GCSE MATHS: Cosine Rule and Area of Triangles LIVE LESSON 5 EDITED

hey guys just before we get into the lesson I just want to let you know that you can catch these Live on YouTube uh every Tuesday from 6 p.m until 7 P.M in the UK time and that means you can be more interactive you can ask your questions and they'll be answered okay so without further Ado let's get into the lesson okay hello welcome guys uh to the fifth lesson in my kind of GCSE maths free tuition live lesson things so over the last two weeks we've actually been doing a lot of trigonometry and uh some of you may be glad to hear this is actually the last lesson of trigonometry we're going to do so today we're going to cover the cosine raw and the area of any triangle okay so in the first week we did sohattan which is just for right angled triangles then we said what about when we don't have right angle triangles and we introduce the sign War now we have the cosine rule which is a little bit harder and also we need to be able to work out the error of these triangles so to start us off with we're going to just focus on the cosine rule now let's first draw out a generic looking triangle okay oh God okay perfect triangle and if you remember we can label the angles inside to a b and c and it doesn't matter so long as the correct letters are opposite each other so for example I can call this a and that's totally fine but and I can call this B if I want to totally fine I can call this C now when we label these sides we have to make sure that the letters actually match up okay so if I zoom in a little bit here as you can see the C the capital c is opposite the lowercase C and the capital B is opposite the lowest B and the capital A is opposite the lowercase a so other than that it doesn't actually matter I can swap all of these letters around and so long as the same letters are opposite each other it's fine now the cosine rule is a bit trickier and it basically says that a squared is equal to B squared plus c squared minus 2 BC cause hey yeah pretty big and the good news is is that you for most examples you do get this on your formula sheet if you're unsure if your example does then just look up your formula sheet whenever you can and basically you don't need to remember this that's pretty good okay so if we have a look at what this is actually saying it's saying that if I take one side so a and I Square it that's equal to both of the other sides added together and squared so squared and added together minus two times those sides times COS of the angle between them okay so this would be useful if you want to work out so if you have two sides plus angle between and you want the third side which is also going to be the side opposite the angle right because it has to be okay so now you should be able to realize that I can actually rewrite this formula for both of the other sides now right but keep in mind the actual letters don't matter right so if I wanted to find out let's say I wanted to find out this side here instead of rewriting the whole formula I can just call this a and change this angle to a and so on right there's nothing stopping me doing that there's something that you might want to consider is the fact that you know all it's saying is the side opposite the angle is equal to both of the other sides added and squared minus two times both those sides times the angle between them okay so the actual letters don't matter that much okay one thing you might realize is that you might actually want to work out an angle so another way you want to use this formula is to work out an angle when you have three sides okay now this does get a bit tricky okay so if I rearrange this for cos a which is the angle then what I want to do first is move this thing over to the other side so just flip the um sign so a squared plus 2bc cos a equals B squared plus c squared so again you don't need this middle bit minus a squared from both sides gives me 2bc cosine a equals B squared plus c squared minus a squared and this is where it gets the order is going to matter divide both sides by 2bc and you get cos a equals B squared plus c squared minus a squared over you see and this is when you have three sides want an angle now you have to be really careful because if you look B and C are interchangeable because B squared plus c squared is the same as c squared plus b squared doesn't matter right and likewise 2 times b c is the same as 2 times C beam and vice versa but the angle you're working out is the one opposite to this negative side whoops okay so in this case it would be this angle because this side is negative okay just be very careful of that now I'm going to show we're going to do some practice right now I'm going to show you it's really not that bad the equation's long but like I said earlier in most exam boards in virtually all of them you get given this formula and you can just check your formula sheet to see if it is on there okay so you don't really need to worry about remembering it the thing I want you to actually learn is the fact that the angle that you have has to be opposite the side and then just label that a and you're done everything else falls into place so hopefully that makes sense and what I'm going to do is actually do some questions with you now okay so let's go down and look for the first exam question so we have here we have an angle and we get the sides that make up that angle good we can use the cosine rule okay so again with questions like this you're always going to ask yourself are you going to use the sine rule or the cosine rule and honestly it's up to you you can normally use both so in this side in this case sorry I have two sides the angle between them so it implies cosine rule and it says to work out the value of x so what I'm going to do is say well this is my angle so I'm going to label it capital A which makes