 Hello friends. I hope you're able to hear me. Okay, so today we are going to discuss trigonometry and how we have done or structured this as is we will be doing quick revision of the introductory part and then we will be dealing with heights and distances so or the application of trigonometry that's the plan so we will quickly revise our basics of trigonometry first we are not dealing with identities we have kept a dedicated class for identity identities so hence we will be discussing identities there so in this class we are basically going to focus on so yeah so today we are going to deal with trigonometry okay and in trigonometry we are we know there are these things which are there in your syllabus first is the introduction part that is basics so basics in which we talk about the t ratios and all right so trigonometric ratios t ratios will be there will be problems related to that as well in your board exams and though they will not be you know they will be around two maximum two to three marks but mostly one marker and two marker will be there in t ratios now we also will deal with t ratios of standard standard angles standard angles that is zero degrees 30 degrees 45 and then 60 degrees and then 90 degrees yeah we will deal with this then we will also deal with the complementary angles so t ratios t ratios of complementary complementary angles right where sin theta is equal to cos 90 minus theta and all that we will see that after that we will be not dealing with identities today we will be dealing with identities in one dedicated session so after that we will go to application of trigonometry which is basically the word problems around heights and distances application of trigonometry now this is one part trigonometry this is one part where you will see a lot of previous year questions you know or let's say hi or high-valued question in terms of four markers five markers and things like that so this is our plan today and we will quickly revise and we will be solving lots of questions I've already taken some questions from previous year question papers and we'll try to solve those questions here and the expectation is you guys please take note of how and what methods have to be adopted and try to follow that and as you know in board exam steps will be very important so do not ignore them so hence all those things which we are going to do will be important today okay so let's start our session in case you have any trouble any question or you know you want to discuss any particular question you can always post it on the group box whichever way you feel like okay so let's start today's session okay so we will go by the basics first so let's say the first is an angle is considered as the figure obtained by rotating a given ray about its end point so they are defining angle the revolving ray is called the generating line of the angle the initial position O is called the is called the vertex and the and the initial side in the final position OB is called the terminal side of the angle so basically this is what they're trying to say so you take a ray this is O and this is a let's say this is a and you rotate this ray by some angle let's say this angle and this is B so this theta is angled right so there is nothing much in it so we know that theta is measured is either in degrees either in degrees and it ranges from zero to 360 degrees and or we also measure angle in radians which varies from zero to 2 pi the basic you know the standard values are between zero to 2 pi or zero to 360 degrees okay now next the measure of an angle is the amount of rotation from the initial side to the terminal side there is nothing much to it so you know these are all basic basics of angle so we'll quickly skip that and go to the next slide and now this is where it all starts so we say that in a given triangle so though trigonometry has its genesis in circle but and we have discussed that in a class also but here will be restricted to our right angle triangle and the thing is that in a right angle to triangle that means if you have a right angle triangle a vc let's say this is a b c and this angle is theta this angle is theta then we know that there are six trigonometric ratios possible starting from sine theta which is nothing but let us write this as p which is perpendicular this is this is base and this is hypotenuse okay now p is also called perpendicular is also called opposite side opposite side and b is adjacent side adjacent side now adjacent side to the angle theta and p is the opposite side to the angle theta now h is not considered adjacent side h is unique because it is the longest among the three and it is hypotenuse hypotenuse okay and you all know there are six ratios possible out of three so sine theta is equal to p upon h or it's adjacent or sorry opposite opposite by hypotenuse right and sine cos is cos theta is b upon h equals adjacent upon hypotenuse then it's tan and tan theta is p upon b which is opposite by opposite by adjacent you know this and then there are reciprocals to this so cosik theta cosik and theta is nothing but one upon sine theta one upon sine theta and this is equal to hypotenuse hypotenuse by opposite side okay and then there is secant theta which is nothing but one upon cos so you have to remember this is equal to hypotenuse by adjacent adjacent and then there is a cotangent theta cot is equal to one upon tan and this is equal to adjacent upon opposite and we know that there is a there is a trick to remember these ratios and that's called so where sine is opposite by hypotenuse cos is adjacent by hypotenuse and tan is opposite by adjacent so if you remember this you know remember all this ratio always remember sine is reciprocal of cosik cos is reciprocal of secant and tan is reciprocal of cot yeah this is what you must learn okay now opposite and hypotenuse you can always put p and b and h all the values okay moment then you know all the trigonometric ratios are changed so for example if I take this angle as theta or let's say phi then you see sine phi will be now now now this b will be opposite side to phi and p with p would be adjacent to phi so please be very careful when the question comes the question would be definitely easy but they will they will trick you in let's say you know I you you would be making mistakes in identifying the right theta so always this helps so you know so katoa helps so please first then the theta will be given as this as the vertical angle and then you will get confused and you will use wrong ratio so please be very very careful about it this is a common mistake which we have observed over a period of time again b of the theta we commonly find the ratios so just don't go blindly here okay because the problems would not be very difficult they will trick you on in your concentration and your you know really uh you're really paying attention to the question okay so this is what the basic of trigonometric ratio these are all trigonometric ratios okay next next is okay we have cosik is equal to sine theta and all that we know these are the basics of trigonometry of you know this but still let's revise and well I'll I'll write it once in a different manner all together okay so it is sine theta into cosik theta is always one or two secant theta is always and uh tan theta into cot theta is equal to one right these are all reciprocal so hence you don't need to uh pay or let's say be much worried about it okay and the first tan theta if you see tan theta is sine by cos now these are two basic for you to spend more ways instead of where you become over confident hence please be very very careful so cos theta upon uh sine theta so what changes and then the entire question will go for a toss okay so let's move on next is yeah this is very important please remember the value sine theta and cos theta never exceed one so hence we can say minus one less than equal to sine theta less than equal to right and or you can say mod of sine theta is always greater than equal to zero but less than equal to one yeah so mod the the mod value of sine theta will always be equal to one so I'll always remember similarly for cos of one if you remember zero is less than mod of cos theta mod is nothing but the absolute value of sine and cos and if you see uh it was written graph so let's say this is theta as theta progresses here and let's say if you have to plot sine theta so you will so this is sine theta this is sine theta right so if you see this is the upper limit this is plus one this is my load and cos my friend starts from here you know that sorry for a bad drawing so you start from here and then you you go up like that yeah so this is cos cos theta this is also between negative one and plus one additionally for is this also that negative infinity is equal to theta is less than equal to positive or you can't have equal to actually for a infinity symbol so you can choose not logical or legitimate okay so hence you have right less than minus infinity less than tan theta is less than positive infinity okay now what what about the other