[Music] [Music] okay so looking at trigonometry now the hint we're gonna be using trigonometry is we've got a right-angled triangle and there's an angle within them so looking at this one on the left to start with because I know we're gonna work out the missing links and these two triangles are going to start with this one on the left so got our angle of forty-three and the only thing I have to remember here is help dispel sohcahtoa so yes oh hey see a H T Oh a there we go now that's just gonna help me to know which I'm gonna use what I'm going to use sin cos Alton so the first thing I need to do is I need to label this triangle to see what sides we've got so opposite the angle that we're using that's called the opposite the longest side which is opposite the right angle is always called the hypotenuse and the final side there is called the adjacent now there's one of the sides here I don't need to be thinking about I need to use the opposite I'm looking for the hypotenuse so I don't need the adjacent so let's have a look at what formula triangle we're gonna use well oh and hate shop is it in hypotenuse is within must sign triangle there Soh we've got O&H within that triangle so I'm going to use sohcahtoa cut the parts okay I'm going to use is sign now then the side I'm looking for is the hypotenuse up here with the X next to it so hopefully you used to using formula triangles I'm gonna cover up the H okay so the Sun that I need to do to find the hypotenuse is o / s or oh / s so that's own divided by sine so to find the hypotenuse I'm going to do the opposite which is 6 and I'm gonna divide it by sine all I need to remember is to put the angle here 43 with sine so sine 43 so sine 43 tells me the relationship between the opposite and hypotenuse and in this case we're gonna do a divide oh one that's off base on the bottom don't forget to put 43 in the bracket there so many calculators so six divided by sine 43 on the calculator 6 / 9 43 is eight point seven nine centimeters I'm going to round that to two decimal places and it actually comes out as if I write a few down it's eight point seven nine seven six the question will tell you how to round it so the rounded to two decimal places at the eight point eight actually eight point eight or eight point eighty let's round it to one decimal place so we've got eight point eight centimeters for that first question now if we have a look at the one on the right so question two are very similar process here let's have a look so we still have a look at our angle label the sides so opposite that is our opposite and then we've got the hypotenuse here and again the adjacent ends are being down the bottom based on my that angle is now again we need the opposite because I've stormed by looking for we need the hypotenuse because that's given to us but we don't need the adjacent again so again I'm going to be using my sine triangle because I've got O&H I'm gonna rewrite it again so I've got this Oh H but this time we are looking for the opposite so I'm gonna do the same process again I'm gonna cross off the opposite and that tells me I'm gonna do a times I'm gonna do s time and sage now s again stands for sine I just need to put the angle with it so this time it's 36 so sine 36 making sure you put it in a bracket again times by the hypotenuse which is here which is nine so sine 36 times 9 and again typing that into the calculator sine 36 times nine gives me an answer of five points and if I round it to one decimal place again while it comes out as five point two nine so it's a one decimal place that'd be five point three centimeters and again just thinking logically here about whether your answer makes sense because it can't be longer than the hypotenuse there so it can't be longer than nine so you should know if you've done it wrong because it might end up coming out longer than nine so that's finding side lengths using sohcahtoa when it comes to finding an angle we got a very similar process I'm gonna have a look at this question it says find the sides of angle ABC so from A to B to C that gives us this angle down here and again I'm gonna follow the same process opposite that angle is over here our hypotenuse is opposite the right angle and our adjacent side is down here so it looks like I've got a different bit of sohcahtoa to use this time because this time I don't need the opposite but I do need the hypotenuse and the adjacent so again just write down sohcahtoa and we're looking for the form with a triangle this time that has to hate chin the alien the hypotenuse in the adjacent so the H and the a is in my cos triangle so I'm going to be using this one now we're looking for the angle this time so if I cover up the angle part which is the C it tells me that I need to do a over H now if we just do that a is 7 and H is 13 now finished type back to the calculator 7/13 just gives me a decimal naught point 5 3 8 now the calculator doesn't know that this time I'm looking for an angle so what I'm going to do to tell the calculating for an angle is first and then I press shift and then press cos so shift cuz and on your calculator it will come up with and let's get on my calculator here you get it saying cos minus 1 and that disturbs your calculator this time I'm looking for the angle and this is the fraction I want to put in so you type in cos minus 1 and then 7 over 13 and if you calculate us the fraction function you can put it as a fraction if not in the bracket there just put 7 divided by 13 so let's do that cause -1 7 divided by 13 or 7 over 13 is a fraction and you get the answer here 57.4 2 and if we round that's one decimal place it'd be 57 point four degrees there we go for an angle so it's the same process no matter which formula triangle you use you just have to identify the sides and then when you're looking for an angle you're always gonna press shift cause shift sign or shift um and you'll get this little symbol here cos minus one now there are a few non calculator questions and again there's more of these on the higher but two here that we really need to know and that is sine 30 and cos 60 now the value of sine 30 is 1/2 as is the value of cos 60 so they're both equal 1/2 and they're really good ones to make sure that you know for these non calculator