radians rather than converting from radians to sorry degrees to radians by using the formula. Ok how much is 12 degree? Anybody can tell me 12 degree fast. 2 seconds. 12 degree. 12 degree is 60 divided by 5 so it's 5 by 15 radians. Are you getting it? Tell me 7 and a half degrees. Fast. Fast. Ok 15 degrees is this by 12 so this by 24 would be your answer. Ok so these things should come to you automatically. You should not sit with a calculator to calculate degrees into radians. Is that fine? Ok now since we have talked about radians being calculated as the r-length by radius will take some problems based on this concept. But before that a small thing that I would like to add. Let's say we have this r and this is theta. Now this theta is mentioned in radians. Ok so we know that your r-length can be calculated by using this formula l is equal to r theta. So this length would be r theta. Again as I told you I would not write this symbol so by mistake I wrote it now but going forward I will not write it. Ok don't use this formula when theta is in degrees. This is only to be used when theta is in radians. This formula doesn't work so I am writing it here. Don't use theta in degrees. Ok next thing that I would like you all to understand here. If I give you the radius and if I give you the angle theta in radians. Ok remember theta in radians. What is the area of the sector? What is the area of the sector? Please find out the area of the sector in terms of r and theta and tell me the answer. Just type it out on your screen. On the chat box. Anyone? It's just unitary method nothing else. Ok very simple. If you had 2 pi radians ok the area would have been the area of the circle. So if you have theta radians then how much is the area? Unitary method. So theta into pi r square divided by 2 pi. So pi pi goes off so it becomes half r square theta. So this answer here would be half r square theta. So going forward please remember this result also. So the area of the sector when the angle subtended by the arc is theta radians is half r square theta. The length of the arc would be r theta. So these 2 results should be known to you for the rest of your life. Ok let's take problems on this. Let's understand what type of questions will come on this. I will start with a very simple question. So all of you please listen I will not write the question here. So there is a horse which is tied to a post at the origin of this at the center of the circle. Now this horse starts moving on the circle and it covers 88 meters on the circle. Ok so it starts moving from this position till it reaches this position h dash let's say. So it has covered 88 meters on the circumference of the circle thereby covering an angle of 72 degrees. Ok. Find the length of the rope which is tying the horse to the post. Find the length of the rope which is tying the horse to the post. Is the question clear? So there is a horse which has been tied to a post at the center of the circle. And this horse starts to move along the circumference and let's say it moves 88 meters on the circumference thereby tracing an angle of 72 degrees at the center of the circle. You have to find the length of the rope which is tying the horse to the center. You can use an approximation of pi as 22 by 7. Please type in your answer on the chat box once you're done. 1.2 meters. No. No. That's wrong. You have to take it in radiance Mac. I think you took it in degrees. See there is a horse which is moving along a circle. Ok. It has been tied to a rope of length r. Ok. When it moves 88 meters on the circle circumference this angle is 72 degrees. Tell me the length of the rope. No. Wrong. That's also wrong. Amla. See what is the formula? The formula says theta is equal to L by r. Right. But here the theta should be in radiance my dear. So 72 degree has to be first written in radiance. So 72 degree in radiance is 72 into pi by 180. Right. L is nothing but 88. Capital r is nothing but your small r. Ok. So basically this becomes 2 and this becomes a 5. So it's 2 by 5. Pi is nothing but 22 over 7 is 88 by r. So 44 and this will become a 2. So r will become 2 into 5 into 7. That is 70 meters. Absolutely. 70 meters. Again caution. What did I write in the previous slide? Let me check. You have not taken the... Do not use theta in radiance. I have clearly stated over here. Say you people use theta in degree... Sorry. Don't use theta in degrees. You have used theta in degrees. That's why your answer became wrong. Ok. Next question. Next question. Let's say there is an equilateral triangle. Ok. Let's say there is an equilateral triangle of sides 4 centimeter. Ok. Now with vertex, one of the vertex let's say A as the center you sketch an arc like this. Ok. Ok. With this as the radius. This arc divides the circle into two areas. One is this and other is this. So one I have shaded with horizontal lines and other I have shaded with vertical lines. Both these areas are same. Let me call them as A1 and A2. Area A1, area A2. Ok. So if area A1 and area A2 are same, find the value of r. Is the question clear? So basically there is an equilateral triangle of length 4 units or 4 centimeter. With vertex A as the center you have sketched an arc like this. Right. Arc of a circle. This arc divides the triangle into two area, area A1 and area A2 and both the areas are equal. Both the areas are equal. You have to tell me what should be the radius of this circle that you have taken to sketch this arc. Express your answer in terms of pi and if at all you are getting under root sign, leave it in under root. Don't try to calculate the exact value. Ok. 24 root 3 by pi. I am not saying right or wrong for it. Let's see the response of others. Again Amla, don't calculate in terms of decimals. Just give me the expression in terms of pi and root because you won't get a calculator in commutative exams. No. Alright. Let's discuss this. If A1 and A2 both are equal, can I say each one of them would be equal to half the area of the triangle? Correct. And I think all of you know that the area of an equilateral triangle, area of an equilateral triangle is given as root 3 by 4 A square. Right. Where A is the side? Where A is the side of the triangle? Okay. So in this case, I can say A1 area which is nothing but half R square theta. Theta here will be pi by 3 because this angle is 60 degrees. Right. 