one so let's say sine theta is between minus one and one then this is what it is so negative infinity is less than cosecant theta and this is less than and or or not and either this will be true this will be true or one less than equal to cosecant theta similarly minus infinity you know secant theta less than equal to negative one and one less than equal to secant theta less than positive infinity these are the things you must remember and tan and cot behaves so minus infinity is less than cot okay and is less than yes okay so please careful about it so though these inequalities will not be asked in the exam but this is important but you be please careful about the ranges of the two or the all six t ratios okay so next let's move on next is okay so this is complementary angle question complementary angle question and or let's say what is complementary angle basically is this that in a triangle and it's very easy to prove also so what is sine 90 minus theta so if you see this is my theta this is theta and let's say this is p this is b this is h and let's say this is a and b and c okay so you know this is let's say this angle is five this angle is five right this angle is five angle c is five so clearly theta plus phi is equal to 90 degrees because this angle is 90 degrees so hence theta is equal to 90 degrees minus five or phi is equal to 90 degrees minus theta okay so let us say sine of phi sine of phi is equal to sine of 90 degrees minus theta okay now from from trigonometric this thing knowledge we know sine phi is opposite by hypotenuse so i opposite is equal to b and hypotenuse is h which is nothing but cos of theta so hence what do i get sine of 90 degrees minus theta is nothing but cos of theta this will be very easy to see that even cos 90 minus theta is going to be sine of theta sine of theta and tan of 90 minus theta is equal to cot of theta cot of theta and vice versa that means cot 90 minus theta is also similarly if this is true then the reciprocals will also be true so hence we say the other reciprocals are let's say cosec cosec 90 minus theta is equal to secant theta and and secant 90 minus theta is equal to cosecant theta right and hence cot of 90 minus theta is equal to tan theta okay now this is where will you use it so for example sine of 1 degree now you know sine of 1 degree is equal to sine of 90 minus 89 degrees which is equal to cos of 89 degrees so cos 89 is sine 1 similarly sine of random let's say 29 degrees is nothing but cos of cos so this is cos of how much 61 degrees right so like that so this is what so cosecant cosecant 23 degrees is is nothing but cosecant of 90 minus how much 67 degrees is it yeah so hence it is nothing but secant 67 degrees yeah so whenever you see in any question some random angles are given right two types of angles two types of operations on different angles in trigonometry have learned one is the standard angle 45 60 and all that and another one is this yeah where the moment you see random angles like that your brain should say that it is a this is a question related to complementary angle t issues so be yeah very easy you can you can immediately understand how to solve that problem okay so let's go ahead now okay now these are the questions but before we take up the questions I will just try to give you this knowledge so so what are we going to discuss is how do we find out you know the values of standard angles so hence you must know a trick to remember though you might have learned it by now you must be by hearted by now but here is the table which I'm going to give you for standard angle so t ratios so let me yeah so t ratios t ratio t ratios of standard angle and we learned a trick also to remember it isn't it how was it so this is how you have to remember if you remember one then you remember all of them so let us say I have I'll make a table now so basically let's say this is zero degree this is or yeah I'll make a vertical table instead of a horizontal table so just a minute yes so what I'll do is I will write the angles here and functions here so sine then cosine then tangent then cosecant cosecant then secant and then cotangent okay so these are the six t ratios and I am trying to write the ratios ratios of let's say zero degree then 30 degrees then 45 degrees and then 60 degrees and then eventually 90 degrees so these are the five standard values for each t ratio you must know now it's good practice to remember remember these numbers actually it is six into five 30 numbers but then just in case if you don't remember and you you forget in the exam paper so what you can do is you can always make this chart very easily okay so you have to it will take two minutes of yours to make the chart and quickly yeah and then it's better practice to make one chart any which way so that you can refer to that chart again and again but make sure that you have made a correct chart okay in case this is only when you don't want to or you don't you can't you know remember those numbers so hence how do we start so it's basically zero one two three four hence what you need to do see there are five right so zero one two three four so here are the other you know this thing I'll write this is the method of zero one two three four okay and this is what it is it is nothing but if you see root of zero by four what happened to the color here just a minute yeah okay so root of zero by four then this is root of one by four then this is root of two by four and this is root of three by four and this is root of four by four this is the best way to remember right so you just put root and then write zero by four one by four two by four three by four four by four yeah so if you see these are the values zero one by two one by root two put three by two and one correct and now whatever you wrote here you just write upside down so let's say in in the reverse order so zero by four here one by four here two by four here three by four and it is four by four okay and now you know uh this is nothing but tan is nothing but ratio sign and cost so it is zero this is uh root over one by three so roots are maintained all the time yeah root if this is a root of one which is one and this is root of three and this is not defined correct and then cosec you know is the reverse of sign so it's always a good practice right one by sign here so that you don't make a mistake one by cost here and this is one by tan here and then it's done so hence if it is zero this is n defined not defined this is nothing but root of four which is two isn't it there's nothing but root of four by two which is root two is nothing but root of so root of four by three and root of one like that okay here what will it be it is nothing but again the opposite or or you can just reverse this order so nd root four root two root of four by three and root one right and then here it is the ratio again so uh got or rather the opposite of tan so nd root three one one upon root three and this is zero this is how it has to be so you know you can remember this is a good trick so for sign what do you need to write root of zero by four one by four two by four three by four four by four in this order zero thirty forty five and then what you can do is if you don't want to put this root signs all over so what you can do is you can simply write zero and then simplify you can do this in the side or a rough sheet this is one by root two this is a root three by two and this is one and then you can just propagate ratio take ratio and all that right so please please revise this i hope you remember this table the moment you remember this table half of your pain will be gone because in heights and distance again you will be needing these ratios so please remember those ratios okay so these are t ratios of some standard angle now some questions could be of the form of this that they will give you a triangle and they will ask you to write all the t ratios so it is very trivial and very simple question so they will give you triangles and in fact they will try and give you some different shapes so that you know in so that you get confused so they will give you theta here or 90 degrees here and they will say okay this is my angle theta and many times they will not say theta they will call it something else and then they will give you let's say this is four this is three and now you have to find out all the six ratios okay so what will be sine theta so in this case sine theta so the first thing is how do you solve this so you'll have to find out the question is find all so let me write the question the question will be something like that in triangle a b c in triangle a b c if angle a c b is equal to theta find all find find all t ratios for theta and they're also given that a c is four and a v is three so you know this is nothing but you have to write four square plus three square which is five so you have to write like this so b c square is equal to a c or rather a b square plus a c square which is nothing but three square plus four square is equal if it is a one marker question you don't need to