GCSE Maths Free Tuition Lesson - Trigonometry SOHCAHTOA

with my time foreign foreign okay welcome back to the third of my uh kind of free tuition lessons on YouTube today uh as per request we're going to start going through trigonometry okay so just so you know that if you are kind of either commenting or in the chat you can request different topics so with trigonometry essentially it's what we use when we have a right angled triangle so you may know that if we have two sides let's say four and two we can work out whatever the X side is by using Pythagoras right so x squared equals 4 squared plus 2 squared and then just go on and we can work it out that's all well and good but what if we want to work out an angle or what if we have a net angle let's say 30 degrees and we have one side and we want to work out any of the other two sides what can we do then well that's when we use something called trigonometry and in terms of trigonometry there are three kind of things that you may have seen on your calculator we have sine of cos and we have 10. so if you have a look at your calculators you should see sine cos and tan buttons somewhere on your calculator so for mine is just over there and that's what we're going to be using so you may have seen sohcahtoa before as well so just as a reminder let's go through it so we have these sign causes and tans we need to know when we can use them okay so we can use them or kind of which ones we use in a given situation so the so part stands for so so stands for sine equals opposite over hypotenuse the KA part so we can design opposite hypotenuse the cat stands for cos equals adjacent over hypotenuse okay and normally we put like an uh the angle sign here and lastly we have TOA which you can probably guess gives me tan Theta the O again stands for opposite and the hate uh the a stands for adjacent so knowing this you can actually do all trigonometry questions okay except for the sine 1 cosine rule which we'll come on to next week but we can basically work out a bunch of different stuff from that but what we need to do first is we need to be able to label our triangles so I've used these words adjacent hypotenuse and opposite but I don't actually know what those mean yet so I'm going to do is I'm going to draw a very generic looking triangle it's going to be right angle because keep in mind we can only use this only for right angled triangles okay so anytime we can use it and let's say our angle that we get given okay we use this kind of Greek Theta sign is here so how will I label my sides well this is the really important thing you need to be able to do so let's start off with the opposite because it's the first one here the opposite side is the side that's opposite the angle so it's always going to be the one that is directly opposite the angle so I can also just write this as an O okay the adjacent one is the one that's adjacent so touching the uh angle however we'll come back to the adjacent because there's something slightly tricky the next one you want to label so you first label the opposite now we're going to label the hypotenuse now the hypotenuse is the longest side in a triangle so how will I know well you can look but that's not exactly um very mathematical but it's always going to be opposite the right angle is the hypotenuse okay now the adjacent is just the only other one that is left okay so hypotenuse we sometimes just label it as h and the adjacent is the last one left right A triangle is made of three sides and we have three names okay so that's where that comes from now if I tried it again let's draw another triangle okay and I'm going to start giving it some numbers okay we have 20 degrees here okay we have three centimeters let's label up our sides well again we start with the opposite so what I'm going to do is draw a dotted line going directly opposite my angle and I hit the three centimeter side which is my o for opposite the hypotenuse is always opposite the right angle so I'm going to put a h on the opposite side and therefore the a is the one next to it that's as easy as it is now of course we haven't actually done any maths yet okay we just labeled the triangle but labeling the triangle is the first step if you get the first step wrong you get the rest of the question wrong so it's really really important that you remember this opposite the right angle hypotenuse opposite the angle you're given because again keep in mind if I give you a different angle here of let's say 30 degrees that's impossible but anyways my opposite and adjacent switch positions so it's opposite the angle you're given or that you want to use okay in terms of which one you want to use you can use any one you want just make sure you've labeled a triangle properly okay so assuming that makes sense let's go on to do an example here okay so if I label this question number one let's bring question number one up here so again what's the first step the first step is to label your sides so what I do is I draw a dotted line going from the angle they gave me and that's my opposite side because it's opposite the angle that I've been given the next I label the hypotenuse well okay I do this again a dotted line going from the right angle means that this side is the hypotenuse the last side has to be the adjacent okay so now we've leveled up our opposite hypotenuse and adjacent respectively we now need to