A Level Maths Made Easy - Solving Trigonometric Equations

hey guys welcome back to another a level maths video this time we're going to go through solving trigonometric equations and this is one of the kind of first topics that we've gone through that are is really really quite tricky depending on how you want to solve these equations that we're going to be doing there are three main ways I'm going to show you in my opinion the best way to do it which is something called the cast diagram but I know that some people prer to use the either the rules in terms of just adding and subtracting certain numbers and some people like to use the graphs by person think that the cast irum is definitely the best uh it's something that even AEV students struggle with so it's quite involved this is going to be a very long video and this is one of those kind of topics that I would definitely recommend doing a bunch of practice outside of this worksheet as well otherwise this worksheet would be about 500 questions long you can find basically infinite questions online because it's trig equation so s cos tan anything squares and all that kind of fun stuff as always the link in the description for the worksheet again it says PID Paton but it is completely free it's just that I use that to host it because it's very convenient for me there's also the AEV maths playlist which has all of the live lessons I did last year and there's also going to be um kind of everything else that you need for the Discord and all of that fun stuff so I'd recommend doing this worksheet again after you've watched it a day or two after watching the video but also once I've gone through one question have a go at the next question yourself that way we're doing kind of active learning it's like it's an actual lesson and you'll actually get a lot of benefit from it so without further Ado let's just get started so solving trigger equations you're probably going to think it's not that bad it's quite easy you've done it before well let's actually have a think if I said to you could you work out answer this question sinx is equal to let me give a good example of a half okay what you'd probably do and rightly so is whip out your calculator and just do inverse sign of a half so 30° cool and that is good enough for gcsc but if we draw out the sign graph you're probably going to notice something so if I draw you know we got my y- Axis here and we're going to have a pretty long x axis here the sign graph looks like this and goes on in both directions forever so if I'm looking for so if you remember it varies between 1 and minus one so let's say a half is over here this value here is 30° however it also hits a half over here and then over here and also over here so there's actually one two three four answers but actually because the graph goes on forever there's actually an infinite answer a number of answers for this infinite goes on forever so when I ask okay sin x is a half what is X I don't give you enough information to solve the question because you could rightly say it's 30 and also this value and this value and this value and you could write values going on forever so we need to work out first how do we work out these other values my calculator only gives me one value and if I press equals again it just keeps giving me 30 interesting right so how do I work out the other values is step one but step two is they also have to give me a Range okay and that is the key Point here now using the graph you can probably work out what the other values are so for example I know that this is zero over here and I know that this is 180 so that means this should be 30° less than 180 so it should be 150 should be however if that doesn't make sense to you I've got some good news there is a diagram that does all of the work for us now I'm going to use this as an example question this is a genuine this is the exact layout for a question that might be about four marks in an exam the difference is that it's not usually as simple as this they will have some numbers in the front and some other stuff but as a base it will say solve insert trigonometry for all values of X between two sets of values in this case 0 and 360 so in this case we're limiting ourselves to only values over here which is quite useful okay because we need to have a limit otherwise the answerers in just an infinite number of solutions so how would you do it well the first thing you do is exactly like you would expect you would simply do x equal to the inverse sign of one simple as so once again 90° cool fine right so those steps are the same so at the same point so if I use this as the first example here as kind of just continuing this let's work out how we get to the other values we do something called the cast diagram what does the diagram look like well the first thing you're going to do is you're going to draw out your axes okay and then you're going to write c a s t in a counterclockwise notation okay I'm going to put this out there this won't make sense straight away this will take time because it seems somewhat arbitrary okay I'm just going to point that out it is a bit arbitrary so what does this mean what do these things stand for well it stands for cosine you don't need to write these red bits all s and tangent or tan now what this actually means so we know what it stands for what does it mean if you look our value here is positive okay we have a positive value so we can draw a line in these regions for different kind of trig values so this is sign isn't it so I'm going to draw a line for sign in the sign region okay and I'm also going to draw sign in the All Region because s is positive in the sign region and the All Region why do I care about it being positive because it's a half it's a positive value okay might not make much sense I fully understand that but I hope you can at least remember the rules and with practice you'll be a it will make sense I promise okay so once again these signs these symbols here these regions represent where the trig value is positive so where this right hand side is positive so all means sign cos and tan are all positive in this region s means s is only positive in this region this region only tan is positive this region only cos is positive because we have S and