write so many steps you can just you know for example if it is a two marker or three marker please write by Pythagoras theorem Pythagoras theorem and then it is 25 so hence b c is equal to five okay or if you're you don't want to write in terms of b c a and all that you can mention that let it be p or let it be b and let it be h like that and then you can you know reduce your effort on writings so many names okay so all t ratios so sine theta clearly is it's also always a good practice even if you know it please write down so cut over somewhere even if you know because easier problems attract more careless mistakes but at the same time you also don't need to spend too much of time or don't need to overdo right so sine theta if you're confident go ahead sine theta opposite opposite is three and this is five so sine theta is three by five then cos theta is adjacent by so I just in my heart to so I just it is nothing but four four upon five isn't it and then tan theta is nothing but opposite by adjacent so it is three upon four and then rest all is very easy so cosecant is you don't need to go back to the triangle you can do it from here itself so cosecant theta is equal to five upon three secant theta is equal to five upon four and and cot theta is four upon three okay it's always a good check to see if you know the ratios are making sense so tan theta three by four cot theta four by three four by three and their multiplication should be one so it is one cos theta into sec theta should be one it is one sine theta and cosec theta should be when when multiplied should be one and it's one so hence these are some one marker two marker questions could be like that okay now so let's move back to our this so let us solve this problem the problem is sine 50 degrees so based on complementary angles very easy you can write this as sine of sine of 90 degree minus 40 degrees divided by cos of 40 degrees so as I told you these are unconventional angles so hence it has to do something with complementary angles so you know what to do so cosec 40 can be written as 90 minus 40 90 minus 50 sorry and divided by secant 50 degrees okay and then here it is minus four cos 50 can be written as sine of 90 degree minus 40 into cosecant 40 degrees okay so this please write all the steps like that now sine 90 minus 40 is equal to and then here it is advisable instead of writing in each step you write here itself that sine of 90 minus theta is cos theta okay and then it is cos of 90 so you are not you are going to use cosecant so write cosecant cosecant 90 minus theta is equal to secant theta and then yeah these are the two things you are going to use so in brackets like those write these down so hence what will happen sine 90 minus 40 is nothing but cos 40 degrees divided by cos of 40 degrees so you know this is going to be one then this is cosecant 90 so in the secant 50 degrees and this is secant 50 degrees and then minus four times sine 90 minus 40 is cos I'm sorry did I do something cosine I'm sorry cos 50 degrees sine I'm sorry I'm sorry this is not this is not the case here it should be sine of 90 minus this is cos sorry this is cos yeah this one is cos cos of 90 minus 40 degrees so now you can write this is nothing but so be careful these mistakes are going to be there so sine 40 into cosecant 40 right and then now you know there is another identity which you are going to use or that is nothing but sine theta into cosecant theta is equal to one yeah right so hence you have to use this so hence it is one plus one minus four into one which is negative two so this is what the question would entail okay so it's good practice to write all the please be careful to write the reason you don't miss this step they will look for this step as well yeah if you miss it and if it is a one marker question then you don't need to do all that but if it is a two marker or more probably this question would be a two marker so hence write the steps okay now next again similar question so no problem here so let's go back and solve let us say cosecant 60 again there are you know odd odd looking numbers ugly looking in angles so you know what to do you have to use complementary angles so hence cosecant 65 plus theta is okay so how will you how will you you know you have to get some 90 here 90 you know how do you use or let's say make use of complementary angle bring down 90 somewhere somehow so hence what can I say so I can say so cosecant 65 degrees can be written as 90 degree minus 25 degrees plus theta okay now this theta comes as it is 90 minus 25 is 65 degrees okay then minus secant you have to write um let it be 25 minus theta here itself why because there's a purpose if you change it 90 is here then cosecant will get changed into secant right so let it be like that now minus tan of 55 can be written as 90 degree minus 45 no not 45 35 degree the hint is here 35 so it will be 35 degrees plus sorry here it is minus theta and then next is let it be as it is 35 degrees plus theta okay so now what can you say you can say cosec cosec 90 degrees minus of 25 minus theta isn't it this is the first term you get you took minus common so minus common when you minus theta minus secant 25 minus theta let it be as it is then third is negative tan and then this will take months within back 35 this know that it is going towards zero okay so plus cot cot 35 plus theta okay okay so now if that is the case then what is the next step so you have to now reason it out so you will say cosecant 90 minus theta is equal to or usually don't use the variable which is you know you should not be using the variable which has been given in the question so right 90 minus a is nothing but secant a and then tan of 90 minus a is cot a okay so you mentioned that within brackets and then go for the next step so next step is cosecant 90 minus 25 minus theta is secant 25 minus theta then the the other secant theta is as it is 25 minus theta then it is negative tan 35 plus theta and then plus plus sorry after tan tan 90 will become cot right so this will become cot after applying the after applying what this will get cot 35 plus theta and then plus cot so hence net net is zero okay so no brainer not much of an effort is required in such problems so let's move on to the next question okay next question is again so ugly looking numbers so obviously you cannot now in this question there is a mix of both standard angle as well as non-stand non-stand angle okay so hence you'll use in this question both the or let's say both the um concepts would be used so 2 sine 68 degree yeah so the question is 2 sine 68 and cos 22 the question itself is inviting you to use complementary angles isn't it so hence you can write you can say 2 sine 68 degrees can be written as 90 degree minus 22 degree isn't it divided by cos 22 degrees so you you are you know what it is going to be then again 15 and 75 so again the question itself inviting you to write that but be be careful there is 2 sine 68 cos 22 there is 2 upon 5 right so 2 by 5 you separate out the constants now cot 15 can be written as cot 90 degree minus 75 divided by tan 75 degree okay and then last is negative 3 so please be careful there is a 3 in the hurry of solving don't miss out on that so tan 45 you know tan 45 is 1 okay and then you write tan 20 tan 20 can you see there can be a pairing done tan 20 and stand 70 so tan 20 can be written as tan of 90 degree minus 70 degrees then tan of 40 can be written as tan of 90 degrees minus 50 degrees and the rest is tan 50 don't touch it if you if you change this also then you the purpose gets defeated but don't don't use complementary here and then tan 70 degrees okay divided by 5 so the entire thing gets divided by 5 and yes as i told you you have to you mentioned the reasons as well so you write you will write now sine of 90 minus theta is equal to cos theta then you'll write cot of 90 minus theta is equal to tan theta and then you'll write tan of 90 degree minus theta is equal to cot theta and then you'll write tan of 45 degree is equal to 1 all this you have to mention okay again if it is a one market question you know what to do okay now it is if you see next step would be 2 into sine 90 minus theta is cos 22 degree divided by cos 22 degree okay minus 2 upon 5 minus 2 upon 5 this is tan 75 degrees 25 degrees you must be tempted many attacks would be tempted to skip these steps but ideally i would recommend that you should write hello yeah so minus 3 by 5 into what into tan 90 minus 70 is cot 70 isn't it yes cot 70 degrees and then you tan 70 you write here itself and then into you write here tan cot 50 into tan 50 and one thing which we forgot oh sorry one thing which you forgot to write mention here is tan theta is equal to 1 upon cot theta or you can write tan theta into cot theta is 1 okay using all these now what will i say this is nothing but 2 because cos 20 to cos 20 gets cancelled 2 by 5 tan tan gets cancelled minus 3 by 5 why because this is 1 and this is 1 okay so hence it comes out to be 2 minus 5 by 5 which is 2 minus 1 2 minus 1 equals 1 so