work out what well let's see what the question is asking us it's asking us what the value of x is in other words we need to find so we're finding H which is the hypotenuse okay so in terms of how we do that we look up here at our so Ka TOA and we look at what we have and what we want okay so we want to find H and we have o which means you want to use o and H so which one of these three has o and H in it well it's Soul has Owen H in it so we're going to be using sine theta equals opposite which so up over hypotenuse okay the other way to do this is to use a formula triangle so we're going to have sine in the bottom here we have hypotenuse and above we have opposite and if you have this triangle we can use that to work it out the answer as well so now if we want to find the hypotenuse we need to rearrange this formula okay or what you can do is if you cover up the hype part of this we get that our value is H is equal to the opposite over sine of the angle okay because again if we covered that up so again you can just put your hand on your screen cover that up you get opposite over sine so that's going to be the opposite side is 20 centimeters divided by sine Open Bracket 35. and all you do is you'd enter that into your calculator okay so on your calculator press the fraction button first 20 on top then press the down arrow to go to the next line and if you find the sin button s i n press it it opens a bracket for you you type 35 and then just close the bracket you need to make sure you close the bracket otherwise bad things can happen press equals and we get quite a long number but it's 34 0.9 if I do it to three significant figures so the full answer is uh 34.8689 okay if you want to do it with um rearranging instead all you need to do is if you make this into letters you have sine theta equals o over H rearrange for H times both sides by H you get H sine Theta equals O then divide by sine Theta you get H is equal to opposite over sine Theta which is the exact same equation that we got with the formula triangle so there's no biggie there at all so this should be a question one but let's have a look at question two to see if we can practice what we've learned First Step you need to label your sides so draw a dotted line going from the angle they give you and that is going to be your opposite because it's opposite the angle they give you you draw a dotted line going directly opposite the right angle and that's going to be your hypotenuse whatever's left is your adjacent so in this sense in this case we want o and we have a so which one of my three things here has o and a in it well it's not this one that's o h that's a h but that's o and a it's tan so what we're going to be using is TOA which is tan theta equals opposite over adjacent again if you want this as a formula triangle you have o a and tan it's a very interesting triangle I've just drawn now which one do we want well we need to cover up the o right or we need to rearrange the O for o but times in both sides by a which means we have a 10 theta equals o if we cover this up we just have that o is equal to a times tan Theta it's the exact same thing rearranging or the formula triangles works the exact same way and then all we need to do is substitute it in so o is equal to what is a it's 11. and then you can put the time sign if you want but you don't need it so if you want I'll ignore the time sign for now and that's 10 Open Bracket 42 degrees close bracket type that into your calculator so 11 10 42 again exactly as it appears I didn't put the time sign but you're more than welcome to and you get 9.90 to 3 significant figures okay give me a second just to work through that make sure that makes sense foreign okay so that was for question number two next week I will space out the questions a bit more so just cut up the pictures and uh move them around a bit I'd recommend if you want to go back and kind of write down the questions just as examples um if you need to do revision or anything that would be a good idea but now let's do a question number three once again what do I need to do the angle I want so in this case they haven't given me what the angle is but this is the angle I want this side opposite is the opposite side go dotted line across from the right angle that's your hypotenuse and the one that's left is adjacent okay so this is interesting because I want X I want the angle so if you want to call it Theta or X and we have a and H so once again go back to this which one has a and H in it it's cat down so we're going to be using Ka in other words we have cos Theta equals a over h now this is where things get a bit more interesting because we don't want cos Theta we want Theta or X whatever you want to call it so what we need to do is we need to do something called the inverse cos so we're going to actually work out if I replace I'll just leave it like that we're going to be typing this into our calculator because we're working out an angle whenever you're working on angle you're going to be doing cos to the minus 1. now before you panic okay if you look at your calculator and you look where cosine is where cos is if you pre can you see just above it in a different colored text it has cos to the minus one the way you get that is you press shift or the same color button so in my case shift is yellow and the COS the minus one is yellow I press shift cards and that