this side lowercase a that makes me happy because the formula here has a equals a squared equals technically but a equals right now it doesn't matter why I label the other two I can call this one b in this one C or the other or the other way around because it doesn't matter okay so what we're going to do here is we're going to say that a squared is equal to B squared 13 squared plus c squared 10 squared minus 2 times 13 times 10. again it really doesn't matter the audio which you do these cosine 100 okay so just as I'm running to the formula I'm using is a squared equals B squared plus c squared minus 2 b c cause capital A okay so all we need to do is use a calculator now uh just as a um something that might hopefully uh soften the blow a bit this only comes up on calculator okay and the reason for that is it's a very complicated formula they're not that mean okay so we have 13 squared plus 10 squared and again you just enter it exactly as I've entered it minus two which also means using um brackets as well just to be 100 sure you're gonna get the right answer and then cos I was about to type because there that's a bit embarrassing 100 and again remember if I make this bigger the one thing you want to check is that this capital D is highlighted it means you're in degrees if not shift menu angle unit degrees number one easy peasy so if I make this smaller again because that's a bit ridiculous we don't need that big equals I mean 314.148 oh all right so I'm going to write a squared and this is the really important bit this is a mistake that I see a few students make is they leave this as their answer because they just write equals 314.14 if you look that's what a squared equals so a is actually equal to the square root of 314.14. okay so just keep that in mind now on your calculator instead of having to write that whole number in I'm going to show you a bit of a trick so if I just press square root there's a button next to the equal sign that says A and S okay now ANS is really really important and useful to know right what that stands for is answer and what it does is it stores whatever you previously had after you pressed equals so in this case my ANS is now 314.14 I think it was eight blah blah so there's equals that is the same thing let's do it to one decimal place so 17.7 centimeters and again remember your units and to me and that's the one DP okay now just to prove it right uh prove the point is this A and S is now equal to 17.72423556 so reference equals again is square root standard plus equals getting square roots that square root square root squared and then so on okay so it's a useful thing to have for iteration as well which tends to come up uh towards the end of paper three normally and paper two but anyway that's another topic for another time so let's get on to question two and see if we can have a look well now it's labeled a b and c for us right but I don't care about their labeling okay I have an angle and the sides that make it up right so I'm going to label this focus a I'm going to label this B in this C I don't care about the way they've labeled it however I've actually matched it up uh subconsciously I guess and that works fine we have an angle the two sides that make up that angle and we want the opposite angle BC right so we're 100 cool so once again we have a squared equals B squared so I'm going to make that 7 squared plus c squared 5 squared minus 2 7 5 because it's 2 BC cos 42. okay and then I'm gonna write a squared equals so again if you want you can do this all on one in one actually I'm going to show you how to do it all in one step right so if I were actually doing this question or I wanted to work out what the answer was I guess those are the same thing I'll do 7 squared plus 5 squared Why am I minus two uh bracket seven close bracket Open Bracket five close bracket because two right I press equals and I wouldn't write that down I'll just do square root answer equals done and it says to give it to three significant figures that's 4.69 so the reason why this is quite useful and that's the 3sf and that's in meters now the reason why this is quite useful is sometimes why I see pupils do it so if we go back that's my answer right and what they will do is they'll say Okay a squared equals 21.9 then they'll square root that and write that answer which can sometimes make this value slightly wrong right so to prove my point let's say if I was a student I'd probably write 21.9 so if I write the square root of 21.9 I actually don't know if this will be slightly different ah there we are so to three significant figures I get 4.68 instead of 4.69 okay normally doesn't matter but again it's just showing you that the answer is technically slightly different because you're rounding okay so let's just keep going uh we've got quite a lot to get through so let's just Hammer through as much as we can perfect there we are and we have a different style of question so what do I want to label a b and c in this case um well if I don't know the angle and I know all three sides I want the side opposite the angle to be a which makes the angle a okay now I you may have seen online kind of three cosine rule formulas where what they do is they change this to eight from a squared to B squared they change all the letters around right which might be why you might be slightly scared it's not necessary I've never seen it really you the fact that you need it but if you do all they're doing is swapping the letters around so if I make this B squared equals all I do is I've replaced a with B so I replace a with b and then C squared stays exactly the same so it doesn't do anything right it doesn't make a difference really all I'm doing is replacing the letters but I personally have never seen a question where that's required to to do oh sorry voicemail anyways so I want us I want an angle instead of a