GCSE Maths Free Tuition Lesson - Sine Rule

anymore foreign foreign thank you okay welcome to the next lesson I think this is the fourth lesson for the GCSE maths content so last week we covered kind of basics of trigonometry so that's when you have a right angled triangle and you want to work out an angle or a side and you get given either two sides or maybe again angle on the side okay and I was using sohcahtoa then we did a lot of now sometimes and this is actually most of the time you get a triangle that isn't a right angle triangle okay so not right angle triangles okay and what we're going to be doing this week and next week is covering exactly how we can work with non-right angle triangles now if they're isosceles so you can spin down the middle and get two right angle triangles but again not all triangles are isosceles either so we have to think of something that we need to use there so today what we'll cover is something called the sine rule okay this is one of two rules that we're going to be using to work out sides and angles and the sine rule is kinda fancy what it states is that sine a over a is equal to sine B over B which is equal to sine C over C and the top three are all angles okay so if you want to work out an angle this is the one to use and you can rearrange this for sides too but some people prefer to also remember the side version so this is the angle version again you don't need to remember both but if you want to work out the sides instead you just flip the whole thing on its head so a over sine a equals B over sine B equals c over sine C okay so again you don't need to remember the two different versions just remember one version and it's literally sine of an angle over the side it was that so you might be thinking what on Earth am I talking about well let's draw a generic non-right angle triangle okay so this is our triangle here in fact let me draw that a little bit better make it bigger okay that's not a right angle triangle it's just a generic triangle okay now the question is what are angles a b and c it doesn't matter okay you can just label them whatever you want okay so I can label this one a this one b this one C doesn't make a difference okay so you might think well what's the point in labeling them well I'm actually going to show you a bit later on I would always recommend using the letters A and B Because and just write them as the sides or angles that you have okay now what does matter now is where you put the lowercase letters a b and c okay so capital A is always opposite side a capital B is always opposite side B and capital c is always opposite side C that matters everything else doesn't so if you label this b a c instead so rejig all the letters around as long as the capital letter is opposite the lowercase letter you're fine okay so you can easily label it backwards if you want okay so Swap all of the B's and C's around and then all the A's and B's around you still get something that works perfectly fine okay so you might think well how the hell do I use this in an actual calculation then so let's start going through it okay so this might look complicated now but watch okay the important thing that the sine rule says is that the sine of an angle divided by the side that's opposite it equals the sine of another angle divided by the side that's opposite so all you really need is you need one angle side pair and then plus one angle or side and you want to work out one and you want to work what one of those out is so if they give you the angle you're trying to work out the side if you're trying to work out the side you'll get given the angle okay but other than that you need one pair okay so an angle there's opposite a side both of which you know then you either want an angle or a side and you want to work out the opposite one so let's go through an example okay so I'm looking at this now I do have a pair I have 13 and 100. they're opposite each other so I have a pair so why don't I label this angle a and this side a yeah I have another angle so let's label that b again it doesn't matter if it's B or C you could label it C if you want to and the side opposite is X that's it so now what we can do and again if you want you can use the side version or you can use the angle version so if I do this the angle version First again if you mind it doesn't matter which way you do it we have sine of 100 divided by 13 equals sine of 38 over X and they're asking me to work out what x is that's it now if you did this in terms of sides then you flip the whole thing over 13 over sine 100 again it's just the angle side pair equals x over sine 38 okay now I actually do have a calculator here so I can do the calculation for in front of you okay so I actually always use just the angle side because rearranging is second nature to me okay so for example if I were to rearrange this I would literally just get x equals 13 sine 38 over sine 100 okay but I only know that because I've rearranged about four million times in my brain okay so that's my answer but let's see how we get to that so if we go down here I want to times both sides by X to bring x to the top and I get X sine 100 over 13. equals sine 38. then our Times by 13. I get x equals 13 sine 38 sorry X sine 100 in fact let's start that bit again because that's a bit messy so we have X sine 100 equals 13 sine 38. divide by sine 100 and remember sum 100 is just a number so it doesn't actually matter and we get x equals 13 sine 38 over sine 100 and I hope that you can see that if I rearrange this equation I would get the exact same thing because all I'd need to do is times both sides by sine 38. okay so the value of sine 100 will they ask it in the GCSE so you won't ever have to kind of do it in your head