60 degree means pi by 3. This angle is pi by 3 degrees or pi by 3 radians. Right. So this is going to be half the area of the triangle. Half the area of the triangle. So root 3 by 4 A square. Correct. So from this equation, we are going to find out what is my R value. So first of all, half cancel it off. Correct. One of the fours you can cancel it off. So R square pi by 3, you can say it's 4 root 3, which means R square is going to be 12 root 3 by pi. So this means R is equal to under root of 12 root 3 by pi. Okay. So this centimeter would be your answer. Is that fine? Now let us talk about finding the trigonometric ratios for, for allied angles. How many trigonometric ratios are there? Six. Right. Sine theta, cos theta, tan theta, cosec theta, seek theta and cot theta. Okay. Now before I start talking about anything else, there are certain misconceptions which I want to clear over here. Many of us believe that trigonometry is applied only for a right angle triangle. Because when we were very young and we were learning trigonometry, we used to make a triangle which is a right angle to find out the trigonometric ratios of sine, cos, tan, etc. Right. So we used to make a right angle triangle like this. Right. And theta. Okay. And we used to say that sine of theta here would be, let's say if I want sine of theta, it would be opposite by hypotenuse. Isn't it? Now let me tell you, trigonometry goes beyond a right angle triangle. You can also apply the concept of sine theta, cos theta, etc. to even a triangle which is not right angle. In fact, the entire concept of trigonometry goes beyond triangles also. I will be learning how to find sine of 1000 degree also. Okay. So don't have this misconception that trigonometry is basically applied only to triangles. It goes much beyond triangles. As you can see, we could generate 720 degrees also. We could generate 390 degrees also. These are all beyond a triangle, angles of a triangle. Are you getting this point? Now the misconception we students have is when I ask them, if you have a triangle like this and this angle is theta, what they know is this is always opposite by hypotenuse. Okay. And when I ask them what is sine theta in this case, they say that sine theta is nothing but, let me call this as A, B, C. Sine theta is nothing but A, B by A, C. Now is this correct? Because they have learned opposite by hypotenuse. So the opposite for angle theta is A, B side, hypotenuse, they think is A, C. Is this correct? I would like to know from you. Okay, Ma'ak is saying no, it's not correct. What about others? Okay. Yes, definitely it is not correct. Right. In this case, the opposite and hypotenuse terms, in fact, this cannot be called as a hypotenuse. Hypotenuse is meant only to be used for a right angle triangle. Right? So this word itself, hypotenuse, is not going to fit over here. And even though A, B is the opposite, please note that when we are finding sine of theta, opposite actually meant for a right angle triangle. So if you have to still find sine theta in this, you have to create a reference triangle. Please note that this triangle which I am shading in yellow is what we call as a reference triangle. We refer to it to get the value of sine theta. This triangle will be called as a reference triangle. Even though the original triangle is not a right angle, we create a right angle triangle like this for our reference. Are you getting? And then we say, let's say this is M, then we say sine of theta would be AM by AC. Now, this would be correct, my dear. And not this. Is that fine? So even though the triangle was not right angle, we could find out sine of theta by creating a right angle triangle for our reference. So don't be under this misconception that it is always opposite by hypotenuse and opposite can be any opposite. Is that fine? Now, you can also create a right angle triangle by doing this. Drop a perpendicular from B onto this. Let's say I call it as BN. Okay. So sine of theta can also be written as BN by BC. You can also do this. It'll give you the same answer. This will be same as your AM by AC. It won't make a difference because now this is my reference triangle. This is my reference triangle. Are you getting this point? Now, if you have an angle like this, let's say, now I'll give you a different scenario. Let's say I give you this as your theta. And now I ask you, what is sine theta? What will you do? What will you do to get sine theta? Please tell me. Will you find sine theta in this case? Yes. All are surprised. How will you find sine theta here? Now, in this case, again, we make a reference triangle, but this time we use something called the reference angle. We call it something like, let's say, phi here. This phi is actually your reference angle. You would soon realize that sine theta and sine phi would give you the same result and that would be nothing but AM by AB. So we'll come to all these things in this present subtopic, which is called the trigonometric ratios for allied angles. Is that fine? Now, please note, when you write sine theta as opposite by hypotenuse or cos theta as base by hypotenuse or adjacent by hypotenuse, note that this can be positive or negative quantities. Now, we are dealing with directed lengths. We are not dealing with their magnitudes. They can be directed lengths also. Okay. Even your base can be positive or negative. But remember, H will always be positive. H is always positive. H is always positive. Why? Because H is obtained by the formula p square plus b square. So, under root of anything, I have already discussed with you in the bridge course that under root of anything is always a positive quantity. Okay. So, we prepared now to find these trigonometric ratios as a ratio of two sides where one of the sides or both of the sides can be negative also. Okay. So, here this can be positive or negative. This can be positive or negative. Okay. When it is positive, when it is negative, I will just confirm in some time. But right now, I am setting the agenda that whenever you are dealing with these trigonometric ratios, be prepared to get a negative answer also from these ratios. Right. In our junior classes, when we talked about 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, we never had any negative ratio. Right. Did you have any answer like minus half? Did you have any answer like minus root 3 by 2? Did you have any answer like minus 1? No. Right. But now you would be surprised that these trigonometric ratios can give you a negative