side so what you can do is you can either memorize this formula which they don't give you always or derive it so rearrange the expression it's not too bad to rearrange okay so I did the rearranging slightly earlier if you are interested but if you want to remember the formula no harm no foul so again let's sub our values in so we have 15 squared and this it does matter which one goes first right because we want to make sure we're subtracting a we have to subtract a otherwise we're going to get something um hideously wrong and 2 15 and 16. okay now bear in mind this gives me cos a it doesn't give me a which is the angle cos a is not the angle okay so you can do one of two things you can work out what this is write equals some answer and then do a is equal to cos inverse that answer that's fine you get the same answer all good and Hood or what you can do is be really fancy and do the whole thing in one line like some kind of mad scientist okay like that it will give you the same answer okay so if I do this in steps again if you're braver than I am you're more than welcome to uh do it's a 15 squared plus 16 squared minus 26 squared over 2 15 and 16 right so again equals that gives me that cool then remember I can use the answer button which is exactly what I wrote here so inverse cos remember when we're trying to find an angle we always have to inverse cos and that's the yellow text above the COS button so to get the yellow text I press the yellow button yeah fairly fairly um self-explanatory and oh God this is way too huge and then just press answer close bracket done DZ 80.28 blah blah blah three significant figures means it's 80.3 degrees again remember your units so you may notice that these questions are all roughly three marks right so three three so far of working out and that's good you get method marks nice and easy but I could do the whole thing on my calculator in one line and get three marks so although the cosine always tends to be something very very difficult it's not that bad once you practice with it and in my opinion it's easy marks okay once you know the kind of setting of it perfect okay so a few people have joined late and a few people have also left um so if you have missed anything please just let me know in the chat oh good question yeah so where it turns up so the way your paper is structured is that the last few questions will tend to be a kind of unseen questions okay so questions that they can kind of throw something weird at you so this can come up this kind of question here that I'm doing right now is going to be roughly the middle of the paper of the high paper however I have a question later on which is the very end of the higher paper and it's one of those kind of unseen parts so 10 of your marks needs to be unseen so where they throw you in a situation that you haven't seen before okay so to give you a little preview of that it would be a question like this towards the end but something that's right at the end is something like this this is literally a five mark question it would be literally the last question or two okay but the ones I'm doing at the moment roughly the middle yeah we're going to get into some uh more interesting ones in a second okay so question six so they give us three sides so that means we have to use the cosine rule because we don't have any angles for the sine rule and they want us to work out the size of angle ABC so if you remember the second letter is the one where the angle actually is so we're looking at this angle here and they want us to work out a b c so that's this angle here now they've labeled it B I'm going to label it a because they don't make the rules I do so my side opposite is here B and C really doesn't matter which way around you do it we want the angle so again we can just use the formula we've either remembered or derived if you want me to go through how we got to this formula from the cosine rule then please just let me know and that's over sorry this is actually incorrect I was writing the cosine rule formula for a second so it should be B squared plus c squared minus a squared over to b c I apologize for that and if we sub in the numbers now we have 5 squared plus 9 squared minus 10 squared for two five nine that's it and then our angle is going to just be what we type it in okay so hopefully that's not too bad um again do let me know if you're finding this tricky but this is kind of the layout that you're expecting so shift cause to get the investing and I'm going to do it all in one step like like an absolute madman so plus 9 squared minus 10 squared stop for some water in a second to five nine I was back here so I'm just going to show you that it does work if you do the whole thing in one step 86 and it's to the nearest degree such as 86 degrees okay and that's it that's three marks so it should take you so the examiner expects you to take up to three minutes to do this question okay so again take your time figure out where you want to put things because honestly it is kind of up to you at the end of the day for one second let's get some water foreign just to run something by you guys I suspect some of you are making notes or you're kind of re-watching the videos to kind of go through the answers so I can actually make all of these PDFs available so I can change these into PDFs and you can download them and print them or look at them or whatever thing is I was looking at a way to kind of distribute them so one option was Google drive but it's a bit messy so I am in the process of making a Discord channel so if you guys would like that then just let me know and I'll kind of make sure I get it done before next week and what I do is I put all the PDFs for all of the lesson content on there so you can redo the questions or gonna do that okay easy exactly not too bad right again there is a question later on that I