or anything like that so what you'll do is use a calculator so the sine rule is going to be calculator only okay so in terms of the ones that you have to know sign 100 isn't one of them you need to know uh 45 and 1690 okay yeah perfect so either way you want to rearrange it is perfectly fine okay so you can either do it the sideway or the angle way it doesn't make a difference but now if we go to the calculator all we're going to do is enter that exactly as it looks okay oh now player did you want me to repeat anything or did you manage to catch everything okay no worries cool I've entered this exactly into my calculator I press equals and I get 8.127 blah blah blah chances are you're going to get a long decimal so yeah no worries that's fine it's also you get a long decimal so I would always round to three significant figures which is 8.13 thank you very much I appreciate that 8.13 centimeters okay and again this is the three significant figures okay so we did get a much larger um decimal a longer decimal not larger but what I've done is uh kind of rounded it okay so you might think this is quite a lot but this is three marks and keep in mind I did manage to get to this step almost immediately again I don't expect you to be able to do that just yet but by the time of your exam I guarantee you will be because you would have done 100 of these questions okay but this is a calculator only topic so uh I think we should just give it another go then so if we look at the next question again this is a triangle it's not a right angle triangle um question how do we know if it is a right angle triangle or not well if they don't tell you then you can't assume unless you can prove it okay so for example if I could take both of these away from 180 and it was 90 then I can but it's not so again do we have a side and angle pair well I have this five meters which is opposite the 53 degrees so let's label it a opposite side a so you might be thinking well what if we just use the letters they've given us doesn't matter if you want to use the letters they give you totally fine so we can change this to a c because the letters don't mean anything the only thing that matters is you have an opposite side okay and the other angle we have again I'm going to call it B because it really doesn't matter and the side opposite is lowercase b which happens to be the thing that they want us to work out so this is perfect so all we need to do is again use our formula we're working out aside so if you want to rearrange the whole thing or just use the side version it's up to you so again I'd write the side version over here it is quicker to rearrange but some people don't like memorizing multiple formulas I don't blame them so a over sine a equals B over sine B but I'm going to rearrange it as well just for those of you that might prefer just learning one formula so we have sine a for a equals sine B over B and again we're looking for B so for this side all we do is times both sides by sine capital B and we get B is equal to lowercase a sine Big B over sine a also if you prefer substituting in first as well that's also perfectly fine it doesn't really make a difference for this one again with times both sides by B and we get B sine capital A over lowercase a equals sine capital B then I'm going to times both sides by a to get rid of the fractions because we will hate fractions B sine capital A equals lowercase a sine capital B lastly divide by sign a and as you can see if I draw this back up the top we get the exact same formula as what we got with the um other method okay so you might have noticed that I don't really use C and sine C despite it being in the formula all it really means is that all three pairs are equal I've never seen a case where you need to use all three so realistically you just need to know a and b anyway now to do the substitution a is five meters in this case uh sign B again make sure you don't get the angles confused because it's very easy to do that it's 42 degrees in this case and a is 53. with your calculators as well um some students get the wrong answer and they're not sure why so if I make this a bit bigger if I don't make it a lot bigger if you see this capital D that's highlighted that means you are in degrees mode which you should be okay if you don't see this capital D you're going to get the wrong answer whenever you use trigonometry okay and if you want to kind of work kind of get back to normal you need to just change the the units so you press shift menu and if you see number two angle unit make sure you press number one for degrees the other two things so radians are what you use in a level so you don't need to worry about that yet gradients is something that only is in the military so you probably won't ever use that yeah just press number one and we're definitely in degrees mode and that D should be highlighted okay so now what I'm going to do is literally type in exactly what I've written out and again this is why I like having working out five actually I make the calculator a bit bigger it might be hard to see sign 42 close bracket of assign 53. and again I don't need to think about it I'm just putting the numbers in 4.1892 blah blah blah so let's again do it to three significant figures and again we need to use units it's meters in this case um so a question that you might have is how do you know how many single different figures to do it to well in this case they give me the number of significant figures they say three so do it to three sometimes they ask you for two decimal places so don't just always do three significant figures because you'll drop a Mark if you don't have the right number just double check the question because again if they say two two decimal