answer. Right. Now, how do we get these negative answers? Let me explain you with the help of a unit circle. Most of you would have seen a unit circle while studying trigonometry in junior classes. So, basically it's a circle. Okay. This circle has a radius of 1. So, this is a unit circle. Unit circle. Unit circle means having a radius of 1. Okay. Radius of 1 unit. Okay. Now, let's say I make an angle of 30 degrees. Okay. And I ask you what is sign 30 degree. Right. Now, normally if somebody doesn't know the value of it, what will he do? He will make a construction. He will make a reference triangle like this. He'll drop a perpendicular and he will make a reference triangle OPQ. Okay. So, please note triangle OPQ is called a reference triangle. Correct. In reference triangle, he would try to measure the length of PQ and just do PQ by 1. That would be his answer. Right. Sir, how is sine pi equal to sine phi is in sine pi. Okay. I'll come to that, Manav. I knew the confusion is how is how is sine theta and sine 180 minus theta the same. Right. This is the confusion right now. Right. Wait for some time. I will clear up this confusion. Okay. Just wait for some time. Find Manav. Okay. Now, coming to this question. So, you'll find PQ. Okay. And you'll divide it by 1. Isn't it? This is your 1. Okay. So, if you measure your PQ by your ruler, you would realize that PQ length is coming out to be 0.5. Okay. So, this is how you get your answer as half by 1, which is 0.5. So, therefore, you know the value of sine 30 degree is 1 by 2. Right. Now, note one thing here that when you say half, it is not half. It is plus half. Read this as plus half. Because your length is measured in this direction, please note everything measured from origin upwards is positive. Anything measured from origin downwards is negative. Anything measured from origin to the right is positive. Anything measured from the origin to the left is negative. Just like we have it in the Cartesian coordinates. Is that fine? So, when you're measuring PQ, it's a positive length. Okay. Hypotenuse is always positive. So, it's positive half by positive 1. Right. That's how you got the answer as plus half. Okay. But let us now see a separate case where you have a unit circle. Okay. I'm quickly drawing a unit circle. And now I take an angle like 210 degrees. So, hold this here. This is not 210 degrees. Is that fine? Okay. Now, if you make a reference triangle here, it would be a triangle like this. O, let's say P dash, Q dash. Correct. Reference triangle is always made with the X axis as the base. Is that fine? Now, try to listen to this carefully, especially Marav. Now, this triangle that you see, O, P dash, Q dash. Will you agree that this is congruent to the triangle O, P, Q that we had drawn in the earlier case? So, this triangle and this triangle, are they congruent? Yes or no? Very simple reason because this angle here would be 30 degrees. This is also one. So, these two triangles, they are congruent to each other. Yes or no? Everyone, I would like you to say yes, if you agree that O, P, Q and O, P dash, Q dash are congruent. Yes, correct. So, can I say P dash, Q dash will have the same length half. Length is half. But remember, you are measuring this length towards the negative Y direction. So, I would call it as negative half. Right? Because you are measuring it downwards. Something which is measured in the, from the origin down is a negative quantity. From the origin up is a positive quantity. From origin right is a positive quantity. Origin left is a negative quantity. Just like you have positive X, negative X, positive Y, negative Y. Is that fine? So, if I ask you what is sign of 210 degrees, your answer would be opposite which is minus half by hypotenuse which is always plus 1. Remember, I told you hypotenuse is always positive. I'll write it down over here. Hypotenuse is always positive. So, your answer will now become a minus half. Is that fine? So, see how your answer can now be negative also. Magnitude-wise, nothing has changed. Magnitude-wise, it is still one, one by two. But now it has an attached sign with it. That is because now your reference triangle sites are taken as negative also. If I ask you what is cost 210 degree, what will you say? Okay, by the way, OQ length would be, all of you would agree, OQ length would be root 3 by 2. So, OQ dash will also be root 3 by 2. But with a negative sign, my dear, remember that, because it is measured in the negative direction of X. So, you will say this is minus root 3 by 2 divided by 1. So, that's minus root 3 by 2. Are you getting it? If I ask you what is tan 210 degree, what will you say? You will say P by B. Now, this P is minus half and base is minus root 3 by 2. So, your answer will become plus 1 by root 3. Now, see, tan has not changed its sign. Are you getting it? So, here, just one small thing I would like to say is that when you are referring to your reference triangle sites, refer it along with the signs of it. Don't refer just to its magnitude. When we were young, when we were small, when we were in class 9th and 10th, we always had P and B to be positives because we will, we restricted ourselves only to 0 to 90 degrees where both P and both B and H, of course, are all positive. So, we never felt a need of a negative trigonometric ratio answer, isn't it? But now, the time has come that we appreciate the negative values of these trigonometric ratios. Now, coming to your question, Manu. So, Manu had a question about how is sine theta and sine 180 minus theta the same? Okay. It's a very simple concept, Manu. So, again, let me make a unit circle. Okay. And since I was talking about 30 degrees, now let me talk about 150 degrees. Okay. So, if I have to create a reference triangle, so if I have to create a reference triangle, I would create it like this. Okay. And this would become my reference angle theta. Okay. And this theta value is this. So, basically, now remember, Manu, this was your theta that you have taken. And this is your phi that you have taken. And you know that phi is 180 degree minus theta. Correct. Okay. Now, if I ask you, let's say OP double dash Q double dash, would you all agree if I say OP double dash Q double dash triangle will be congruent to the first triangle that we had made? This triangle OPQ, isn't it? So, will you all agree that this would be congruent to triangle OPQ? Yes. Correct. Now, so if this is one, I