this is what it will be okay so let's move on to the next problem yeah now again this is simple so how to do this sign 18 again the same ideology same funda same philosophy will be written so hence it is sign of 18 degrees so it is given as sign of sign of 19 minus 72 degrees divided by cos of 72 degrees then it can be written as sign of 22 it can be written as 90 minus don't go and subtract it you see you know this number itself can be used to write so you save your time don't subtract from 90 and also 90 minus 68 in cos of 68 degrees but yes do check that both of them are adding up to 90 degrees cos 38 again you know cos 38 is cos of 90 degree minus you can check 52 so 52 here and then into cosecant 52 cosecant 52 degree is it and divided by tan 18 so hence what you can do is what you can do is tan 18 can be written as tan of 90 minus 72 degrees then tan 35 and then you write this in with tan 72 here okay and then here again tan of 35 degrees can be written as tan of 90 minus 55 into tan of 55 tan of 55 into tan 60 degrees this is the whole thing correct so hence i'm writing here now so signs at night you know it now it is cos 72 degrees by cos 72 degrees plus cos of 68 degrees by cos of 68 degrees minus it is sin of 52 degrees into cosecant of cosecant of 52 degrees divided by cot 72 into tan 72 into cot 55 into tan 55 into root 3 what because tan of 60 degree is root 3 so you have to guys you have to please remember i'm not writing all that you know but you have to write all the uh rules which you used right so you have to write sin of 90 degree minus theta is equal to cos theta then sin of no sorry then tan of 90 minus theta is equal to cot theta and tan 60 is equal to root 3 and here yeah sin theta into cosecant theta is equal to 1 so this you mentioned in the right side of the you know answer script 1 plus 1 minus sign in this cosec is 1 divided by root 3 1 by root 3 correct yeah so hence what will you get you will get 2 minus 1 by root 3 you can leave it here or you can write 2 root 3 minus 1 by root 3 so this is how it will be solved next okay so now the question says if sin theta is cos theta minus 45 where theta and theta minus 45 are acute angles find the degree measure of theta very good so you know this is a question based on complementary angle again so it is said that sin theta is cos theta minus 45 so sin theta can be written as again it could be a two marker question can be written as cos of cos of 90 minus theta right and now sin theta given is sin theta is equal to cos of theta degree minus 45 degrees okay so from one and from two from one and two one and two you can write 90 minus theta is equal to theta minus 45 isn't it so that means this implies now actually there is a acute angle is mentioned right acute angle if this was not mentioned then you will have you can have multiple such possibilities so you can have this plus 360 also is also correct why because cos of 360 plus anything is as good as cos of anything so anyway so we are not going to that detail this is left for let's say your higher grades so till the point there I mentioned that it is an acute angle then you can easily write this so hence if you saw this in this one linear equation in one variable so this is nothing but two theta is 135 degrees so theta becomes 135 upon two which is 67.5 degrees okay so this is how you'll have to solve it right okay find the degree measure of theta so theta is 67.5 degrees okay so let's go to the next problem yeah if ABC are the interior angles of a triangle ABC all these questions have been asked in the previous year board exam so it can be a good practice for you to go through it once again if ABC are the interior angles of a triangle ABC show that sine b plus c by 2 is cos a by 2 right so a so it's it's always a good practice to draw a quick diagram so a plus a b and c so you can say a angle a plus angle b plus angle c is equal to 180 degrees or you can drop this angle sign you can say a plus b plus c is 180 degrees okay so hence b plus c by b plus c is equal to 180 degrees minus a isn't it so b plus c by 2 will be equal to 180 degree minus a by 2 which is 90 degrees minus a upon 2 okay so sine of sine of b plus c by 2 can be written as sine of 90 degrees minus a by 2 why because b plus c by 2 is 90 minus a by 2 and hence it is cos of a by 2 why why because sine 90 minus theta is equal to cos yeah the second part is again very similar to the first part itself so cos of again you can write cos of this is 1 and then 2 cos of b plus c by 2 is equal to cos of 90 minus a by 2 is equal to sine a by 2 so whatever you are getting as a question in 10th board becomes a common rule or you know common knowledge in 11th grade this is how it is okay fair enough let's move on to the next again so this question says if 2 theta plus 45 and 30 minus theta a q tangles again see a q tangles find the degree measure of theta satisfying sine 2 theta plus 45 so again similar it's a combination of c linear equation and trigonometry okay so what does it mean sine 2 theta plus 45 sine of 2 theta plus 45 degrees is equal to cos of 30 degrees minus theta this implies right this implies cos of 90 degree minus 2 theta plus 45 degrees is equal to cos of 90 degree minus theta so since both the functions are same on the left hand side and that's inside right hand side you can equate the argument that is the angles so 90 degree minus 2 theta plus 45 is equal to 90 degree minus theta isn't it so hence this 90 degrees and this 90 degree gets stuck this minus sign goes and hence it is 2 theta oh no uh cos of oh sorry sorry sorry my bad my bad this is not this is what it happens so hence please be very very careful while writing right so this is cos of 30 not 90 okay so anyway so let me just just just just delete this step okay so yeah i was started why because i was getting theta as negative so hence this is how you check also so if you see cos of 90 minus 2 theta plus 45 is cos of 30 minus theta so you can say 90 degree minus 2 theta plus 45 is equal to 30 degree minus theta okay so hence now it is nothing but 90 degrees minus 2 theta minus 45 degrees is equal to 30 degrees minus theta now club all the theta together and all the angles together so if you take 2 theta on the other side you'll get 2 theta minus theta so 2 theta is taken as on the right hand side and i'm just uh simplifying it further so it is nothing but 90 degrees minus 45 degrees minus 30 degrees okay so hence you will get 15 degrees theta is equal to 15 degrees yeah check it if it is true so if 2 theta sine theta is 15 so it will become 2 into 15 is 30 so sine 75 and sine 75 is equal to cos 15 that's correct so hence theta is equal to 50 degrees correct so this question is correctly solved okay okay let's move to the next question okay theta is positive acute angle such that sec theta is cosecant 60 degrees find the value of 2 cos square theta minus 1 okay so let's do this so the question says theta is a positive acute angle again such that secant theta equals cosecant 60 degrees so if secant theta is equal to 2 cosecant 60 degrees you have to find out the value of 2 cos square theta minus 1 so secant theta is cosecant 60 that means you can write secant secant theta is secant 90 minus 30 degrees sorry 90 minus 60 degrees am i right hold on no in such questions instead of yeah so yeah you write the same function so that you don't make a mistake so let us say secant theta right so secant theta is cosecant theta 60 can be written as cosecant theta 90 minus 30 degrees okay now you change so this is secant theta change the angle first and then change the function don't change the function first otherwise you'll get confused so 60 can be clearly written as 90 minus 30 now cosecant 90 minus 30 can be written as secant 30 degrees okay so mention this why you have to mention the reason reason is cosecant 90 minus theta is equal to secant theta okay so secant 30 degrees so now secant theta is secant 30 degrees so theta is equal to 30 degrees for acute acute values theta is 30 degrees now what is the value of so 2 cos square 30 degrees minus 1 is equal to 2 into cos square 30 cos 30 is root 3 by 2 so here is where you have to remember that the values so cos 30 is root 3 by 2 so root 3 by 2 whole square minus 1 which is equal to 2 into 3 by 4 minus 1 which is 3 by 2 minus 1 is equal to 1 by 2 am I right oh wait a minute no 3 by 4 it is oh no it was correct sorry 3 by 2 3 by 2 minus 1 yeah so half okay so yes half half is the answer and those who know let's say as I as when we discussed trigonometry I remember you I remember I tell you you know so 2 cos square theta minus 1 is nothing but cos 2 theta so if you remember this you'll be able to solve it you check at least you can check so cos 2 theta is what was theta 30 degrees so cos 2 theta is cos 60 degrees so cos 60 is anyways half so this is correct so this value is correct okay next let's go to the next problem okay if cos 2 theta is sine 4 theta where 2 theta and 4 theta are acute angles sine the value of theta again so what can you say so cos 2 theta can be