a positive we've drawn a line in the sign region and in the All Region because that's where it's positive now this x value represents this angle here and it also represents this angle here so what's our alternative value for x well the other thing to remember about the cast diagram is you read angles counterclockwise just like we wrote counterclockwise it's also drawing the angles counterclockwise so our other value I told you this was 150 well let's uh do another color let's do it in green what would the value of that angle be well angles on a straight line add up to 180 so it's 180- 30 which is 150° so our two answers in this case are 30° and 150° that is four marks again this is assuming that the question was that it's between 0 and 360 okay if you change these limits you change what values you can get that will come on to later take a second to digest that but we're going to do a ton of examples this video is this video is already quite long as you can see because this is a very difficult concept okay but with practice this becomes very very easy and the cast diagram is the best way and the the reason is kind of hard for me to explain now without giving spoilers but just to keep it simple you don't need to remember anything with the cast diagram except that it goes counterclockwise if you try and use the rules the rules seem really appealing all you do as you subtract this or add this but when it becomes negative or the regions change like the the boundaries it goes out the window and you need to do different rules you'll end up needing to remember about 20 or 30 rules that's not a smart thing to do cast diagram you don't need to remember anything you just need to learn how to use it which comes with practice which is why I prefer it so let's go back to our other example let's quickly discuss all the different ways that I'm going to help you in your GCC or a level maths careers first of all I wanted to talk to you about the YouTube members this is a really cost- effective way of getting tutoring so I'm going to do weekly live streams with gcsc and a level maths and that all comes for £18 a month and you get at minimum an hour of tutoring per week in these live streams at the moment they're quite small so you basically get tuition for about £450 an hour which is really really cheap the second thing is the patreon for those of you doing a level maths I have already completed as stats on my patreon it is a full CLA with worksheets mark schemes videos I answer all the questions we're going to do exam papers and everything that needs doing we're halfway through mechanics as of yet and lastly there's tuition I still have places in my group classes for a level maths and GCSE Maths for GCSE it's £30 an hour and for a level it's 40 per hour so if you want to let me know if you're interested just email me or join the Discord where I'm very very active all the time anyways back to the video so going back here our first value for x if I just write it underneath just to keep everything nice and and neat is 90° now we've got sign again which fair enough so what what what's our next step well we need to draw our cast diagram so let's draw our cast diagram again it's just a big quadrant and remember you're writing c a s t don't bother attempting this by yourself yet I've got another example where you can attempt it by yourself okay this one we'll do together again again our answer here one is positive so we're going to draw a line where s is positive and this is where things going to get a bit interesting because it says 90° right so technically speaking I'm going to draw in another color just to make it clear it's going to be here cuz that's 90° can you see that's still technically in the all quadrant and it's also going to be here 90° okay or in this case it would technically be 0 deg but it's 90° from the last line okay so what are the other values that we have well we need to work out what this angle is here well it's 180 minus nothing right because it's along the line which is just 180° again four marks these questions worth three or four marks every single time they're pretty quick once you know how to the cast diagram what I want you to do is have a go the next example so this one down here and then I'm going to go through it as well but if you notice the regions are different okay because I want to expose you to another type of question so I expect you only to get one of the two answers which is totally fine okay with this example I have thrown at you a few new things first of all're using cos instead of s that should not really change anything okay but if you look at the region instead of it being 0 to 3 60 it's- 180 to plus 180 don't fret okay it's going to be the same thing however we also need to work out a negative answer this will make sense cuz before negative angles doesn't really make sense does it because angles are a one-dimensional measurement or at least you th they would only really be they Scala not Vector well in this case we do have negative values Okay so once again what we're going to do is do x equal to the inverse uh COs in this case of a half so X is equal to and then let's just change it around so we have inverse sin cos a half so do a fraction 60° that's a nice number to have again I have picked these so the numbers are quite nice at the moment but technically they can be long decimals but that again that shouldn't change anything c a St now our answer is positive so I'm looking for where C is positive once again that is in the all section you're almost always going to have a line in the all section it's also in the COS section so now we have 60° and 60° cool if you notice if I draw a little dotted line going around to work out this value what would that be well this is a reflex angle isn't it so these angles add up to 360 so it's 360 minus 60 so in other words it's 300° but look at this it says minus 180 to 180 so it's too big now I actually like to write out all of the answers and then cancel them out afterwards so what I might do is do this and then draw a little line through it I'm not scribbling it out I'm showing the examiner that I've worked out the other value and then I'm getting rid of it and that's a very smart thing to do so you have the value and then you're saying