gives me cos to the minus one that's all you have to do detail Json was 16. hypotenuse is 24. equals and again I'm putting the exact same thing in my calculator so again the COS to the minus one comes from doing shift cos okay so if you have a Casio it's going to be shift any other calculator look what the color of the COS minus one is and find the button that corresponds to that it'd be shift or alpha or function equals and we get 48.2 degrees to three significant figures not too bad foreign going on to question four once again we start by labeling the sides so opposite the angle we want or given is going to be your opposite opposite the right angle is going to be the hypotenuse and then the last side is going to be your adjacent okay that's it now let's have a look at what we want slash have we have so we want X or Theta or whatever you want to call it and we have the opposite and the adjacent well let's go back up o and a appear in the tan formula so TOA opposite over adjacent so you can probably guess already what's going to happen excuse me so looking at the first part when we had cos Theta we wanted an angle we put cos the minus one and we have tan Theta and we want an angle we do tan to the minus one exact same thing so I'm going to write this all down and again if you look at your calculator episode 15. Jason's 11. look at your calculator where it says 10 in a different color also say times the minus ones you both shift to 10. opposite is 15 11 is the adjacent so we have 53.7 degrees again to three significant figures and that should be for that answer give me a second just to make sure you're okay with that okay let's go into question five and once again we're doing the same thing okay start your angle draw a line going opposite that is going to be your opposite side look at the right angle dotted line that's your hypotenuse the one that's left is your adjacent now again we have we have an angle and we have adjacent and we want what does it say we want the length a b which is the opposite side okay so if we're looking at o and a in this case let's go back up to our triangles yeah so katoa o and a is part of the TOA name okay so we're going to be using tower which is tan Theta equals opposite over adjacent or again if you want to use the formula triangles can okay and we want o so if we cover up o or again multiply by both sides by a we get a 10 Theta right both both ways we do it so if we sub that in so that's o equals and that'll be o equals so if we substitute that in adjacent was 20. and then Theta was 35 degrees so 20 10 35 so again we're not doing the um inverse 10. we get 14 the 3so that's it technically we should put centimeters here as well hopefully that is okay okay so let's keep going uh it's going to get a bit more tricky uh shortly because um again this is the most basic that they can give you if you look there's only two Mark questions so far look at the next two they go three four and they can even have six Mark questions so these can get a lot more tricky so if you have any questions make sure you ask them now because again it's going to be kind of assumed later on assume that you are okay with this okay so let's keep going so once again we need to label our side this is the first step for every human being that does trigonometry look at the angle they give you or you want draw a dotted line going across from it and that is your opposite side if you draw a dotted line going across from the right angle that is your hypotenuse whatever's left is your adjacent nice and easy so what we're going to do here is if you have a look at what we want what we have we oh I didn't even level the question my bad so to put six here and here so what do we want or sorry what do we have first so we have an angle and we have the hypotenuse and what do we want well it says calculate the length a b this is a b and I've labeled it a okay so that's why I want so far so if I have Theta and H and I want a let's scroll back up to see which one has a and H in it so no uh no this one has a and H in it cat so that's right cat and that is cos Theta equals adjacent over hypotenuse okay so if you want it as a formula triangle you can put the A on top cos here H here up to which method you want to use I personally prefer rearranging but that's because I'm a nerd I and we want a so if I cover up a what am I left with or if I want a here I'm going to multiply both sides by H to get H cos Theta equals a if I cover this up I also get a equals H cos Theta it's the same thing for both ways okay so now all I need to do is substitute in what they've given me so H cos Theta well H is 15. so 15 and then we have cos and what's our Theta it's whatever angle they've given us or the one we want which in this case is 42 and again make sure this is in Brackets equals a and again it's the same for both into a calculator 15 cos 42. again it doesn't matter if you put a Time sign between the 15 and the COS or not whichever way fine and three significant figures it's 11.1 centimeters for next time I'm going to actually um pay for a virtual calculator so I actually had one before but the license ran out which was fun and I didn't realize that until I was about to start the lesson so that's a bit of lack of preparation on my part but for next week we're going to be able to see this calculator on screen so if you're unsure now in the next lesson I'll be able