definitely want to go through because it is tricky um it's a five marker okay once again calculate the value of x so I'm going to make X my a because it means I don't have to worry about doing anything else and that means the other two sides can be labeled whatever the hell I want so once again we have cos a again let's rinse and repeat okay so B squared plus c squared again make sure I don't make that mistake again minus 24 squared over 2bc so 221 13 then a will just equal whatever the inverse COS of that is so I'm going to do again another tip is just to go through and just change the numbers like this so 21 of course that would be 13. this one is 24. that's in oops and 21. uh that looks correct perfect and it says to one decimal place 86.4 degrees three marks remembering all the cycle theorems yeah that's a bit tricky uh one thing you can do is actually practice using them in questions so if you find every single question that has a circle in it in the mark scheme you'll see that there are Circle theorems referenced so those are the big ones so practice is the best way to do it for the ones that you don't really apply in questions it's just a bit of memory work so what I like to do is write out all of the circle theorems and then write them out again and again okay if you do that 10 times you'll remember them forever but mainly if you just use them so you use most of them in the actual maths uh practice that we're going to do perfect okay how many more of these do I have okay so after this we do have a somewhat interesting question so again I suspect some of you might be even be even be getting bored BAC well that is this angle right here so I'm going to label that a because I want to so the opposite side has to be lowercase a and then B is going to be here B and C 100 on every test that's practice right the more you do maths the less likely you are to make mistakes so just practice as much as your math teacher does or has and again work out the size of the angle so a is equal to again I'm going to write this as the inverse Causeway just for the people that are doing it that way two sides are adjacent to it so 5.2 squared plus 7.3 squared minus 6.9 squared all over 2 and then again times the adjacent sides seven three all done and all we have to do is type the whole thing in so I think sweet so on the bottom we have 7.3 5.2 on top we have so the half a b sine C formula is one they're actually about to come on to that works at the area of triangles okay so what this is used for specifically is working out the um angles or sides in a triangle [Music] yeah equals and to three significant figures so 64.5 degrees there we are and that is to 3s all done okay we're making good time we do have to go through one more formula but we still have over half an hour left so it should be okay and I've got to close that bracket that's embarrassing okay so far there any questions at all perfect sounds good perfect okay glad to hear it so again this one is slightly more interesting because we have a four-sided shape split into two triangles okay and we have to work out what the angle X is so bit of a problem right because the angle X is in this other triangle here okay so here's what we should do in this case actually let me just turn on the light it's a bit better make it a bit easier to see anyways I really don't think that's improved it much but whatever so we have to do here is this is a right angle triangle which means we're going to be using sohcahtoa I've got the H in cat whoops socatoa for this right-handed right angled triangle on the right hand side okay the problem is that to work out the angle using socato we need two sides we only have one but if you look at this triangle over here we can use the cosine rule to find out what this side is over here okay so if I label this angle a this side a we're going to work out what a is okay so let's get started we have a squared equals the two other sides squared and added so 10 squared plus 13 squared minus 2 times those same sides so 10 and 13. COS of 40. okay and that is going to give us our value after we square root it of course okay so again go through it again 10 squared now you might be wondering how accurate do we need our value and honestly three significant figures is fine but using the answer button you can actually keep it exactly right so I'll show you how to do it that way as well the answer won't differ by much and so 69.82 don't forget square root the answer right because we want it to be a so what we can do is make this 8.36 right and that's going to give us a pretty good answer okay because we only need to go to one decimal place anyway but this is now saved as the answer button so you can also use that okay but let's do a little bit of recap on two weeks ago the lesson we want this angle here this is trigonometry so specifically we're doing the cosine rule and we're about to come on to doing the areas of triangles okay but the overall topic is trigonometry okay we need to label up this triangle so the side opposite the angle we want so if I draw a little guideline going here that's our opposite side opposite the right angle is always the hypotenuse and the other side is the adjacent side okay so if we look at this let me rewrite this out just to make it a bit more clearer so okay we have the opposite side and the adjacent side and we want the angle so we're looking for opposite and adjacent that's using the Torah part so we're going to be using tan theta equals opposite over adjacent so in this case it's 8.36 over 5. okay now because we want the angle not tan of the angle we're going to also do the inverse 10 okay so we can call it X if we want to tan inverse 8.36 now on your calculators you can just write 8.36 over 5 or you can just do shift 10 answer button over five you get the same answer okay to one decimal place 59.1 degrees that's it