written as so you write cos 2 theta is equal to sine 4 theta so that means what does it mean sine of 90 minus 2 theta is equal to sine 4 theta right sine of 90 minus 2 theta is sine 4 theta right cos 2 sin of 90 minus 2 theta is cos 2 theta you have to write the region sine of 90 minus 2 theta is yeah you have to write sine of 90 degree minus theta is equal to cos theta okay so here what will you say you will say sine and sine are now equal so you can say angles must be equal 90 minus 2 theta is equal to 4 theta so this implies 6 theta is 90 degrees so this implies theta is 90 by 6 which is 15 degrees you can check also cos of 30 is root 3 by 2 and sine of 60 is root 3 by 2 so hence theta is 15 degrees also always try to check how did it how did I check I put the value of cos 2 theta so cos 2 theta is cos 2 into 15 degrees which is cos 60 sorry cos 30 so cos 30 is root 3 by 2 and there it is sine of 4 theta which is sine of 4 into 15 degrees so sine 60 which is 3 by 2 root 3 by 2 so check okay done so quick let's now move on to height and distances I believe yeah so this is a similar type of question so I'm not going to pick that up let's move on to height and distances because there are lots of yeah so similar similar all these are previous boardier papers but they are all similar questions you can solve them now let's come to our heights and distances problems okay now there are a few definitions you must know the line drawn from the eye of an observer to a point in the object where the person is viewing is called the line of sight right so let's say this is a tree let's say this is a tree and this is the ground level so let me draw a proper line here so yes let me so let's say this is my ground level and sorry and let's say now let us say here is the observer at this height okay this is the and so hence this is the line of sight of this tip okay so angle formed by the line of sight with the horizontal when the object is above the horizontal level is called the angle of orchard this is the next slide actually wait a minute I'll go back to the next slide previous side yeah so what is it it says the line drawn from the eye of an observer to a point in the object where the person is viewing is called the line of sight so this is line of sight let's say the person is viewing this point on the tree so this is line of sight okay line of sight next next is next what does it say the angle formed by the line of sight with the horizontal when the object is above the horizontal level is called the angle of elevation so hence you now know what is angle of elevation so if I this is the horizontal level and hence this angle is called angle of elevation theta theta is angle of angle of elevation right the same line can be extended from here to the ground level if you see though the the observer is not at the ground level but then this can be done and let's say if you extend this also here sorry yeah and let's say now if you you do this let's say so this angle this angle also is if you see angle of elevation so all similar triangles and basic proportionality theorem and all trigonometry all basic geometry will be applicable over here okay so this is angle of elevation very good now the angle formed by the line of sight with the horizontal when the object is below the horizontal level is called the angle of depression so if you see excuse me now let's say the person is looking at here the bottom or the foot of the tree or whatever it is then this angle phi phi is called angle of depression phi is called angle of angle of depression you know this angle of depression so depression so depression is always below the horizontal level elevation is always above the horizontal level okay now next the height of an object or the distance between distant objects can be determined with the help of trigonometric ratios yes we know that and now let us apply this okay so first is first question a tree is broken by the wind very good the top struck the ground at an angle of 30 degrees the top struck the ground at an angle of 30 degrees that means let's say this was the original tree and it broke from somewhere so let's say after breaking down after breaking what happens this part this is the tree sorry now now the tree broke so hence let's say it broke from this point this point this point so from there let's say it broke like that okay and they are saying that now it is touching the ground now this angle okay so in height and distance always make drawing so let's say this is theta is equal to 30 degrees okay so an angle of 30 degree and a distance of 30 meter from the root so this is given as 30 meters name it let's say a b c d right find the whole height of the tree so whole height of the tree so you can whole height height of height of of tree height of tree is how much is nothing but a b plus b c right which is equal to a b plus b d y because the same portion has now touched the ground so a b plus b d now now a b a b plus a b plus so hence you have to do this x and this y you have to find out okay right and let's say this is z okay you mentioned all this so hence what can you say you can say x upon 30 x upon z is equal to 30 isn't it x upon z x upon z is tan 30 right which is nothing but 1 by root 3 you have to like see you have to remember the values 1 upon root 3 if you make it root 3 then the entire problem will be wrong wrongly solved so x equals z root z upon root 3 okay so what is z z was 30 so 30 by root 3 meter which is equal to rationalize it you get 30 root 3 by 3 meters which is nothing but 10 root 3 meters now either you can find y using Pythagoras theorem because x now you know is nothing but 10 root 3 10 root 3 meter either you can use Pythagoras theorem or you can use sine third sine theta sine theta is half sine third theta is nothing but x upon y right opposite by hypotenuse so y is equal to sine x upon sine theta x upon sine theta now x is 10 root 3 upon sine theta is sine 30 okay so 10 root 3 upon 1 by 2 sine 30 is half which is equal to 20 root 3 and the brackets you write sine of 30 degree equals to half fair enough so hence you got y right so hence height of the tree height original height of tree is x plus y is equal to 10 root 3 plus 20 root 3 is equal to 30 root 3 meters okay this is this is the total height of the tree okay next let's move to the next problem these are all questions asked in previous year board papers now the angles of elevation of the top of a tower from two points at a distances a and b so here is how you should you know proceed with the question so angles of elevation that means there are two points from where something is being looked at the top of a tower okay so that means there is a tower so let's make a tower quickly this is a tower and let's say this is the this is a ground level okay fair enough now now from two points from two points at a distance a and b meters from the base and in the same straight line with it are complementary okay the angles of elevation of top of a tower from two points at a distance a and b from the base so let us say this is one one angle angle of elevation from here another is this okay and it is being said that from two points at a distance a so let's say this is a what is a this distance is a and let us say this distance is b okay so tower from two points at a distance a and b meters from the base and in the same straight line with it are complementary so this angle is less angle of elevation is theta and this is phi let us say this is a b c and d right now you have to in this question you have to what is that prove that the height of the tower is root a b so also mentioned few variables like h okay right so you start with saying that let h be the h be the height height of the tower you start with this assertion okay h is the height of the tower now you also mentioned these are the given things so it's a good idea good practice to mention that let let a be at a distance of b units or b meters it's mentioned in meters meters from c and let at a distance of a meters from c let us say angle d a c is equal to theta and angle d b c d b c is equal to phi phi so we know that theta plus phi is equal to 90 degrees right because i'm sorry so theta plus phi is 90 degrees that is given because they are complimentary we have to find h okay so let us try and find h in terms a and b so clearly if you see um you have to basically eliminate theta and phi okay so hence let's start with tan theta why tan i'm not signed because i don't want to engage with ad which is unknown so i will engage with whatever i know so i know i have to basically find h and i know the basis so hence tan and cot will be ideal trigonometric ratios to deal with here so hence i will say tan theta tan theta is h the opposite opposite divided by the adjacent so for theta the adjacent is b so h by b tan theta is h by b similarly tan phi is h upon a h upon a now i have to somehow eliminate theta and theta and phi isn't it now tan theta into tan phi right tan theta into tan phi is equal to i can write h square by ab right why am i multiplying there's a