to do it and you know step by step on a calculator itself but this would be our answer for part six okay take a few moments just to make sure that's all okay before we go on to the harder and more fun questions Okay so what I'm going to is remove the last two parts of this for now just so we can properly have a look at this question so they give us a trapezium instead of a triangle so you might think well hang on a minute I thought we could only use it for right angle triangles well you can make a right angle triangle from this right hypothetically if I was if one wanted to if I drew a straight ish line along here would this top thing be a right angle triangle yeah it would okay so the question is how would you know this in real life right like if this was just an exam question then it wasn't a trigonometry lesson how would you know well if we have a look it says work out the size of the angle b a d so something that I've noticed a few people kind of get stuck on is the fact that they aren't 100 sure just give myself some space they aren't 100 sure which angle they're talking about right so here's an easy way to tell the angle they're talking about is always the letter in the middle okay so find the letter in the middle that's a and it's made up of sides beta a so this four meters and a to d so what's the oh God I'm sorry for my handwriting what's the angle made up from line a b and a d it's this whole angle here so if you notice it's made up of two parts if I write this out here we have a rectangle right where this is B a uh there's no letter on this side C here and there is a triangle part on top so if I separate that out a bit where this is also a and that's d so the angle they want is this angle plus this angle okay let's give you a second to digest the true genius of what I just did so if I label some of the sides as well make it nice and neat and 15 meters for now if you have a look I hope that makes sense so all I've done is I've basically cut off a triangle off the top but the total angle is this right angle here so we already know what that is that's 90 degrees that's baby stuff plus whatever this angle is here that's not baby stuff so my question to you and this is a right angle triangle because it it is right it tells me here so this length here should still be 15 meters right so go back to the original question there's no difference between the length of BC and the length from a to there no difference now what is this length here well I know that from here to here is four meters because this other side's four meters and the total length is six which means this must be two meters any questions okay assume that's okay I'm gonna just calf that part so now you can probably see where trigonometry is going to come into play right I now have a right angle triangle and I want an angle so what's the first step label your sides draw a dotted line going straight across from the angle we want that is your opposite dotted line from the right angle gives me my hypotenuse the other thing is the adjacent so I have o and a and I want Theta so o and a let's go all the way back up to the top of the document O and A is tan uh there we are so we have Torah here which is tan Theta equals op over adjacent remember we do when you're looking for an angle we have to just do the inverse which again you should be fine with so theta equals tan inverse make sure you sub in your numbers correctly too opposite is 2 adjacent is 15 in this case and then if you're doing the tan inverse you have again if you look above your tan button there is like a different colored tan inverse you just need to press shift and press that button and that gives me an answer of it says give your answer to one decimal place so 7.6 degrees the full answer is 7.594643369 that's it not too bad is that the answer though nope give mind b-a-d is equal to the right angle 90 plus the angle we just worked out so in total it's 97.6 degrees to one decimal place so you can probably see exactly where students lose the final mark because they think that this is the answer read the question it's the whole angle okay so if you slipped up on that bit that should give you some motivation to make sure you're reading the question extremely carefully okay let's go to the next one so this is a weird looking shape some kind of quadrilateral uh it's question eight and we don't actually need to break this up if we don't want to this can be done normally we want to find the value of x let's have a look what we have okay and if you have a look at the right hand triangle we have no information whatsoever which sucks however can we work out some information so what do we need to work out X so to find X if we progression a up here to find X I need another side right so either this one or this one because logically speaking right if I want to use trigonometry I need a side and an angle or I need two sides at this point I have just an angle which is completely worthless to me so I need another side so the question is can I work out this top side no I don't think so can I work out this side well I have two sides of a right angled triangle and I want to work out the Third so if I call this so let's have it actually level it up first right dot dot dot this side is my opposite side so it's already an O friendly enough obviously the right angle is my hypotenuse and this is the adjacent so to find o use