papers okay so do remember those ones sine 30 and cos 60 and they're both equal 1/2 and let's see how we can use that so this question says given that sine 30 is 1/2 which hopefully we know anyway work how the length of a B and a B is this side here so let's label this triangle up we've got the opposite we've got the hypotenuse which is asking us to look for and this is our adjacent again now again we're not going to use the adjacent here we're just going to use onh and if we remember our little formula look s Oh H has the oh and the H M and we're looking for the hypotenuse there we go so if we're looking for the hypotenuse would cover up the H and we need to do o divided by s so s is my sine 30 so if I was to do this on the calculator I'd have to do 4 and I'd have divided it by sine 30 okay but this questions actually told us that sine 30 equals 1/2 so actually what I'm doing is for would have I did by 1/2 I'll write that as a decimal it makes it a little bit nicer to see but 4 divided by 0.5 so actually it's just asking me to do in the muscle note which is 4 divided by 1/2 and if you divide by 1/2 it doubles them so they're doubles two number so 4 divided by 1/2 is 8 demands would be 8 centimeters there we go of course that could go with the other way around as well I could have to do a times so have it not given me this 4 here maybe it had already told me that this was a then instead I would do 8 times 1/2 doing s times H so 8 times 1/2 and that would give me the 4 here that's been given as the opposite so we could have it either way okay so going up to some missing links in this triangle there's not a right-angled triangles if I'm missing lengths and angles we can use either the sine role of the cosine role and these are rules are you gonna have to remember well I've got a look at one to start with and how we know when to use it now the first thing you do is you try and identify in a triangle for first of all what we're looking for which is a B let's label that X now what I look for straight ways do I have pairs of opposites so I have this pair of opposites here and I've got both of those and then I've also got X opposite to this angle so I don't have X but it's in one of my pairs of opposites so when we've got this scenario where there's two pairs of opposites we can use the sine rule and we only ever need part of the sine rule so I'm just gonna use a over sine a equals B over sine B also equals C over sine C but we only ever use only ever need to use two of them here so let's have a look we need to label this up and I'm gonna completely ignore the letters that are on the actual triangle itself I'm gonna label this angle a which it already is and this thing on this side little a opposite that and then this angle B and then one opposite little B and I'm going to do now is stick all the numbers into the formula so let's have a look a is 12 so it's a 12 over sine the one opposite that 55 is equal to B which is our x over sine B obviously you should already know there's two for two variations of this formula we could have it flipped over so we could have sine a very equal sine B over B but this is our one for side lengths and we know that because our unknown pieces on the top so we're able to isolate this now quite easily so sorry I've written B there it should be 20 ok sorry there we are 20 so what we need to do is times both sides now by sine 20 what you could do is work this out when your calculator and times your answer by sine 20 but I'm gonna multiply it straight over so I can type it all in one go so times by sine 20 and if we do that we get 12 sine 20 it goes onto the top there over sine 55 there we go and all we have to do is type that into your calculator obviously just being careful that you put these angles in brackets some calculators are gonna need you to do that so if we cycled into the calculator again not forgetting you could just work out 12 over sine fifty-five first and then times it by sine 20 but I'm just gonna go for it like this so 12 sine 20 on this up closing your bracket over sine 55 and on the calculator just writing down what you got you got five point zero one zero three five three nine nine eight now a question would normally say how to round this so if we imagine it was two decimal places for this one it would be five point zero one and it's a length so centimeters okay so just obviously would just be careful the question says let's have a look at an idea whether we've got to find the angle okay so in this question let's have a look work out the size of angle BAC so let's identify that that is here BAC okay so we're gonna use the formula the other way round this time so sine a over a equals sine B that does not say same B sine B over B okay so plugging in our numbers let's just label it up so let's call this one a as the ADEs next to it that's fine and again I'm just gonna drive over this one I'm just gonna put b and b okay just because the way I've written my formula so then sticking in all the numbers what have we got sine a is sine X so we have sine X over 20 equaling sine 43 over 14 okay so exactly the same approaches we did before we can isolate the sine X by x in both sides by 20 and again you could work that right hand side out and times by 20 but I'm just gonna go stick it up the top there so we end up with sine x equals 20 sine 43 over 14 if we type that into the calculator now what do we get 20 assign 43 over 14 and we get an answer here let's write it over here so we get sine x equals zero point nine seven four two eight and a few more decimals and obviously just like normal trigonometry when you're doing sohcahtoa to get the actual X here we have to the inverse of sine so we're leaving that number on your calculator you do ship sine which gets you sine minus one type in that answer or just press your answer button so shift sign answer press equals and I get an answer here of seventy six point nine seven seven nine and again a question what our system round it here just depends so let's just go to the nearest degree we'd go for 77 degrees obviously just making sure what the question says here but let's just round it to one the nearest degree there so 77 okay so that's how you use the sine rule right let's say all these questions different then so work out