can say this is half. What will be the denominator? It will be minus root 3 by 2. Correct. But I don't need the denominator right now, because I am finding sine of 150 degree. So, sine of 150 degree, I can say it's opposite by hypotenuse, which is half. And half is same as sine of 30 degree, which we had learned earlier. Correct. So, isn't sine theta equal to sine 180 degree minus theta? And hence, proof man of clear, is this understood? I know the triangle has got completely changed. That is why we call that as a reference triangle, because the triangle is helping me and I am taking the reference of the triangle to solve my trigonometric ratios. Understood. Now, many people ask me, sir, if I am asked, let's say something like tan 500 and let's say 10 degrees. Okay. Will I sit and make this on a unit circle and try to figure out the values? Will I do it like this? So, I'll have to make a unit circle. I have to make 510 degrees. By the way, 510 degrees for that, you have to take one complete revolution, the one complete revolution and then you have to go 150 degrees more. So, basically, you are here. This is 510 degrees. Okay. So, if I make a reference triangle, this would be my reference triangle. The reference triangle would be OPQ. So, we all know that this angle would now be 30 degrees. Correct. So, if this is a unit circle, this is going to be half. This is going to be root 3 by 2. Yes or no? Okay. Minus root 3 by 2. Remember, don't forget the sign. Since you are calculating in the negative direction, this is your negative direction, OQ would be negative root 3 by 2. So, I'll write over here. PQ is positive half. OQ is negative root 3 by 2 and OP is positive one. Remember, hypotenuse is always positive. Okay. So, here if I have to answer this, I would say opposite by base. So, my answer will be minus 1 by root 3. So, do I have to always solve it like this? No. I will tell you an algorithm for that. I will tell you a shortcut for this, which I will discuss after a small break of 7 minutes. Let's meet at 10.55 a.m. You can see the time here on my MacBook. It is 10.47. We'll meet at 10.55. So, just have your water, snack, whatever you want. We'll meet again after 7-8 minutes. All right. Welcome back after the break. Hope you guys are back after the break. So, now we are going to begin a very, very important concept. So, now I'm going to tell you how to find out how to find out trigonometric ratios for any angle theta. Any angle theta means theta should be an allied angle. As I already told you, allied angle is what? Angle which are related to angles which are related to zero degree, 30 degrees, 45 degrees, 60 degrees or 90 degrees. Okay. Because these are the only angles that we had learned in our junior classes. Isn't it? So, if you have been given a problem of finding a trigonometric ratio, I have purposely written it STR because it can be any one of the six values. It can be any one of the sine, cos, tan, c, cos, cot. So, how do we find trigonometric ratios for any allied angles? Okay. Now, we have, if you're finding the value of trigonometric ratio for any angle theta, we should have two things with us. One is the sine and second is the value or magnitude, you can say. Okay. I'm writing a value for it. For example, little while ago, we did sine 120 degree, right? Sine, sorry, sine 210 degree which was minus half. So, this is your sine and half is your value or you can just write it as magnitude rather than value. Magnitude is more appropriate word here. A magnitude. So, how do you find these two? So, first, let us talk about sine. How to find the sine without actually making the diagrams or without actually constructing a reference triangle. Okay. All of you, please listen to this very, very carefully. In order to find the sine, whether it's positive or negative, we follow a simple limonics or schematic diagram which is like this. A, S, T, C. A stands for all trigonometric ratios. So, any angle which falls in 0 to 90, that is the first quadrant or let's say 360 to 450 degrees or 720 to 810 degrees. So, A means all trigonometric ratios. So, any trigonometric ratio which is calculated for an angle for lying between 0 to 90 or 360 to 450 or 720 to 810 etc., they will all be positive. Okay. So, this is called the first quadrant. By the way, any angle which lies on the edge, they're called quadrantal angles. For example, 90 degree exactly is a quadrantal angle. It doesn't lie in the first quadrant. It is in the border of the first and the second quadrant. Those angles are called quadrantal angles. Okay. So, all angles which lie on the borders, I'll write it down over here for your reference, angles lying on the border, on the border of two quadrants is called a quadrantal angle. Okay. I'll talk about quadrantal angle little later on. Okay. So, for quadrantal angle, it will be difficult to decide the sign. Okay. So, for that, I will talk about later in this particular course and deal with it later on. As of now, we are going to talk only about those angles which fall completely in one of the quadrants. So, let me ask you a few questions. In which quadrant 120 degree lies? All of you please answer to this. 120 degree lies in which quadrant? Just type it in your chat box. This is first quadrant. This is second quadrant. This is third quadrant. And this is fourth quadrant. Very good. Second quadrant. Okay. Tell me 330 degree lies in which quadrant? 330 lies in which quadrant? Fourth quadrant. Absolutely. Tell me minus 510 degree lies in which quadrant? Minus 510 degree lies in which quadrant? Third quadrant. Absolutely. See, 510 degree means you're going, minus 510 degree means you're going clockwise. So, this is 360. This is minus 450 and you reach here. So, this is your minus 510 degrees. So, that lies in the third quadrant. Getting this point? Okay. So, this is in the third quadrant. Tell me 810 degrees lies in which quadrant? 