yeah absolutely so just the letters this formula is cosine Raw it's a squared equals B squared plus c squared minus 2bc cos a again you don't need to remember this formula it's given in your formula sheets um but it's always faster more efficient to just remember the formula the best way to remember is just to keep using it in the questions so what I used to do is literally write this formula out every time I used it uh growth and decay the maths topic or if it's the math topic and you're talking about the kind of like interest and depreciation that kind of thing I can definitely do that that's coming up soon yeah so that's like the percentage um increase over time and percentage decrease over time right so like interest and depreciation that's that's actually probably going to be the next lesson so yeah yeah no worries that's fine absolutely so for that question that was a five mark question so this question came up towards the end of the paper and we did it in two big steps it is meant to be difficult so if you found that difficult that's totally fine this if this is especially if this is the first time you're seeing this but it's the end of paper question so it's meant to be hard there any questions on that cool perfect let's keep going then so this is probably the hardest cosine rule question I've seen a hold of the abstract as in the ones that are like the Unseen parts there's not that many of them but if you go to the end of every single paper and take those last few questions those are the abstract questions but what I would also recommend you do is go to the old spec now the old spec is basically every exam before 2015. the actual content is the same it's just they re they changed around where the um questions appear and things like that but if you do those questions as well that'll be an excellent practice but the actual content is the same the cosine rule was discovered in 2000 BC so like it hasn't changed okay it's all the same stuff anyways let's get on to this question this uh somewhat scary looking question I'll give you a second just to process how interesting it is I mean I say interesting okay work out the value of x now they've actually given me all of the sides so I'm going to label it the same way I've labeled everything before I'm going to say this is a okay and the opposite side is going to be a it doesn't matter which one b or c is that's what I'm going to start with because it's the same way I've started every other cosine rule question just as a little side with these abstract questions just writing out formulas get to a couple of marks an attempt gets you marks okay but you'll find that you can work out what to do so I'm going to do now is I don't know what to do right that's confusing so I'm just going to write out the cosine rule okay and just as a reminder the cosine row is a squared equals B squared plus c squared minus 2bc cos a okay I'm just gonna write now I'm going to sub it in right this actually gives me one mark the second mark can actually come from subbing in the values so we have square root 73 oh my God I'll rewrite this with better handwriting I promise B squared says X plus 2 squared plus c squared which is 2x minus 3 squared minus 2 B which is X plus two C two x minus three cos 60. now this gives me a mark already and all I've done no maths right all I've done is substitute into a formula but I've got marks even if I don't know what I'm doing okay so that's a very key thing I want you guys to remember okay now what is the square root of 73 squared well if you're unsure by the way this question it could have appeared in the non-calculator paper it appeared in the calculator one so don't worry but if you won't actually need to use I'll show you you won't need to use the actual trig functions so whenever we have a square root of seven of a number and we Square it it just gives us the number so as proof this is just going to give me 73. I'm hoping that's somewhat familiar to you so I have 73 right now we have this bit here which is two times a bunch of algebra times cos 60. now hopefully you know that cos 60 is equal to a half so if I do 2 times cos 60 what's 2 times a half it's one so two cos 60 just equals one so in other words I can just get rid of that I'm gonna give you a few seconds I'm going to do this question very slowly just to make sure you're following each step I think that's okay yeah perfect so um I can expand these brackets in my head but given that you guys aren't near your exams yet you probably haven't had that much practice but if you do X plus 2 squared Let's uh work all of this out okay so X plus 2 squared that's the same as writing X plus 2 X plus two use foil right do you all know what foil is we have first outside last if you look at this first thing it spells out the word foil yeah good so if we look at the first two terms x times x gives me x squared the outside terms is 2x plus 2X plus 4 right outside inside loss so we have x squared plus 4X plus four so I'm going to do is work out all of these yeah perfect why don't you be nice I'm going to work out all of these individual brackets and then just sub the whole thing in again so 2x minus 3 . again if you can do this in your head you're more than welcome to but just for those of you that might not be able to 2 times 2 is 4 x times x is x squared minus 6X minus 6x Plus 9. again please make sure you're checking my work by the way it's very easy to make a mistake when you're trying to explain stuff and lastly we have that will give me 2x squared outside we're giving minus 3x 4X 6 that's 2x squared plus x i oh I think you can tell how much algebra is going to be in this question right so if I write the whole expression out now after I've expanded and simplified everything I have 73 on the left hand side equals X plus 2 squared we said was x squared plus 4X Plus 4. 