trick behind it why am i multiplying because the question is asking you know see how do i get a into b so i have to multiply tan theta and tan phi then you'll get ab in in one side now this this attempt will also tell you that okay tan theta and tan phi since theta and phi are complimentary will be one how so you can write tan theta so is tan of 90 degree minus phi into tan phi is equal to h square by ab why is that so you'll write from one this let's say put it as one so from one from one okay and then next step is cot phi into tan phi is equal to h square by ab why because tan of 90 minus phi is cot phi you have to mention okay now it is nothing but one is equal to h square by ab so this implies h is equal to root of ab yeah you have to also mention here the reason is tan theta into cot theta is equal to one okay this is another question let's go to next problem this was easier so maybe tree marker max not beyond that next is a man on top of a vertical tower observes a car moving at a uniform speed coming coming there's something missing i think here coming after 12 minutes for the angle okay it takes 12 minutes i think for the angle of depression to change from 30 degree to 45 degree how soon after this will the car reach the tower let's see if you can solve with this what will happen if let's say the question is you know somewhat not correct then also you don't leave the question draw the diagram and try and solve as much as possible many a times if the question is wrong if someone has attempted they get points for that so hence don't leave the question thinking that question is incomplete or the or information is not being supplied so a man on the top of a vertical tower observes a car moving at a uniform speed coming towards him this will be this question basically so after 12 minutes for the angle of depression it takes um i think 12 minutes to for the angle of sorry angle of depression to change so let's so let's say this was initially angle of depression was so how do i measure angle of depression so let me just yeah take uh this yeah so let's say from so let me just yeah take this as my horizontal okay now so this angle was 30 degrees theta is equal to 30 degrees and then after some time what happens the theta changes to 45 degrees right so let us say this angle is phi let me change the color here so that doesn't confuse you so phi phi no this is not done wait yeah yeah so i'll write phi here okay so phi is 45 degrees okay now they're saying it takes 12 minutes for the angle of depression to change from so if this is 30 degree my dear friend this is also 30 degree okay and this is 45 degrees okay 45 is more than 30 makes sense here so 30 45 and it is saying let us say this distance is x right this distance is x and let's say this distance is y and let's say this distance is h the height of the tower okay the question is how soon after this will the car reach the tower so i have to find out okay so it takes 12 minutes for yeah so hence let us say uniform speed right so let us say speed speed of car is v meter per second okay so x will be nothing but distance is speed into time so v into 12 minutes 12 minutes means 12 into 60 seconds be very careful why because you have taken speed in meter per second so minutes will have to be changed too so 720 v x is 720 v okay now let us see yeah if we can yeah so hence x plus 1 let's say total time total time taken for covering while let us say let us write here so time taken let's say time taken to cover distance y is equal to t seconds okay so y can be written as v into t why because velocity into time is right okay now i know now let us use 30 and 45 this is what it is indicating so you'll have to use 30 and 45 somewhere so how do we use it so tan 30 i know tan 30 degrees is h upon x plus y isn't it right and which is so hence this implies h upon x plus y is equal to 1 upon root 3 isn't it now since it is 45 degrees either you use trigonometry or geometry you can say h is equal to y why because because i have a name also let's say this is a b c and d and you can write triangle b c d is an isosceles triangle right so hence h equals to y so hence you can you can you can write you can you know from 1 and 2 let's say this is 1 and this is 2 so from 1 and 2 from 1 and 2 what can i say i can say i can say y upon x plus y equals 1 by root 3 isn't it so this implies what was y v into t v into t upon what is x x is 720 v plus v into t is y is equal to 1 by root 3 so for the want of space i'm writing it here you should actually carry it down right so let us let me write it here in this space here so i'll be right i'll be utilizing this space okay so if you see v can be can you know can be stuck off so v goes so it becomes t or i write here t upon 720 plus t is equal to 1 upon root 3 now it boils down to solve this you know you just need to solve this linear equation so it's root 3 t is equal to 720 plus t so root 3 minus 1 t is equal to 720 okay so t is equal to 720 by root 3 minus 1 now root 3 is our 1.73 so hence 1.7 1 point root 3 is 1.73 so 1.73 minus 1 is 720 upon 0.73 right which can be written as 73 into 100 73 by 100 right so now it is nothing but if you see this is around 0.99 so around how many seconds i think it is 90 so nine around 9000 seconds so i have to do some calculations every so hence i will need some space so let's say i utilize this space so if you see this is nothing but 720 by 730 in 2000 second okay so it's approximately if you see this is approximately one so approximately you can say thousand seconds so if you give your answer to the nearest second it says you have to actually do a you know calculation so 730 you have to divide 7 so 72 by 73 you have to see let us see what is that you know let me just yeah use this so 730 73 and then 72 you have to do so hence it is nine times if you see so nine times three is 27 to 97 60 to 65 right so it is 43 plus 23 is 63 63 year yeah so hence again 630 you have to put so nine and then nine just right so it is i think eight eight times so 83 that would be four two 56 58 yes so six and then 46 right and then 460 so hence you have to do one more one more time so 73 right so 76 so 18 43 so it is nothing but 22 right so around 989 86 in 2000 into so it is 0.986 986 in 2000 so approximately your answer should be 986 seconds so it is nothing but equal to 986 seconds so answer is t is somewhere around 986 second so this much time will be taken so around you divide by 60 you will get into minute so looks fair enough okay this is how you have to you guys please check you know at least that if the calculations are done correctly and always in in high marker questions you please revise the solution once again so that there is no calculation mistakes so so far looks good but just in case I have made a mistake you can highlight or you can check once again yeah never mind so let's move on so this is how the problem will be solved next is let's see what is the question next question is yeah 26 question number 26 the angle of elevation of a jet plane from a point a on the ground is 60 degrees okay very good after a flight of 30 seconds the angle of elevation changes to 30 degrees okay so again you have to make one ground and there is a plane so I will so the angle of elevation jet plane from a point a so let's say this is point a and here is a jet plane which is like that let's say this is a jet plane okay jet plane right so angle of elevation of jet plane is 60 degrees okay now after a flight of 30 seconds the angle of elevation changes to 30 degrees so that means this jet plane would have moved towards right so so let's say it is moving towards right between 30 second it has gone to this place okay and here is where it is standing so let's say this is my new position okay now here they are saying angle of elevation is changed to 30 degrees so this angle is theta is equal to 30 degrees and this angle is five which is 60 degrees okay the jet plane is flying at a constant height of 3600 root 3 meters okay so basically what do we know we know so this height let me write this edge edge is 3600 root 3 meters and find the speed of the jet plane okay so let us say from a and this is let's say b and then c so you write let let the time taken oh it's it's already given so let the time taken oh or let's say um no it's already given that okay you know that the time taken is 30 so the time taken time taken by the plane by the plane to go from b to c is equal to 30 seconds okay so bc or find the speed you want to find the speed so you have to write right let the speed b v so bc which is a distance is nothing but v into 30 correct bc is v into 30 now obviously lot of use trigonometry so hence let us drop perpendicular so let's drop perpendicular from here so let me drop perpendicular right so you drop a perpendicular from here to here and from here to here okay and now let them let these points be d and e okay so again i know phi tan phi why will i use tan phi in most of the uh heights and distance problems you'll use tan and cot only why because they will not generally give you