it's not part of the region so now how do you think I can get Negative values well what I'm going to do is I'm going to erase hopefully not the whole damn thing this part here so I've erased that typical isn't it there we are so how do you think I get Negative angles well if you notice I told you we read angles counterclockwise they always give you positive values so positive angles so to get Negative angles you read clockwise and this always works by the way so our first negative angle well if I read it clockwise it's just 60° or negative 60 de the other negative angle is this and that's just again since it's a circle I should probably continue that here since it's a circle it adds up to 360 360 - 60 is 300 so it's- 300 but if you notice that's also out of the range so draw a line through so my actual final answers two values are 60 and -60 as a hint if it just has X inside the trig value so what I mean by that is there's no 2x or 3x or half an X if it's just X inside s c or tan and the range is 3 60° so - 180 to 180 is 360 0 to 360 is obviously 360 you're usually only going to get two answers just keep that in mind every 360° you only have two answers but this is assuming there's nothing in front of this x so again there's no 3x there's no 2x there's no 5x there's no half an X okay so keep that in mind you should have two answers for every range of 360° minus 180 to 180 is still 360 so I hope that makes sense let's go back through from the beginning okay if you want you can skip the section if you're comfortable um but I'm willing to let most people AR practice practice practice okay and we're going to do a lot more practice so starting from the beginning just Lightning Fast recap first of all we can't solve trigger equations unless we have a range because they go on forever in both directions so instead we use the cast diagram to do a cast diagram you draw a little quadrant so literally just split your page your section in four and you write C A going counterclockwise okay what these represent are cosine all s and tan and what they mean is that in those quadrants those trig identities those trig values are positive so because we've only dealt with positive answers so on the right hand side is positive all we're doing is drawing lines where those trig values are positive so s is positive in the S quadrant and the a quadrant so we draw lines there if it was cos we draw in the C and A if it was tan we would draw in T and A if it was negative we would draw it in the opposite directions but we're going to come to that a bit later with this first example that we had here all you need to do is draw your lines and then read the values off counterclockwise that should be nice and easy hopefully with this second example we actually needed to get Negative values and this is what you need to remember you read positive angles counterclockwise so if you're going to the left with your angles you're reading them as positive angles if you read them to the right or count uh clockwise you're reading negative angles that's all you need to remember so if you want negative angles you read clockwise if you want positive angles you read counterclockwise that's all you need to remember again this is really simplifying it this cast diagram is so powerful because it simplifies it you don't need to remember rules you just need to remember a few key things so which regions are positive and then which way you need to read for positive that's it that's two things as opposed to the list of rules that you might find online so I hope I haven't terrified you too much just yet if you did have a got this question really well done that's a good effort there are two new things in this question right first of all we don't just have X instead we have something plus well Theta in this case plus 30 so we have to do some special stuff and likewise here equals - 2.5 it's a negative but keep calm remember those lit the two rules you need to remember which way you read for positive and which sections are positive that is it so what we're going to do and we should note by the way that this is Theta is between these values not Theta + 30 and and there are a few different things that people do in order to kind of work these out here's the first rule and I would I would lowkey kind of recommend you do this if you think about it at the end what we're going to do is subtract 30 from all of our values that we work out right so let's say I was working this out for the first time so I would say theta plus 30 equals the tan inverse of -2.5 right that's that's fine and I'd work it out just as per usual so shift tan Lord -2.5 NE giv you minus 68 Point blah blah blah and it says to one decimal place so we're going to leave our answers to one decimal place and then what you would do is you'd take away 30 from both sides wouldn't you you would just do 30 minus You' do- 68.2 - 30 right and that' be - 98.2 and that would be kind of your answer in this case because we're doing that we should really change what these boundaries are because what I don't want to do is accidentally miss a value this is not a compulsory step okay mathematically doing this is fine but imagine for a second one of my answers was 200 for Theta + 30 I might cross that out but if you notice if I subtract 30 I get 170 which is actually a correct answer but I've crossed it out because 200's bigger than that this happens to a lot of students and remember believe me when I say that this gets more complicated they could put a two in front of the Theta so in your head it might be a bit tricky to keep track of everything what if it's 3 Theta + 45 are you going to do that in your head to make sure values work or you going to do what I would say is a smart thing and make your life a bit easier this is super easy all you do to change these limits is do exactly what it says so in this case it says plus 30 so I'm going to plus 30 to all parts of the inequality so- 150 cuz you know - 180 + 30 less than equal to Theta + 30 less than 180 + 30 210 all I've done is applied the exact same thing and if it was two Theta I would also double them so I'll double them then add 30 so - 180 double that is - 360 + 30 be - 330 and that way I'm actually looking at these for the beginning part which is very very