places and you do three significant figures you can drop a mark um if they don't say anything three significant figures that's it perfect so so far I don't think this is uh particularly difficult I think this is okay um but keep in mind we're going on to the cosine rule next and then uh doing the areas of triangles you may have to use all of these together at once so be really really sure you're okay with this before we get onto the more complicated stuff okay so again sign rule is the easiest one so if you're not sure then please let me know because um cosign rule is a lot more difficult not to scare you thank you very much for joining in play I appreciate it okay so next we have again an angle side pair so what we need to do here is label it again I'm going to label it a because it doesn't matter so a and then the opposite side has to also be a but in this case lowercase the angle is something we don't know it but it's what we need to work out okay so I'm going to label that b and the opposite side is 5.4 centimeters again please make sure that you are looking at the units you drop them up for units as well they're very very picky okay we do now we do the same thing we have sine a okay equals sine B but you might be uh happy to know that you don't actually need to rearrange this much because look at what we want to find out we want to find out capital B which is already on top so it's a bit easier to rearrange times both sides by B and you're done you have sign B equals lowercase b sine capital A over lower case a good now here's the interesting bit when you work out an angle if I look at my left hand side I don't just have an angle okay what I have is sine B which is not an angle okay B is an angle so I need to get rid of sine and the way you do that is you do the sine inverse okay I'll explain what this is in just two seconds but let me write this out when I inverse sign something that has signed it just becomes B then I do sine to the minus 1 big bracket B sine a over a so you may uh look at this and think wow this is a bit complicated that's good right when you have these big math equations you look smart and you sound smart so if you show your parents an equation like this people will go oh my God or your parents will it's not hard not hard at all I'll show you exactly how to do it okay so we have b equals now let's substitute those numbers in inverse sine lowercase b is 5.4 then sine of big a which is 95 degrees and again I'm going to put those in Brackets too over lowercase a which is 6.7 and I shoved that in my calculator exactly as it looks so fraction button actually before that so Press On refresh your calculator we need to do this sine inverse because it's the thing outermost thing so if you look where it says sign again I'll make this calculator a wee bit bigger oh God there we are you have sine cos and tan sine should be a different color the sine to the minus one bit the way you get that is you press the shift button and sign so the reason why so if you have a different calculator you might be panicking don't worry this sine to the minus one is yellow so I pressed the yellow button if it's red then I press the red button if it's some other color you press the button that corresponds to that color and it'll always be at the top okay so in this case it's in yellow so I pressed shift that's literally as difficult as it gets okay so then all we need to do is fill in the rest of this okay so fraction button because that's the next bit then you put the 5.4 sine 95 so for those of you that have just joined um we're just doing the sign rule if you want me to go over it I'll go over this question one more time um but if you're okay then just let me know go down 6.7 and then close bracket equals done 53.4 and again three significant figures so 53.4 the units are degrees okay and I hope that that was not too shabby it's not too bad once you get the hang of it price and again I'm gonna write three significant figures just to make it seem okay so guess what we're doing now we're doing the exact same thing again okay so again I know it might be a bit tedious now but practice is key if you understand it now that's brilliant but keep in mind if you then came across a question in a month would you remember it what about two months so everyone that's watching this now probably has at least eight months until their exam so just be conscious of what you're going to actually remember in at least eight months so make sure you're practicing this regularly okay or again just do a thousand questions now and you never have to do it again except in your exam of course cool so once again label up the side so this is the one we have so I'm going to call it a this is the side we have something called lowercase a the angle we want let's call capital B so we want this called clock SB nice and easy right work out the size of the angle okay so again let's do sine a over a equals sine B over B multiply both sides by B and we get that sine B equals lowercase b sine Big A all over lowercase a once again we're looking for an angle so let us inverse sine both sides to get minus one B lowercase b sine capital a over lowercase a and we're done now we just need to sub in the numbers okay B is 10. not too bad sine of 60 locations a is 15 so then B is just equal to well let's work out what B is equal to by doing it on our calculator so I could cheat and just edit what's in here but given that it's unlikely you'll get two sine rules in a row I'm going to reset it so again shift sign inverse sign I'm again I'm working outwards in okay fraction button because that's the next bit 10 sine 60 almost 61. over 15. bracket 35.3 because it's rounding to three significant figures so 35.3 okay so even though this