810 degree lies in which quadrant? It's a quadrantal angle. Okay. It's a quadrantal angle. It doesn't lie on any quadrant. Is that fine? So, this is very important. You should make no mistake about identifying whether it's a, whether it's lying in the first quadrant or second quadrant or third quadrant or fourth quadrant or whether it's a quadrantal angle. Okay. So, now this science scheme says all, A for all, all trigonometric ratios for angles lying in the first quadrant, for angles lying in first quadrant is positive. Is that fine? So, all trigonometric ratios for angles lying in the first quadrant would be positive. S means only sine and cosec of angles lying in second quadrant is positive. T stands for only tan and cot of angles lying in third quadrant is positive. And c stands for only cos and sec of angles lying in fourth quadrant is positive. Is that fine? So, these are just simple abbreviated mnemonics you can say which will help you to recall it. A, S, T, C. You can make a word out of it like add sugar to coffee. Okay. Or after school to centum. Okay. Or all silver tea cups. Right. There are so many ways to remember the science scheme. Okay. If you have understood this, I have a small question for you. Just tell me the sign. What is the sign of, what is the sign of tan 840 degrees? Is it positive or is it negative? Positive or negative? Sine. Sine. This you have to tell. Will my answer be a positive answer or will my answer be a negative answer? Just tell the sign. Don't worry about the magnitude. We'll talk about it little later. Negative. Absolutely. Because this will lie in the second quadrant. Right. 840 will lie in the second quadrant. And in second quadrant, we know that only sine and cosec can be positive. Rest all trigonometric ratios would be negative. See what I have written. S means only sine and only sine and cosec are positive. Rest all are negative. Are you getting it? So it would have a negative sign. Very good. Okay. Tell me what about sine of cosec minus 570 degrees? There is a confusion. Vidyutha says positive. Haran says negative. Guys, there has to be a consistency. See, apply your brain. See, if you have to reach 570 degrees, minus 570 degrees, I have to start rotating clockwise. So this is minus 360. This is minus 540. And minus 30 means more. This is your angle that you are looking at. This angle is your minus 570. So you are in the second quadrant. In second quadrant, cosec is positive. So this will be a positive answer. Is that fine? So positive is the right answer. Well, good. So there's no doubt regarding identification of whether the answer would be positive or negative. Now let's talk about how to calculate the magnitude. So let's say I repeat the question here. So we have sine and we have a magnitude. So sine part we have already done. So for that, we'll follow a small schematic diagram ASTC. Okay. Now for magnitude, there are two methods. First method and second method. Both methods have their own importance. Okay. So try to learn both the methods. Method number one, mostly this method many people like to follow. But again, as I told you, the second method has its own importance. Okay. So all of you please listen to these methods very, very carefully. Very, very carefully. If you want to find trigonometric ratio for an angle theta, the first thing that we do while finding the magnitude is we express theta as any integer times 180 degree plus minus a alpha value. Okay. Where n is some integer i and alpha is a value which is, alpha is a value which should belong to 0 to 90 degree interval. Okay. Is this understood? First step, first express theta as n into 180 degree plus minus alpha n is an integer and alpha should be between 0 to 90 degree. For example, if I give you something like this, 210 degrees. How will you write it like this? You will write 1 into 180 degree plus 30 degree. So this becomes your alpha. Okay. Is that fine? If I give you something like let's say 300 and let's say 300 degrees. Okay. How will you write this? Will you write this as 1 into 180 degree plus 120? Will you write it like this? No. You will not write it like this because now your alpha is not in 0 to 90 interval. It has exceeded that. So this is not a wrong way. This is not a right way to write it. So what we'll do is we'll make this as 2. If you make this as 2, then this will become a minus 60 degree. Isn't it? So this will now become your alpha. Please note that don't take this sign along with it. Sign is to be ignored. This sign here is to be ignored. Just take the magnitude of alpha which you get. So here you'll take 30 degree. Here you will take 60 degree. Okay. Don't take minus 60 degree. I'm again repeating you. Let me give you few more examples. Let's say I ask you. I ask you 150 degree. How will you write it? Will you write it as 0 into 180 degree plus 150? No. I will not write it because 150 exceeds 0 to 90. So this is not a right way to write it. So what I'll do is I will write it as 1 into 180, but then I have to subtract 30 degrees, isn't it? So this 30 will become your alpha. So alpha will be 30 here. Don't take this minus. This has to be ignored. Ignore it. Understood? Yes, correct. Very important guys. Many people make a mistake because they take alpha along with this sign over here. I'm again repeating sign has to be ignored. Is that clear? Now coming to the next step. So once you have done this and you have figured out what is your alpha, your magnitude would be, your magnitude would be the same technometric ratio whatever you have in the question for that alpha. Understood? Take an example. Let's say tan 150 degree. I want to find out. Okay. So first thing I would do is I would figure out the sign. So I will see 150 degree lies in which quadrant. 150 degree lies in the second quadrant. Correct? And in second quadrant, tan is negative. Tan is negative in the second quadrant. Correct? So I would write a negative sign here. Okay? Next step what I will do is I will do 150 degree as 1 into 180 degree minus of 30 degree. So this 30 degree becomes my alpha. So magnitude will become tan 30 degree. That's nothing but 1 by root 3. I hope you all know the basic 0 degree, 30 degree, 45 degree, 60 degree, 90 degree result. So this is becoming, this will become your answer. Understood? Please write CLR if it is clear to you. Please write CLR on the chat box if it is clear to you. Great. Great. So almost everybody has done. Okay. Let's take few examples. Then I'll come to second method. Okay. Let's take few examples. Find the value of let's say cos 240 degrees. Would you like to try it? Or should I start explaining it? It would be great if you try it and tell me the result. Just type it in your screen. What do you think is the answer? Be careful while deciding the sign and the magnitude. Okay Vidyuta, let's see what others say. Okay. Okay. Half and minus sign has come later on. Okay. Fine. Minus half, minus half. Excellent guys. Almost everybody