Pythagoras and in this question here what are my sides well this is my hypotenuse because it's opposite the right angle so to work out the missing side o what I'm going to do is do the square root of 20 squared minus 16 squared and the reason why I'm doing it that way is because I don't uh sorry step back normally you add the two smaller sides together and then square root them but if I have the biggest side I subtract them instead okay so I hope that makes sense so now if I wanted to work out sorry that's not o squared I made a mistake there if you sub into calculates 20 squared minus 16 squared right again sub it in Normal I get 12 centimeters so now I have opposite of one hypotenuse I have an angle so if we're looking for so and H is sine so sine Theta of course or again if you want the triangle again I'd recommend learning the algebra especially if you plan on doing a-level maths or even doing any kind of science at a level but yeah if not it's totally fine what do we want uh we want H so cover up h and my equation now becomes H equals opposite over sine Theta uh to do it this way you'd have to times and divide around but it's the same thing and we get H is equal to opposite which was 12. over sine 25 in Brackets which says to one decimal place so we'll leave it to one decimal place 28.4 and don't forget your units which in this case is centimeters that's a four mark question should take you four minutes and I know it's taken me more than four minutes to explain it but in an exam I reckon you guys could do that much faster much much faster and then we have one more question yeah and if no one else has any questions in the chat then I won't gone much further but there are a few kind of announcements we can make so what we'll do is I'll drag this oops okay so I'm just thinking okay got a bit more today so this question work out the size of angle b c d so remember let's mark the C we're looking at this thing here and it's made up of sides B C and C D so we're looking for this angle here which let's call it no let's not call it X we use Theta because we're men and we need to work out the sides of the ankle okay so once again right to find this angle I need two sides right so if I label the sides I've got so far that's the opposite hypotenuse this is the adjacent how many sides do I know one I need two which one can I work out well if we so if we look over here on this side of the board here I could work out this length here and use it to work out this length here which I'll call Y or you can leave it as adjacent to do that we need to label this up as well so this is the opposite this is the hypotenuse and this is the adjacent so if we speed up a little bit now that I think we've got a bit more comfortable we have or let's say we want opposite and adjacent if you remember that that is TOA right so if we again if we write that out so adjacent and both in one second again we want to we want the adjacent so cover up the adjacent so adjacent is equal to the opposite over tan Theta so it equals 9 over 10 uh 37. so again let's do this 9 over 10 37 gives me 11.9 centimeters so if I know that this side's 11.9 centimeters then this other side should be 22 minus 11.9 centimeters which is 10.1 centimeters okay so now I have my adjacent for so if we call this triangle two this triangle one I have the opposite and the adjacent and I want the angle so I'm going to be doing tan Theta equals Observer adjacent so we're going to be taking the tan inverse opposite is nine the adjacent's 10.1 so 41.7 degrees that's probably the hardest question we did so far so if you understand that then you're in a really really good place for the kind of GCSE section of trigonometry in terms of the basic trigonometry because just like everything in maths it gets worse so we'll briefly start with today is something called the sine rule okay um actually remember when I come on to that tomorrow um just for the sake of time to start this with uh secretary we'll save that as that uh the reason for that is the sign rule might take a bit of time so if we just say this lesson was socatoa uh we can reason that we need to continue uh from this because I I don't think we have time to really go through the sign rule as well um just for the just because it's a whole new concept and we only have 15 minutes left so I think we won't do that today um okay so in terms of these lessons again uh thank you for everyone that's joining them and kind of watching them live uh it's better for you if you watch it live because you can message me you can say if you're stuck you can ask for help Etc you can give me feedback you can also ask for specific topics but I do read all the comments on YouTube so if you're ever stuck and you're watching this um after it's been uploaded then please make sure that you are kind of leaving comments and suggestions and everything like that um but overall I think we'll leave it there for today and tomorrow we'll definitely do the sign rule stuff but just before we finish up does anyone want to let me know if there are any anything you want me to go through or if you've joined late in your way to go over something I don't mind doing that for the next 15 minutes okay so I'm just making a note okay well uh if that's the case and everyone's okay then I hope you all have a wonderful evening and I'll look forward to seeing you in the next one thank you very much