2x minus 3 squared gives me 4x squared minus 12x Plus 9. and that last bit on the end gives me bear in mind there's a minus sign so I have to change the sign of all of this to make it 2 at minus 2x squared minus X plus six so just go over that one more time even though I canceled the 2 there's a negative sign so everything in this bracket that I worked out has to change sign okay so let's simplify this a wee bit and I'm wondering if you guys might be kind of coming onto an answer so let's go to the X squares first we have x squared plus 4x squared is 5X squared minus two x squared gives me 3x squared I then have uh plus 4X minus 12x which is minus 8x minus another X gives me minus 9x so one thing I like to do by the way when I'm simplifying is actually cross through the ones that I've done so then we did 4X minus 12x would give me minus 8x minus another X cool then we have four plus nine which is 13 plus 6 which is 19. oh boy okay that looks right to me so far but again you know I'm only human uh minus 73 from both sides but it's actually before I do that hopefully you're looking now that finding out the value of x just means we're solving this quadratic that's it so this unseen question is all mats we have done before but it's all brought together right we've solved quadratics before and we've done the cosine rule before but what about solving a quadratic with the cosine rule same thing so I promise you in an exam if you get these weird questions sub stuff into formulas expand brackets and see what happens and I promise you a solution will present itself you get most of the marks you actually get up to three marks for getting down to here the other two marks of solving so I get most of the marks without even answering the question okay but to solve this we need to make the equation equals zero right on one side so zero equals three x squared minus nine X and then 19 minus 73 is 54. minus 54 that is so to make this a bit easier to solve I'm going to divide the whole thing by 3 because there's zero on the other side so we have 0 equals x squared minus 3x and I know that we can divide that by three it gives us 18. and then all we need to do is shove that into brackets right so what we have here is we want two numbers that times to negative 18. and add to minus three I can think of minus six and plus three so x minus six x plus three equals zero so my values of x are x equals six or x equals minus three up to here I've now got four marks good well done nice one Evie now the question is look here it says value singular so I actually marked a student that did this question before and they got up to this point and they thought they got five marks because they found both Solutions right cool but it says value singular six good uh Evie why did you pick six so six is correct to give you a second thing about why because it's not negative good think about what x is X is the length of a side right now does it make sense for a length to be negative like if I ask you to get a ruler out and draw a line that's negative six centimeters long could you do it no so what you want to do is say x equals 6 not negative and that is probably the hardest cosine rule question um at least one of the hardest five marks all done and bear in mind this middle stuff once you guys get to your exam I guarantee you guys are going to be able to do this in your head so you'll be able to go from well where I've joined the arrow right so you could do it in basically four or five lines get five marks pretty good okay perfect yeah so the only other style of question that um you can find is something like this calculate the perimeter of the triangle okay so I'm going to go through the area and right after this it's really not that tricky is calculate the perimeter of the triangle all you need to do is work out this third side here okay so they give us the area of the triangle so let's now talk about what the area of a triangle is when it comes to a triangle that isn't a right angle one so if we have a look at just a regular triangle okay the formula for the area of a triangle is simply a half a b sine C which Evie mentioned earlier now A and B are two sides and C has to be the angle between them okay so again the letters don't really matter okay so the formula just says you want half a times B times sine of the angle between them that's it okay this question yeah it's a five mark question I want to come back to it I want some practice with this slightly on a slightly easier question okay because this question is a bit it's a bit long okay so if I go down I actually have some questions here so I'm going to rewrite area I promise I'll do that question before we finish okay half a b sine C so using that logic right A and B are the sides that make up the sign make up the angle sorry so that's how I'd label it so now my area super easy is just a half 13 12 sine of 120 degrees that's it not too bad and that will give you the area of any triangle okay so let's do it let's do a half if you want you can write 0.5 come up to you off 13. 