has answered this correctly. So 240 lies in third quadrant. Right? In third quadrant, we know that cos is negative. So we write a negative sign. Right? Next step, 240 can be written as 1 into 180 plus 60. Correct? So I have to take the magnitude as cos 60 and cos 60 is half. So minus is going to be minus half. Absolutely correct. Okay. Okay. Tell me tan of 37 pi by 4. Now, all of a sudden I have changed my degree to radiance. So don't be scared. Instead of 180, now you deal with pi. Okay. So instead of 180 over here, you can also read this as theta can be expressed as n into pi plus minus alpha. So don't get scared of radiance. Just start treating pi as 180 degrees or 180 degree as pi radiance. Okay. Mehak says 1. Okay. Oh, Vidyuta says 0. Okay. Manav and Mehak, they agree with each other. Okay. Plus 1. Others, Chhavi, Aniga, Adweta, Akshita, Harain, please participate. Okay. Mostly people are going with 1. Let's check. Let's check. So for this, we have to first establish in which quadrant does, if we draw a line here, this is a separate question. Yeah. In which quadrant 37 pi by 4 lies. Now, this can be written as 9 pi plus pi by 4. Right? Now, just a small trick for all of you. In order to find the quadrant, you don't have to keep rotating. Many people, they have a habit. Okay. 9 pi. So they'll rotate 1, 2, 3, 4, 5, 6, 7, 8, 9 and then oh, I'll come here. No need to do these rotations. It'll waste your time. This simple trick, which I'm going to tell you, if you have an even into pi, you are at this position. Your arrow will be at this position. Okay. If it is an odd into pi, I'm sorry, if you have an odd number into pi, you are in this position. For example, 9 is odd number into pi, right? So you are in this position currently. Plus pi by 4 means you have to just come down like this, pi by 4. Okay. So you don't have to keep rotating in order to know the axis. That looks so stupid and it takes a lot of time, isn't it? So you're directly in your third quadrant. In third quadrant, tan is known to be positive. So write a plus sign. Correct. And what about the magnitude? Magnitude will be the same as the tan of this angle. This is your alpha now, my dear. This is your alpha. So alpha is pi by 4. So tan pi by 4 is 1. So your answer is plus 1. So congratulations to all of those who said plus 1. Okay. Let's take few more examples. We have to be very confident in this. Then only I'll move forward. Tell me, let me give you a googly question. Cos 540 degrees. Cos 540 degrees and also one more question, tan 540 degrees. So tell me for this. Cos 540 first. Then we'll come to this. Why is this a difference of answer? People are saying minus 1, 1. I should get consistent answer from everyone. Okay. Let's see. Now the problem with 540 is it's actually a quadrantal angle. So basically you are here. This is a no quadrant, right? Yes. 540 is 3 into 180. Since 3 is an odd number, you are in this position. So it is a no man's land. It is neither in the second quadrant nor in the third quadrant, right? But the saving factor over here is that you are talking about cos. Cos is negative here also and negative here also. So anyways you have to put a negative sign. Are you getting this point? If you talk about cos, cos is negative both in the second and the third quadrant. So negative sign will anyhow appear. Yes or no? Advaita? Are you getting it? So you can't get plus 1 as your answer. Getting that? Advaita? Okay. Now what about the magnitude? So 540 you can write it as 3 into 180 plus 0. So this answer here will be cos 0. Cos 0 we all know is 1. So your answer is minus 1. So congratulations to those who have given the answer is minus 1. It cannot be both minus 1 and 0. It has to be only one of them. Akshita. It only has to have one value. Okay. Getting a point. So even for quadrantal angle this method works. Okay. Now try out the B1. So A is done. Let's try out the B1. Okay. So Akshita says 0. What about others? Manu also backs it up. Again the same situation with us. It's a quadrantal angle. Now the problem is if it is considered in the second quadrant it will be negative. But if it is considered in the third quadrant your tan is positive. So with respect to sign I have a confusion whether I will put a plus or whether I will put a minus. Right. But hold on. There is something that will come as a savior for you. We know that 540 is 3 into 180 plus 0. Right. So your magnitude would be tan 0. Tan 0 is anyway 0. So it doesn't matter whether you write plus 0 or minus 0. Ultimately your answer has to be 0. Right. So those who have given 0 as the answer congratulations. That's the right answer. Very good. Is that fine? Okay. Let me give you another one. Let's say cot of 119 pi by 6. Oh okay. Okay. Sorry. I interpreted it as if you are giving two answers for the same question. Okay. Fine then. Yeah. Anyone done with cot 119 pi by 6? cot 540 is undefined. Manu cot 540 is undefined. Okay. Max says root 3. What about others? Root 3. Root 3. Everybody is going for root 3. Okay. Let's check. So 119 pi by 6. I could write this as 20 pi minus pi by 6. Isn't it? Correct. So 20 pi means you are in this position. This is a position of 20 pi. Okay. Minus pi by 6 means you have dropped in the fourth quadrant my dear. Are you getting this? So you have dropped in the fourth quadrant when you are talking about 20 pi minus pi by 6. Are you getting it? So people who are claiming plus root 3 please note it would be minus sign over here. Yes. Because fourth quadrant cot is known to be negative. Yes or no? cot is known to be negative. Now value wise it will be the same as the value of cot pi by 6 which is known to be root 3. So the answer would be negative root 3. Almost everybody got it wrong. Guys be very careful in deciding the quadrant. It will make a lot of difference to your answer. Sign itself is opposite means your answer is wrong. Okay. Now let's come back to the second method. So what is this method? Let's talk about. So first method is clear to all of you. Let's talk about second method. In second method what do we do is we express theta as a multiple of 90 degree plus minus alpha. Okay. Where alpha is somewhere between 0 to 90 open. Okay. You can include 0 here but 90 cannot be included. It will come on the second interval. So once you have done this then follow the following algorithm. Then if n is even okay. Then the magnitude would be then the magnitude would be the same trigonometric