the length of a B so this one over here now straight away looking at this look we've got a pair of opposites there but I don't have any other pairs of opposite opposite so I've not got anything opposite my 15 I've not got anything opposite the 12 so I can't actually use the sign rule here and that is your clue that is your hint here that the sign rules not going to work we're gonna have to use the cosine rule so another rule that you need to know so the cosine rule is he squared equals b squared plus c squared minus 2bc because a okay so a being the side we're looking for so we'll label this little a and this one big a ignoring the letters on the triangle and then labeling the other two sides and they can be B and C however you like and from there all you got to do is stick these numbers in it's actually quite simple to use when you know it so a squared equals B squared which is 15 squared plus C squared which is 12 squared minus and agrestic this bit in brackets two times 15 times 12 cause a which is down here which is 20 there we go alright so sticking that all in the calculator let's have a look what we get so 15 squared plus 12 squared minus 2 times 15 times 12 cause 20 press equals and I get a squared equals 30 point seven one zero six and a few more decimals and obviously that is a squared we don't want a square to one now where a is so we just need to square root both sides now so square root leaving the answer on the calculator square root answer and we get a equals five point five four one seven one nine and again obviously be asked to round this in a particular way in the question let's go for two decimal places so a equals five point five four centimeters all right there we go and there's using the cosine rule okay so working out the size of angle BAC which again is this one at the top and again just having a look there are definitely no opposites because we've got no other angles but the angle that we're looking for is gonna be our a and it is opposite nine there we go so the others can be B and C again now obviously here we're looking for an angle so every sub T if you choose to learn the formula I tend to find that I just learn this formula a squared equals b squared plus c squared minus 2bc cause a and then quite happy just rearranging that to get cos a on its own so in order to do that i'm gonna get this whole minus 2bc cause a i'm gonna add it to the other side so we get a squared plus 2 bc cuz a equals b squared plus c squared now I can get rid of that a squared so I can minus a squared from both sides and minus a squared and you get to be C cos a equals B squared plus C squared minus a squared now and then you can finish off this rearrangement you can divide by two BC just to leave you is cause a so cuz a equals B squared plus C squared minus a squared over 2 BC so to you can choose to just learn that formula if you want but that's the formula we're going to use to find angle so plug in all these numbers then just into my formula there we'll get cos a equals b squared plus c squared so 10 squared plus 5 squared - a squared so minus 9 squared all over 2 - 2 times B times C so 2 times 10 times 5 nice and easy type tonight into the calculator say fraction button 10 squared plus 5 squared minus 9 squared all over 2 times 10 times 5 and that equals sorry look so cause a equals zero point four four and then same process again we need to do the inverse of course so cos minus one of your answer and you get let's have a look chief cause answer and I get 63 0.896 okay degrees and again we could around that so we could just say 63 point and let's just go to one decimal place sixty three point nine degrees again just reading the question and that's how to use the cosine rule for finding angles moving on to the area of a triangle so your over triangle formula obviously not half base times height because we've done up the height here so for any triangle the area is half a B sine C so 1/2 baby signs see another formula there that you need to know and let's just have a look at how we apply that so signs see this time the angle that we're going to use is gonna be our C so we'll call that C this would then be little C and the others will be a and B and they can be a and B in either order and then it's as simple as just sticking those numbers into a formula so it's 1/2 times 5 times 8 times sine 41 there you go type it into the calculator so 1/2 times 5 times 8 times sine 4 C 1 and we get the answer thirteen point one two rounded to two decimal places it's area so that's meters squared just being careful of the unit so thirteen point 12 meters squared there we go there's using the arrow triangle formula question like this though you might be given the area of the triangle so it says the area is 80 so we're working backwards a little bit here so we're going to use half a B sine C again so half a B sine C but this time it gives us the answer so the answer is that half a B sine C has to eat core 80 in all the numbers here again see being my angle and then a and B being my other length so if I was to type this into the formula we'd have 1/2 times 11 times 16 times sine C and it would equal 80 now I can't actually work that out but what I can do is I can work this bit out so if I do 1/2 times 11 times 16 we get the answer 88 so we have 88 times sine C equals 80 so we can rearrange this look we can get both sides and divide them by 88 and they'll just give us sine C on its own so sine C equals an 80 divided by 88 is stick as a fraction actually it is a decimal there it comes out as not 0.9 zero and that is actually nine zero recurring there we go I'll leave on the calculator screen and then obviously to finish that off you do the inverse sine again just like we have on these other questions so I'm gonna do that over here to the left so sine minus one of this number here which is actually also a fraction it's 10 over 11 so I minus 110 over 11 or that note point 9 0 recurring and we get the answer of 65 0.38 degrees there we go so we can find our angle as well just working backwards if we're given the answer and you can apply that logic there's are lots of different questions where it gives you the answer and you can use your formula backwards just dividing everything over to the other side [Music] [Music]