uh the slant or let's say the distance in the line of sight so tan phi is equal to h so obviously you have to also mention that bd is perpendicular to ae and ce is perpendicular to ae right and bd is perpendicular to ae right and so tan theta tan phi is b you can write bd bd upon ad okay which is nothing but h upon ad and phi is how much 60 degrees so hence you know h upon ad is equal to root three in the bracket you'll have to mention tan 60 degrees is equal to root three okay so this means h is equal to root three times ad yeah or basically you know ad why because now this implies h h was 3600 root three is equal to root three ad so this implies ad is equal to 3600 meters ad is 3600 isn't it so 3600 by yes 3600 root three by 3600 is tan 60 but perfect so ad you now know now okay similarly similarly here i will use the space here so yeah so what can i say i can also say in triangle in triangle aec i can say h or h upon ae sorry h upon ae h this is h and ae is tan 30 okay so ae is equal to h upon tan 30 right which is nothing but 3600 root 3 by 1 upon root 3 which is nothing but 30 into 3 which is nothing but 108 00 meters i'm sorry so 108 00 meters okay now so that means i know de is equal to nothing but ae minus ad which is nothing but 108 00 minus 3600 meters which is nothing but 7200 meters okay 7200 meters now v is nothing but distance upon time so 7200 see i would have written all the steps as well but then now just to illustrate because of positive space here so 7200 meter was traveled in 30 seconds so this goes so hence it is 240 meter per second this is what the speed is okay so this is how you have to solve i hope you guys are getting the approach and how to solve questions okay next next next question so next question is next question is the angle of elevation of a cloud from a 0.60 meter above lake is 30 degree and the angle of depression of the reflection of cloud in the lake is 60 60 degrees find the height of the cloud okay so there's a both angle angle of elevation and depression are there so let us make a lake first so let's say this is the water surface this is the water surface um so let us say this is the water surface okay and the angle of elevation for cloud from a point 60 meter above a lake okay so from a point let's say 60 meter above so first of all there has to be a cloud height so let us say this is my lake and from here this is the cloud height let's say the cloud is here okay and this is a point 60 meter above lake okay and the reflection will be let me just take another color so that uh to differentiate so if you see this is the reflection of the cloud here okay so i hope the size these two are same anyway so this is the drawing now uh what do you do you join these two so this is angle of elevation angle of elevation and this is angle of sorry and this is the angle of depression yeah how do you find out so angle of elevation is from always from the point of observation so this was my point of observation let's say oh from here from here okay so this is my horizontal line now this angle they are saying theta is 30 degree angle of elevation is nothing but what did we learn you have to join the point of observation to the point which is being observed and from horizontal what is the angle is the angle of elevation what is angle of depression from the point of observation to the point which are which is being lived upon then from there from from the horizontal to that line line of sight is angle of depression depression is when it is below the horizontal elevation is when it is above the horizontal so this is 60 degrees now h let's say this is oa h h is given as 60 meters okay it may be a building or something so right find the height of the clouds so you have to find out let's say oa let me mention some names b c d all the points this is e and let's say this is f okay so you say point e is the reflection reflection of point d therefore what do i know let's say i have to find out h right now so you write this is h right so this edge yeah so so db is equal to de is equal to h okay now in triangle odc again you'll use tan theta so tan 30 is nothing but dc upon oc dc dc height that is perpendicular upon base right or adjacent or opposite upon adjacent so dc is nothing but let me also mention this as x for the ease of right so oh sorry oh sorry i have taken two edges together so let's say this is h and this is capital h instead of small h you mention it as capital h so hence this is also capital h now you have to write oabc is a rectangle is a rectangle hence hence oa is equal to bc is equal to h so you don't need to write this as h this is simply x sorry this is h okay small h okay now so tan dc so hence you can write tan 30 that is 1 upon root 3 tan 30 is 1 upon root 3 dc so dc is nothing but capital h minus small h divided by oc so this implies oc is equal to h minus h h minus h into root 3 right root 3 h minus h okay why am i doing this because i'll eliminate oc now in triangle oce tan of 60 degrees is ec that is opposite upon oc so this implies root 3 tan 60 is root 3 ec ec getting written as eb eb plus bc divided by oc which is nothing but eb eb is capital h plus bc bc small h divided by oc is equal to root 3 so from here you can say oc is equal to h plus h upon root 3 now from 1 and 2 you can equate both these both are oc both are equal to oc right so from 1 and 2 from 1 and 2 what will happen you can write h minus h root 3 is equal to h plus h by root 3 okay so now it boils down to only simplification so it is root 3 into 3 is 3 so you can write 3 capital h minus 3 small h is equal to capital h plus small h which on simplification 2 capital h is equal to 4 small h 2 capital h equals to 4 small h so capital h is 2 small h okay so hence if it is a 60 meters so it is 2 into 120 it's 2 into 60 meters it is 120 meters this is question number 27 okay fair enough let's go to the next problem so what is the learning so please make neat diagrams and then you know tan 30 tan 60 and tan 45 must be on your fingertips now tan 30 is 1 by root 3 tan tan 60 is root 3 tan 45 is 1 these are the common angles which will be used in heights and distances mostly so please do not make mistake while taking the value of angles okay so let's move to the next problem and it says that a straight highway leads to the foot of a tower so let's now we are now we are comfortable with problem solving on this so straight highway leads to a foot of a tower so that means there has to be a tower like that so this is the tower okay a man standing at the top of the tower observes a car at an angle of depression of 30 degree which is approaching to the foot of the tower with a uniform speed so hence i have to probably take this is 30 degrees okay 30 degrees okay which is approach and then i have to also make a vertical line sorry horizontal line which is angle of 30 degrees this is theta theta is 30 degrees okay next is so the car this is a car here this is a car which is moved with a pv approaching to the foot of the tower with a uniform speed six seconds later the angle of depression of the car is found to be 60 degrees this is very similar problem similar to the previous case this angle is this angle five is 60 degrees okay find the further time taken by the car to reach the foot of the tower exactly similar to the previous question okay so i will not write the steps but i'll solve the problem so that you at least yeah you know you get a hang of it so this is 60 degrees clearly this is 30 degrees and yeah six seconds later the angle of depression so this is six second later so hence this distance is nothing but v into six or six v yeah speed into distance is what distance is equal to speed into time speed into time right now so speed into time six v right where v is in meter per second right now you have to find out what find the further time taken so what is the time taken here so let's see here time taken is t so hence this distance is vt okay so now it's similar to the previous problem so tan 30 tan 30 is let's say this is h i'm not writing the steps you can always you know while in the exam you please don't forget to mention all whatever you are assuming so h is tan 30 is h upon six v plus vt right so tan 30 is this and then tan 60 is h upon h upon vt right so divide both tan 30 by tan 60 so tan 30 by tan 60 is equal to h will get cancelled so vt upon six v plus vt so we also will get cancelled now tan 30 is one by root three divide by root three is tan 60 is nothing but t upon six plus t t upon six plus t so this is nothing but t upon six plus t is one by three so this implies three t is equal to six plus t yeah i hope i'm not making any mistake here but let me mind you just oh wait a minute one by three so three t is equal to six plus t yeah correct so this is two t is equal to six so this implies t is equal to three seconds so in three second three second it will be there at the foot okay so you can check also uh if it is six is to two no but then you don't know the h anyways so yeah so this is how it will be six and t am i so let's go to the next problem