ratio that we have in the question for that angle alpha. And if n is odd the magnitude would be the complementary of the trigonometric ratio for that angle alpha. This is very important. What is the complementary? Let me explain you with example. We all know from our junior classes that complementary of sine is cos. Right. Complementary of sine is cos. Isn't it? Complementary of cos is sine. Complementary of tan is cot. Complementary of cot is tan. Complementary of sec is cosec. Complementary of cosec is sec. Okay. Let me write is is over here not equal to. The name itself says complementary because see how many of you know the full form of cos. Cos means complementary of sine. So this co and s makes cos. Get in the point. What are the meaning of cot? cot means complementary of tan. Okay. So co and t becomes cot. Cos means complementary of secant. Are you getting it? So let's take a question on this. Let's say I want to find sine of let's say 240 degrees. Okay. Let me use the first method first to get the answer. So let me use the first method. So according to first method 240 lies in the third quadrant. Correct. So my sine would be negative. Correct. And 240 can be written as 1 into 180 plus 60. So this is my alpha. So my magnitude will be sine 60 degrees. So that is negative root 3 by 2. Right. Let's use second method now. You'll see we'll get the same answer. So first sign will remain negative. There's no change in the sign because sign is only determined by the quadrant in which the angle falls only in case of magnitude will follow the second method. Sine is same. The sign, the concept of finding the sign remains the same ASTC. Okay. Only for magnitude we are following the second method. Okay. Now listen to this. 240 degrees I can write it as can I write it as 3 into 90 degrees minus 30 degrees. Correct. All of you listen to this very carefully. 3 is an odd number. Correct. So since I'm finding sine the complementary of sine will come over here which is cos. Correct. And for the same angle alpha over here alpha again ignore the sine. This has to be ignored. So you'll get cos 30 whose answer is again root 3 by 2. So do you see both the answers are same. Both the answers are same. So it doesn't matter whether we are using method number one or method number two our answer will come out to be the same. So please write CLR if it is clear to you. If this is clear let us try to do questions with second method. Now please only use second method. Okay. Only use second method. I want you to be very very confident with both the methods. I know you would like the first method over the second because you know that's easier. But second method has its own importance. I'll tell you where it is important. Okay. Tell me tan 840 degrees. That's number one. Tell me seek 11 pi by 6. Tell me sine 37 pi by 4. Okay. So mostly people are answering minus root 3 for the first one. Let's discuss this. So 840 would lie in the second quadrant. So 840 is like 5 into 180 degree minus 60. So 5 180 degree you are here. Minus 60 means you are gone back. So this is your 840 degrees. Correct. So you are in the second quadrant. Okay. So in second quadrant your answer would be negative. Correct. Now I would just use the second method to get the value. So what I will do is I will write 840 degrees as 9 into 90 plus 30 degree. Now since 9 is a odd number, tan will convert to cot. Cot of what angle? Cot of 30 degrees. So cot 30 degree we all know is root 3. So answer is negative root 3. So congratulations to those who got it right. Okay. So seek 11 pi by 6. Some of you have already answered this. Again 11 pi by 6 means it's 2 pi minus pi by 6. So 2 pi you are here minus pi by 6 means you dropped to the fourth quadrant. So this is where your 11 pi by 6 will lie. So seek there is already positive. Okay. Now so I can write this as I can write 11 pi by 6 as 4 into pi by 2 minus pi by 6. Now since 4 is an even number, seek will remain seek. Remember it will become complementary only when this number is odd. When it is even it will remain the same. Right? Seek of what angle? Seek of this angle pi by 6. So your answer will be 2. Sorry my bad. Your answer would be 2 by root 3. I am sure you know the value of seek of 30 degree. Seek 30 degree is 1 by cos 30 degree which is 1 by root 3 by 2. So that becomes 2 by root 3. Nobody has answered this correctly. Why you guys are saying 2 for this? No Adretha question. 1 answer is not minus 1 by root 3. Check. I already solved it over here. This is your answer. Do the last one now. Okay. So third question Amla says minus 1 by root 2. Let's check. Now 37 pi by 4 is 9 pi plus pi by 4. 9 pi plus pi by 4 means you are in the third quadrant. So in the third quadrant 37 pi by 4 will lie. Correct? So sin there is negative. Correct? If you want you can write this as 18 pi by 2 plus pi by 4. Since 18 is an even number, sin will remain sin. Sin will remain sin. Off which angle? Pi by 4. Right? So it will become negative 1 by root 2. So this one will become your answer. Absolutely correct. Is that fine? Now what about negative angles? How to deal with negative angles? So so far we have not discussed about how to find the trigonometric ratios for negative angles. Okay. For negative angles there are certain important identities which we want you to note down and remember. Sin of negative theta is as good as negative sin theta. Now how does this property come? We'll try to answer this when we are looking at the graphs of these functions. So I'll try to answer this from the graph. The graph of these trig functions which is our next subtopic under trigonometry. So all these answers that all these properties that you're going to see here, I would try to answer them from the graph. Second note down, cos of minus theta is same as cos of theta. So please note cos is not affected by the negativity of the angle. Cos is unaffected by the negativity of the angle. So cos minus 120 it would be the same as cos 120. Cos minus let's say 300 degree is as good as cos 300 degree. So this negative sin has no meaning for cos. Third property tan minus theta is negative tan theta. You can directly obtain this by taking 1 divided by 2. If you do 1 by 2 you'll automatically get the answer for this. Cos of negative theta is negative cos theta. Seek of negative theta is as good as seek theta. And finally cot of negative theta is negative cot theta. Please remember this result. They are