rule looks really complicated at the beginning I hope you realize oh you're starting to realize that it's actually not as bad as it might seem and this is true for a lot of maths okay maths can seem very intimidating at first and scary but it's designed to be at least shortcuts to working stuff out okay I mean it may not seem that way and I get that but once you do a bit of practice everything does become easier including maths okay and working out a length again HC here we go right ah yes I've already seen something very interesting in this question very interesting indeed so we're looking for the length AC which is this one here I'm going to label it X for now okay now you might notice we don't have the angle opposite right we don't have this one but can we work it out I think we can so let's label this angle a and this side a I'm going to label this angle B which is the same as what the question says and this side B angle B I can work out okay because all I so the angles in the Triangle again I'm writing like a caveman add to 180 degrees we have two of the angles already so angle B should be 180 minus 20 minus 55. so 180 minus 75 which is 105 degrees if you want you can add it to your diagram if it makes you if it helps you visualize and I'm just looking for the opposite side we've done this a thousand times already today well about five but you know so that should be a okay cool so now whoops I'm going to do sine a over a again and once again you can use the method for working outside sides okay that's completely fine I won't judge you okay and then we can rearrange it for B so if I rearrange this for br times both sides by B and again to do that on this side I just times both sides by sine B times both sides by a divide both sides by sine a b equals eight sine B oops that was a dreadful mistake over sign a so again you might wonder why would you not just use just use the method for remembering two equations but some people don't like doing it and that's totally fine it's up to you I don't think rearranging is that difficult because I can go straight from here to here because if you notice and go back through the video if you don't notice this yet but it actually has the same pattern every time okay so it's a so it's the the side the other side times the same angle over the other angle if that makes sense okay so now what does b equals so b equals side a was 12 centimeters angle B if I remember rightly was 105 degrees and angle a was 55 degrees so now we just shove this all into our calculators 12 sine 105 close bracket sine 55. 14 does it say one decimal place see they changed it although in this case three significant figures in one decimal place would give you the same answer you get 14.2 centimeters remember the units they're very um about it a very intense we don't like it not bad so far making extremely good time for the questions that I have for us but again I encourage you to go and do some questions yourself if you're not sure where to get questions from you have places like math Genie uh Pixie maths as well Corbett maths all of these places are very good or you can just start looking through past papers getting yourself familiar with those okay so if you could go through one second I need some water okay fun times we want to work out the size of angle BAC so I want to label that first if you remember the angle is always at the middle letter so it's this angle here always the reason why there's a BNC there is if you look this angle is made from side ba and AC so that's why it has ba AC in there okay and we need to work out the size of the angle perfect so again we can call this angle a this side a this angle b side B if you call it what I call it C you can really doesn't make a difference and work at an angle which is quite nice so sine a we have a sine B for B and we want angle a so just times both sides by little a we have sine of angle a the Big A equals a sine capital B over lowercase b and again since we work out an angle what we want to do is inverse sine it so we have a equals inverse sine a sine B oh God I don't know I'm struggling to write now let's sub in the numbers a is 20 and it's in meters this time sine of angle B which is 43. over 14. and it says to three Sig fig so let's do this inverse sine is shift sine inverse causes shift cause an inverse 10 is shift 10 shockingly 20 sine 40 whoops three over fourteen 76.97 oh okay okay so this would be 77.0 okay so the reason why that's important it's because we need to show we actually did it 3sf so we put the point zero in there because we actually rounded up to the next unit fair enough and that is everything I have for today so again it might seem like a shorter lesson technically we were supposed to do it last week but um it took a lot longer to do the sohcahtoa stuff um but I'll give you a few moments to ask any questions you may have those of you that are left in here and in the meantime I just want to say that again these lessons are live okay and a few of you are making use of the chat function so to actually ask me questions um don't underestimate how important that is okay so some of you when you're in math class you might realize that you don't you don't always get the chance to speak to your teacher um with this you basically have unlimited access to me I can see the question I can answer it immediately you don't have to wait in line so to speak um but again these are taking place every Tuesday from 6 p.m to 7 P.M between those hours so it should be up to an hour long but this one's only been 30 minutes okay but if there are no other questions then I'm happy to end it here okay sounds good well then I will see you guys in the next one I hope you have a truly wonderful evening and rest of the week