A Level Maths Modelling with Trigonometry Problem Solving Explained

hey guys welcome back to yet another a level mats video this time again we're going through all the problem solving stuff by the way this time it's the very attractive topic of modeling now modeling in general kind of sucks for students and I completely understand why because you have to apply it to situations you may not have know you may not really think about before so in this case it's talking about a ferris will sure I guess um but again we're going to go through modeling with trigonometry now the good news about this is most of the parts stay the same so for example things like the maximum height um and the first part is normally the same as well as Part B Part B when I say normally the same C and A are basically identical B very similar D it's a toss up but essentially they say how would you it's always adapting the model that's what they kind of ask you a is always finding the whole R CA D plus Alpha thing super standard but yeah I would say b c and d um not always answered very well so hopefully when I go through it it'll kind of give you an idea on how to think about these types of things so let's start off with three part A now doing something like this there's I mean there's a few things you can do you can either just use the essentially the equations but the way that I prefer to to do it in an exam anyway is to actually write out using the addition rule what the um answer is so if we do R cos theta plus Alpha you get given the addition rule in the exam so I'm going to show you how to do it this way just so you're not relying on memory because relying on memory especially when you're under pressure and especially when you've have like four different exams in a week it's not very it's not very reliable so again the addition rule just says that you have cos Theta cos Alpha so COS of both of the things minus r sin Theta sin Alpha and the way that this method works is you essentially make these two equal to each other the reason why we can do that is simply because we're trying to write this in the form of this which means they both should be equal to each other right the form on the right should be equal to the form on the left and then all we do is we compare coefficients so if we look at everything in front of cos Theta here is the same as everything in front of cos Theta here so if I compare cos Theta we get R cos Alpha equal 10 cu again everything in front of cos Theta here all the coefficients of cos Theta which is cos Alpha and R is the same as the coefficient of cos Theta on this side which is just 10 do the same thing for sin Theta we get R sin Alpha equals -3 that's it cancel out the negatives and then from here you can work out Alpha and R here's how you work out Alpha you divide these two equations so if I divide them I'll write this out fully but again you will be able to jump steps a little bit you get R sin Alpha R cos Alpha for got the S equals 3 10 the RS of course cancel out because there's r on top and bottom s over cos is tan so we get tan Alpha = 3 10 which means if you use your calculator Alpha equals something so let's grab calculator go I don't know why it lags all the time but that's fine 3 over 10 we get let's call it 16.7 actually no it says to two decimal places so 16.7 0 then 16.7 0° now work out R is a little bit more involved but essentially you're going to use the fact that sin s theta plus cos squ Theta equal 1 so if we Square both of these and add them together it should equal one so we have R 2 let's do sign first just to keep it no we're going to use Alpha okay that was strange sin squ Alpha plus r cos s Alpha equal 1 and then we can essentially just use the fact that R cos Alpha is 10 R sin Alpha is 3 um but essentially oops I forgot to put the squ there probably the fastest way to do it and probably the way that your teacher taught you the shortcut is literally just using the fact that you can square and add the two coefficients together so literally just 10 s + 3^ S so whichever way you want to do it it's it's up to you um actually no you would't make it equal to one so again if I write that out we will get R 2 sin 2 Alpha plus cos Alpha and then of course we know that sin + cos s is 10 + 3 as well so that goes to one and then you get your answer so 10 squ is 100 3 S is 9 so that's Square 109 you can't square root that so that's it so our final answer is it would be R cos I think it's theta plus yeah theta plus Alpha so we have sorry we don't have R anymore we have root one 109 cos Theta + 16.7 and that's it for part A so again I showed you kind of the full way but if you want to use the shortcuts I would recommend knowing how to get there just in case you're a bit unsure of the shortcut on the day of the exam and you can work it out yourself cuz that it makes more sense to me personally to do it that way but let's move on to Part B which is where things start getting a little bit tricky so for part B let's read it the height above the ground H met of a passenger on a ferris wheel T minutes after the will starts turning is modeled by this equation H equals Alpha minus sorry is that a I'm going to say Alpha minus all of this figure three shows the graph for two complete Cycles it says find an complete equation and then hence find the maximum height so finding a complete equation the first thing that you should notice is that this this equation looks almost exactly like this equation where Theta is just equal to 80 T so what I'm going to do to make this a bit easier is I'm going to say h minus and I'm going to write my equation there's a reason why I'm doing this and I'll explain it in one second but again let's use that little meta analysis why would they get us to do that in part A if we didn't need it in the later Parts especially when the equations just happen to be the same right now it says given that the initial height of the passenger above the ground is 1 M okay so what does that mean well initial means that I've just realized I didn't write the full thing at all so H equals I I think that might be an a you know that H equal a minus well that means that t equals z cuz initial means starting so when T equals 0 H has to equal 1 now why does why does do it make more sense to write it in this form well there's two reasons one when we work out the maximum which we can do first by the way but if you wanted to work at maximum it's way easier to do that when there's only one trig function when there's two trig functions it's way harder to work out what the maximum is whereas instead if there's one trick function well the maximum is just one right it's a lot easier the second reason is when it says the initial height imagine subbing in t equals 0 here yeah it's not too bad like that's not too bad to do but again it's easier if there's just one trick function but this was mainly for part two finding the maximum height so part one you could do it without replacing it but in my opinion I think this is a bit easier so H equal 1 we get a minus and what I'm going to do is I'm going to sub all of this onto my calculator just because I'm pretty sure it's going to give me something a little bit ugly little bit ugers so oh careful minus < TK 109 multip by because Theta is I forgot that it's meant to be 8T right because we said up here Theta equal 80t so that would be 80 * Z so I'm not even going to write it it's just going to be 16.7 and we get basically 10 99.9999 cool