very very useful. So by the use of these properties we can even find out trigonometric ratios for negative angles. For example let's say if I ask you what is tan of negative 137 pi by 3. What is the value for this? All of you please find this out and let me know the answer. Sin of minus 137 pi by 3. Which part? It's tan. Please watch these in 720 pixels. It will be very clear in that. Don't watch it in 144. Watch it in 720 or 360 like that. Please type in the answer once you're done. Minus 1 by root 3 okay. What about others? See you don't have to figure the quadrants and all by you know moving clockwise. Many people what they do is the moment they see a negative angle they start rotating clockwise to see which quadrant it lies. No need to do that. You can simply follow this identity here. Tan of minus theta is minus tan theta. So if I were to solve this problem what I'll do is I will first write this as negative tan 137 pi by 3 okay. Now I will focus on now let's find out tan 137 pi by 3. So 137 pi by 3 can I write it as we can write it as 46 pi minus pi by 3 right. Now 46 pi minus pi by 3 means you are at this position or 46 pi is this position minus pi by 3 means you'll fall down or clockwise by pi by 3. So you are in the fourth quadrant correct. So fourth quadrant means sin is going to be negative. So there's a negative sign okay. Only for this I'm finding out. I'm not finding the answer. I'm first finding this out and then I'll go back to the answer okay. So negative sign fine. Now value will be tan of this pi by 3 correct. So it will be negative root 3 but remember my answer here is negative of the negative root 3. So your answer will be plus root 3. Are you getting this point? So your answer will be plus root 3. So what I did is I did nothing extra. I just followed these identities right and I figured out the value of this first and then I attached an extra negative sign externally to it to get my final result. Are you getting it? So you don't have to break your head trying to rotate clockwise to see which quadrant it falls in and all those things. This negativity can be easily dealt by the use of these identities. No how is 45 pi plus pi by 3 going to be 137 pi by 3? Isn't it just 136 pi by 3? So 45 pi plus pi by 3 is not going to give you 137 pi by 3. Just check it out. Let's take a few more questions. Calculate let's say coseak of minus 79 pi by 6. Second calculate coseak of minus 841 pi by 4. Calculate cot of negative 219 pi by 6. If you are done with the first one just write 1 and the answer. Harin do you mean 2 by root 3 or 2 root 3? Also write the question number just before the answer. I think Shankin has joined in. For the first one Shankin is saying minus 2. Hi Shankin. Okay let's discuss the first one. People are giving different varieties of answer. I can see plus 2. I can see minus 2. I can see 2 by root 3. I'm slightly surprised why so many answers are coming. So the first thing that I would do is I would write this as negative coseak 79 pi by 6. By the use of the very same property which we discussed. Now focus on finding let's find coseak of 79 pi by 6. Now 79 pi by 6. Okay we can write it as 13 pi plus pi by 6. Correct. 13 pi means you are at this position. Odd number pi means you are at this position. Plus pi by 6 means you'll go anticlockwise pi by 6. Correct. So you have reached in the third quadrant now. In third quadrant your coseak is supposed to be negative. So sign will be negative. Okay. Value will be the same value. Value will be coseak of pi by 6. That's going to be negative 2. But your final answer would be negative of negative 2. So your answer will be plus 2 plus 2 plus 2. So yes the answer is going to be plus 2. So Manav Mehta is correct. Very good Manav. Who else gave the plus 2 answer? Nobody else. Okay. Second one. Second one people are saying root 2. Give you one more minute to solve the second one. First thing is remove the negativities. Seek doesn't care about the negativities. So it will be like this. Now 841 pi by 4 is as good as saying 220 pi plus pi by 4. Sorry 210 pi by 4. 110 means you are at this position. Pi by 4 means you come in the first quadrant. So the answer would be positive. Okay. And answer will be same as seek pi by 4. So the answer will be plus root 2 or root 2 whatever. So absolutely correct. Shankin is correct. Amla is correct. Manav is correct. Harain is also correct. Last one. Let's see who gets the last one correct fast. Please write the question number so that I know which answer you are referring to. Shankin is saying zero. Harain is saying negative root 3. Okay. Let's figure this out. Now first of all cot negative theta is negative cot theta. Right. So let us first find let's find first cot of 219 pi by 6. Okay. So 219 pi by 6 I can write it as 38 pi minus pi by 38 pi plus pi by 6. I'm sorry one second. 37 pi minus pi by 6. Let me write it like this. Or we can do 36 pi plus pi by 2 also. Yeah. That would be correct. So 36 pi plus pi by 2. Okay. By the way, you can just simply cancel this off as 73 pi by 2. Correct. So 73 pi by 2 is as good as 36 pi plus pi by 2. Correct. Now 36 pi means pi by 2 means you are at this position. You are dealing with a quadrental angle. Correct. Now when you're dealing with a quadrental angle, you know that cot here is positive here is negative. So let it be any one or plus or minus. But you know your answer will be cot 90 degree which is actually zero which is actually zero. So absolutely correct. Those who have answered with zero, your answer is going to be zero in this case. Okay. Well guys, I'm going to stop over here. Next class again, which will be an offline because today your school was closed. That's why we had to have we had to have an online session. I'll talk more about complementary and supplementary angle properties. And thereafter we are going to start with compound trigonometric functions graph. Okay. We are going to talk about their range domain. And then we are going to take up compound angle identities. I think this is what we can do in the next class that we'll be having offline. Okay. You can try from your module page number. I'll write down the homework here itself in case I forget. Page number 313 of your module one. Okay. Hope all of you have your module. Please do question number one to eight only. Okay. This is the homework for the next class. Thank you very much. Had a wonderful session with you. Bye bye. Have a good day.