12 sine of 120 degrees 39 root 3 but it says to one decimal place not as a third so 67.5 centimeters squared two marks because it's easy peasy lemon squeezy okay let's do another one as practice then we'll go back up to do the hard question and then we'll see how much time we have left once again my formula is just a half a b sine C A and B are just two sides C has to be the angle between them so once again area just equals a half that's another way some people write it is they write it with time signs I don't like that very much but just showing you that there are different ways of writing these things so instead of 13 we have five instead of 12 we have eight instead of 120 we have 41. three significant figures 13.1 meters squared this time so again pay attention to the units so okay for the area Lila EV is that okay I think it's not too bad thank you yeah most of the other questions stay the same yeah so let's go back and do that hard question then okay so again this is one of the um only other like really hard questions I found but you don't need the cosine rule technically you can use the cosine rule but you can also use the sign rule I'm gonna do it with the custom rule so let's have a look at the information they've given us okay they've told me and one other thing this is about exam technique okay for any maths exam you do they never give useless information every number we get given every value has to be used in the question in the answer okay we have two sides sweet we have the area that's interesting we need to calculate the perimeter so I'm going to work backwards what do I need for the perimeter I need this side here okay so how can I work out the side do I have enough information well I have two sides and nothing else I don't have enough information okay so how can I get more information well I'm going to use the only other piece of information they've given me there you go okay so the area is a half a b sine C so before we go on does that logic make sense do you see my thought process and this is how you should think in the exam okay you have to use all the information think about what you need think about how you can get it so in this case we get given that 100 equals now what angle am I going to work out ah good question a b are just the two sides that make up the the um triangle yeah the lengths Okay so as I'd say C has to be the angle between them right so in this case my C is over here so my A and B are these two sides because they're the only two lengths I've got given so I have to call the angle between them C I have to ah brilliant question well first of all if they gave you three sides you can just work out the perimeter by adding but I'm assuming you mean like if they give you like if you want something else right whichever one you want okay the brilliant thing about trigonometry is it doesn't matter which one you choose okay so for example if I if I then had let's say these two sides over here and this angle I would get the same result right I will always get the right answer so you if they give you three sides you can choose whichever one you want to discard because no matter which pair you choose you get the right thing okay in this case you know you you have two sides you need to use the angle between the two sides that's all you need to know okay really good question though so in this case we have half a is 19.7 is 15.4 and we want sine C so now I need to actually rearrange to get what I want so times both sides by 2 to get rid of the half I get 200 equals 19.7 15.4 sine C so if I rearrange this I get sine C if I divide both sides by these brackets here I get 200 over 19.7 15.4 okay and then C is going to equal whatever I say it does all okay with that this is a pretty long question by the way I think unironically very long so I need to do the sine inverse because again if I want the actual angle I need to inverse sine to get to that 200 over 19.7 0.4 1.2 I'm going to leave it to three significant figures okay and what I'm going to do is I'm going to get rid of this C now and write 41.2 I'm always going to get rid of this A and B you'll see why now I need to work out the perimeter okay so I have two sides and an angle and I need the side opposite them so this is just a regular cosine more question now I'm going to label the opposite side a and the angle a and then B and C the other two things that may have confused some of you right but keep in mind I control what sides are what okay so I can change it from C to a doesn't matter I'll give you a second just to digest what I just did and if you have any questions please let me know now and then we can go forwards okay it's okay if it doesn't make sense just let me know is it okay that I can just reassign the letters because I'm using a different formula okay okay why that's a very good question the reason why I changed it is I'm about to use the cosine rule which oh never mind uh which looks like this a squared equals B squared plus c squared minus 2bc plus a right so if that's true right in fact I'm gonna okay I think for next time I'm going to turn down the latency so there's a bit of lag by the way between um what you're saying and what I'm when I see it and then when you hear the response so I'm going to change that to make it as slow as as short as possible yeah so if I want to find a side I want to label it a that's the reason but it doesn't actually matter so now if I sub the values in 15.4 squared Plus uh 19.7 squared and doesn't really matter which way around we do this equal to 1.2 so now what we do is just square root it as well this is a long question but we're right at the end okay once we get this on so I'm going to show you where you get some of the marks just so you guys are aware because I think that's always very useful squared P Squared and again please make sure you're checking to see if I'm making any mistakes is possible because equals Square answer 12.89 that's kind of annoying what should we call it let's call it 13. okay so this opposite side is 13. so far we have earned four marks and let me give you a breakdown on where that came from one mark for substituting into the area formula yeah and another Mark for working out the angle another Mark for substituting another Mark for working out the opposite side the last Mark is working out what it tells you to work out the perimeter okay so the perimeter if you remember how to work out the perimeter you just add up all of the sides so 19.7 plus 13 Plus 15.4 and because it's a calculator paper and I'm incredibly lazy let's just add this up by hand 19.7 plus 13 plus 18.4 whoops that was close and it's just a three significant figures so 48.1 meters okay so what'd you think of that