okay next problem is the angle of elevation of the top of a chimney from the foot of a tower is 60 degree and the angle of depression of the foot of the chimney from the top of the tower huh this is a different question so let's solve this so let's say they're saying the angle of elevation of the top of a chimney so this is the top of a chimney right and the angle of depression of the foot of the chimney from the top of the tower is 30 degree okay um the angle of elevation of top of a chimney from the foot of a tower so let's say this is a this is a tower okay and let me join these join these and let me use this color for angle of elevation and depression so this is let's say this one is tower and this one is chimney okay now if you join these so they're saying the angle of elevation of the top of a chimney from the foot of the tower is 60 degrees so this is theta is 60 degrees okay and the angle of depression of the foot of the chimney from the top of a tower is 30 degrees okay so angle of depression so let's say this is a yeah so this angle is five is equal to 30 degree if the height of a tower is 40 meters so let's say small h is 40 find the height of the chimney so h is equal to what okay according according to pollution central norms the minimum height of a smoke emitting chimney should be 100 meters okay state if the height of the above mentioned chimney meets the pollution norms so what value is discussed in the question yeah so that is also a part of question so you have to find out capital h so again uh you know uh i will be avoiding or you know or you can let us write all the steps also this is how you will say so uh a b c d let let a b a d b the chimney sorry tower the tower with height small h equals to 40 meters let b c be the height of the chimney right right chimney and you can mention capital h here now angle a c a b is equal to 60 degrees an angle a sorry it is let's say you if this is five then this is five yeah so let's say point e so angle e d b is equal to angle a b d is equal to 30 degrees now now so a b is common to both so we'll use again the same yeah so tan of five tan of five that is 30 degrees is equal to tan of five is nothing but h upon a b and tan of theta tan of theta is capital h upon a b okay so tan of five is small h upon a b and kept tan of capital uh tan of theta is capital h upon a b so a b is common so you can divide so tan tan phi by tan theta is equal to h upon capital h so this implies capital h is equal to h into tan theta by tan okay so h into capital h is h into tan theta divided by tan phi now h was 40 and theta theta was 60 so root 3 divided by 1 by root 3 so it is 120 meters okay yeah so according to pollution central norms the minimum height of a smoke emitting is 100 meters so it is clearly more than 100 meters so that means uh you know it is well within the legal limits so the value is that the person you know the law must be abided with it must be obeyed so hence uh yeah so the person who has who is owning this chimney is a law abiding the citizen of this country okay great next and this is the last question for the day I think yes yeah so the question says an aeroplane is flying at a height of 210 meters flying at this height at some instant the angle of depression of two points in a line in opposite directions on both the banks of the river are 45 degree and 60 degrees find the width of the river okay so an aeroplane is flying at a height of 210 meters flying at this height at some instant the angles of depression of two points in a line in opposite direction on both the banks of the river okay so let's let's say this is the river, this is the river, okay? And what he, what he is saying is it is, it is flying at this height, let's say at a given point somewhere in the, so here is the aeroplane which is moving, aeroplane is moving like that and flying at this height 210 meters, so H is 210. The angle of depression of 2 point in opposite direction on both the banks of the river are 45 degree and 60 degree, so hence, this is the line of psi, sorry, horizontal and this is my angle of depression for this and this is, yeah, line, okay. So this angle is 60 degrees and this angle is 45 degree, let's say, so hence, this is 45 degrees, alternate angles and this is 60 degrees, alternate angles. Now what is to be found out with the river? Let's say this point is X, this distance is X, this distance is Y, so what do I know? H by X is tan 45, again, I'm not writing the given and all that, I'm just solving the problem. So H by X is tan 45 and so hence, X is H, X is H upon tan 45. You don't need trigonometry for this, from this, from geometry also, this is 45 degree, isosceles, hence it is isosceles triangle and hence you can say H equals to X. Anyway, tan 45 is one, so X equals to H and what about Y upon, sorry, H upon Y, H, H upon Y, H upon Y is tan 60, so root three. So Y is H by root three, right? So hence, hence, river width of river, width of river is X plus Y is equal to H plus H by root three, so H common, one plus one by root three, yeah? So multiplied by root three, you'll get H is 210, so 210 into one plus root three by three, right? So which is nothing but 210 into one plus, the value is given 1.73 by, 1.73 by three, right? So I'm writing now here, so 210, one plus 0.64, right? So 1.73 is 0. I think so, no, it's not 654, 0.5 and 315 and 23, so now 858, so 0.58 approximately, right, 1.58. So 210 into 1.58, yeah? So it is nothing but you can do a calculation. So 158 into 21, you can do, so 158 and this is 16 and 11, three, yep, so it is, this is, please write clearly. So it's eight, eight, 11, three, three, right? So hence it is 20331.8 meters, good enough for a river, okay? So I think we are done for the questions for today, I think there's nothing more, yes, so we are done with the questions for today. So if you have any doubt guys, you can always, please type to me maybe or 331, I think it is 331 point, oh wait a minute, yeah, something on that chart, anyways, nevermind. So if you have any doubt, you can always reach out to me through WhatsApp or any other mechanism, you can post it here also. I think we made a mistake in question number seven, last question where the answer is 160 meters and we got 120, is it? So let me, I don't know if this will come back but this answer is correct, no, it will not be. So just check the question number, the previous question which we did, let me just check once again. The answer says it is 71, it is 160 meters, let us first go once clearly, you will just solve once again, previous question. Yeah, this was, you know, top of a chimney from the foot of a tower is 60 degrees, angle of elevation, right? So this was the chimney, the angle of elevation top of a chimney from the foot of a tower is 60 degrees. So this is 60 degrees and the angle of depression of the foot of the chimney, depression, angle of depression of the foot of the chimney from the top of the tower, let's say this is a tower and this is the angle of depression is 30 degrees. So this is 30 degrees, so this is, if this is 30, so this is 30. If the height of tower is 40 meters, so this edge is 40, right? The height of, find the height of the chimney. So H has to be calculated. Yes, so tan of 30 is one by root three is H upon, let's say X, this one is X, okay? And tan 60 is root three is equal to capital H upon X. So if you divide both sides, one by root three, divide by root three, so one by three is equal to H upon H. Why is it one by four? I don't know why it's coming out to be 160 meters in the answer, anyways, please check. I think if you have, if you think we have made a mistake then please rectify it. So theory wise, it looks correct. I don't think it is wrong. So H comes, is coming out to be three H. What value, find the height of the chimney, yes. There should not be any factor of four, I think so. H upon X is one by root three and yeah, I don't see much of a problem here. Guys, do you think this question was solved wrongly? Angle of elevation of top of the chimney from top of a tower is 60 degree. Oh, I see, yes, yes, yes, we made a basic, no, okay, okay, okay, okay. So the thing is in this question, it is fine. In actual question, this is fine, but in the actual question from the foot of a tower is 60 degree, instead of this here, it was top of a tower in the actual question. But for this question, we have done it correctly, no problem at all. Okay, so I hope this session was useful for you guys in case, again, as we keep saying, if you have any trouble, you can always reach out to us and keep working hard, keep solving all the question, please attend all the mocks which we are conducting and for instance, our people anyways, the corrected copies are lying at our center, so please collect it. And we just need a time for distribution to RGNR people as well. So I hope this session was useful for you. So let's call it a day and we will be back again with rest of the revision as time progresses. So thanks a lot, guys, and nice weekend, don't waste your time and happy Republic Day to you. Bye, signing off.