which means a is equal to 1 + 99.9999 6 so we're going to say a is equal to 10 which is indeed the answer in the mark scheme so we have 10 minus the < TK of 109 let's try not make any mistakes this time when I'll write it C 80 t + 16.7 what does part two say part two says hence find the maximum height of the passenger above the ground now sometimes they ask for the time taken this in this case they've said the time taken is part C but in part well in this part of the question they can some them say find the maximum height and find the time that this happens but again they've separated into two questions here but it's the same principle so maximum height this is where making it one equation is a lot easier cuz all you need to do is have a look at this equation and think about what varies so what part of this actual equation is varying the 10 always stays the same the rooot one9 stays the same it's the C function that's varying and what is if I look at the whole of a c function what does it vary between it varies between min-1 and 1 so the biggest this C function can be is one so if I make that equal to one what's my greatest height well again that's the biggest this can be right is one but we're looking at min over here so for this to be as big as humanly possible I want C to be as small as humanly possible which means I want CA the ca function to equal -1 CU that'll make it as big as possible so if they ask for the minimum height I would say cos is equal to 1 because then 10us Ro 109 well rooot 109 was well it's probably going to be what 10 point something so that' be negative which obviously is not allowed um but in this case we want the highest possible value so we're going to make that minus one which means H will equal 10us < 109 time -1 so it's going to be 10 +un 109 which of course is extremely useful so again if they wanted the minimum value we would then make cuse not necessarily as small as possible but it would make the overall thing as small as possible by making cause as big as possible so 10 I'm going just say plusun 109 cuz I know it's going to be minus and minus thanks calcul 20 does it say how many significant figures I just going to say 2044 so when you're looking at these kind of Maximum minimum questions you have to be really careful think about what the maximum minimum of the function is and then work out which value you want for the overall function so in this case the minimum of C is minus one the maximum of cause is plus one but because I want the whole thing to be a maximum I need to make cause a minimum because it's being subtracted and I want to subtract by the smallest number possible for part three there's two ways to do this and I'm actually going to use the way that's very mathematical because I think that way will apply to the most number of people but again I'll explain the other way it says here find the time taken to the nearest Second for the passenger to reach the maximum height on the second cycle so in other words if we look at this graph they want that time now maximum height don't forget is when this is a minimum okay it's a minimum value so we're basically solving equals -1 okay so get out your calculators do this this now I know some of you use the cast diagram personally I would not use the cast diagram at all it's absolutely heinous but you may know that this will happen at 180 now I will use the graph so if we draw the graph and this is the C graph here so we're not using the graph at the beginning because well why because this is the graph of H so it's 10 minus this the COS graph starts at one and it goes down up and then it will keep doing this forever and that happens every 360° so if we label the values in fact let's do it in Black we have 90 180 270 360 450 40 now if we think about what it said this isn't the value we're looking for because it says on the second cycle the second cycle means it's already gone round once and it's going to go around again so we worked out this value when it equals -1 but we want the other value when it equals -1 which is actually 540 again normally in questions like this they're a little bit nicer and the other value that you work out is just by doing the whole 180 minus or whatever fun stuff that you get to do in this case they've been a little bit nasty by saying it's the second cycle again second cycle means it's gone around once and it's starting to go around again so in other words the angle has to be bigger than 360° so we're going to accept this as our answer and then all we need to do is work out T because it says find the find the time taken to the nearest a second so 80t + 16.7 = 540 so we're going to minus minus 16.7 from both sides and then afterwards we're going to divide by 80 but that's 5233 so then T is equal to love having to slide this around the whole time 6.54 seconds actually do we know it's seconds it might be minutes tenous am I blind I don't think it tells you T minutes there we are so that's a minute and it says give it to the nearest seconds so what I'm going to do is we have six minutes I'm going to take that 0.54 I'm going to times it by 60 that gives me 32.4 seconds so that is 6 minutes and 32 seconds that's how long it takes the yeah the second I saw 6.5 seconds I was like that is a very fast moving feris then that means it does a whole Loop and a half in 6 seconds which is pretty funny to think about so again for part C finding the maximum or the minimum the two main things I want you to think about is the maximum minimum of the function that actually changes so again the 10 the root 109 doesn't change then when you're solving it think about what value you're actually looking for but part D again is the other part where I said some people get a bit um a little bit antsy they get a little bit annoyed about it is decided to increase profits the speed of the wheel is to be increased how could you model that so if we look at in fact if we look at the equation that we actually have here how would we show an increase in speed well let's think about what an increase in speed actually like functionally means right if it's moving faster it means it should take less time to make a full Revolution so in other words we're looking at this part of the equation because we want this is the only part that varies and we want this part to happen faster so when it go around faster the way you would model that is by whatever your time is you multiply it by a bigger number because that will make it go around faster because again remember 180 will be half a cycle 360 be half a cycle so if you want this to get to 360 as fast as possible all you need to do is make whatever's in here bigger right because that will make this faster hopefully that makes sense because again cause of cause when it gets to 360 is when it's gone round a full time time so we want it to get to 360 faster so the best way to do is increase the value of 80 so maybe make it 100 for example that will make it faster that was quite a long question even though there's only nine marks again the whole modeling with trigonometry section is a little bit tricky it's not the easiest thing especially when you have to apply it but now that I've gone through one you should be able to use this to help you with a lot of the other questions so if you found any if that useful please do consider subscribing and liking all that fun stuff cuz it really does help me out but I'll leave you to your work and I'll see you in the next one