useful so that is why I'd recommend you do is you should change the limits just so you don't you have to think as as little as possible remember in an exam you're under pressure you're panicked your adrenaline is pumping through your body so you're going to make small mistakes so if you limit those by reducing the amount of thinking you have to do as weird as that sounds you will make fewer mistakes okay so there are a few things in here which are a bit spooky a little bit spooky first of all it's a negative answer second of all it's a negative angle so let's draw our cast diagram out before we go any further because that is what we should always do because it's a negative answer you would be right in saying we need to put it in the areas where tan is not positive so we're not putting it in tan and we're not putting it in a we're putting it in these two but remember it's a negative angle so instead of it being this way around it's actually going this way because it's a negative angle okay just to be super transparent and now what we're going to do is we're going to go both ways around and figure out what on Earth is happening now this angle still is 68.2 that's no harm no foul now we want all values between -50 and 210 so if I keep going negative for a second so if I try and work out what and again I'm going to do these in different colors just to make it a bit more transparent if I try and do that what I'm going to have is 180 plus 68.2 so 180 + 68.2 but remember I've read this clockwise so this is a negative value that would be negative 2 48.2 but if I look at my limits - 15 is the smallest I'll go to so I'm going to cross that out that one doesn't count okay now what about if I read counterclockwise that means I'm getting positive angles so let's go what would that be well that would be 180 minus 68.2 which is 11.8 is that inside my limit it more than certainly is cool we can keep going we can now go and read this reflex angle here what that be well that is a angles in a point add up to 360 that would be 360 minus 68.2 which is every 360° can you see the range here is 360° you're going to get two answers so far that's proven true only if there is no number in front of the angle thing the Theta then more interesting things happen if there's a two in front we have twice as many answers there's a three in front we have three times as many answers there's a four in front we have four times as many answers likewise if there's a half we have half as many answers okay what I'm going to do from now on is I'm going to do I'm going to draw Green for reading clock uh counterclockwise so the positive values okay so counter clockwise I'm going to leave this on the mark scheme just so it makes it a bit more easy we be negative because red is negative green positive right that makes sense to me like that blue is going to be the we get from the calculator because it's just a value we get from the calculator perfect are we done no we are not because that's Theta + 30 we're looking for Theta so now we need to subtract 30 from all of these values again these two we don't need to bother because we know they're going to be outside the range because we adjusted our range smart so minus 30 from this you can use a calculator if you like I'm not going to judge you and the green one if we subtract uh 30 lost my calculator for a second there 11 1.8 - 30° is 81.8 De those are our two values that is a four mark question not bad not bad I'm hoping not bad anyway we're going to see the next one is our first five marker it only has one additional thing there's going to be a number in front of the Theta but that's it no more scary than that so we going to my recording software it's been about half an hour already and we've done like four questions so this is fun we can start speeding up slowly though so with this I'm hoping you're thinking yeah not not not that bad not that bad there's there's a two in front of the X but who cares it's just a two right two is a weasel stuff five mark question just because there's a number in front of the X okay easy but remember for every 360° we have two values we only have 180° here so that's one value but because of the two we're actually going to have two values again but anyways let's forget that so first things first I would change the range so we're going to times all of these values by two and then add 15 so 0 * 2 is 0 add 15 is 15 it's remember to it's remember it's good to remember bid Mass you have to times first if I add 15 then double I get different boundaries okay then that's going to be less than equals to by the way one other thing which is absolutely devilish it's so bad that they do this right sometimes they don't put the equal sign sometimes they do we have to be so careful in our examples so far because I'm not a bad person I've made them so that the values are between this anyway but there have been questions where it's gone from 0 to let's say 360 and it includes 0 and 360 but people cross out the 0 and 360 likewise there have been times with 0 to 360 not inclusive there's no equals and you drop marks and it's so harsh please for the love of God remember this okay keep looking back keep double checking and for the upper value doubling 180 gives me 360 adding 15 would give me 375 perfect now we can actually get started and let's have some fun I mean I say fun all we're going to do is inverse cause 0.3 so inverse cause 0.3 and again I'm going to write this in blue because it's just a number 72.5 it says one decimal place let's leave it 72.5 now let's draw our little cast diagram you know what let's a bit straighter than that these shaky disease ridden hands that I have man honestly so we have cast now the answer is positive and the angle is positive so because the answer is positive we're going to draw lines where the cosine is positive which is in C and A remember a stands for all so C and A get a line and because it's a positive angle I'm going to draw it counterclockwise so again remember those rules first of all if the answer is positive you draw lines in the positive quadrants you don't need to think about the angle just draw the lines the second thing you think about is if your angle is positive or negative if the angle's positive you draw it counter clockwise if it was - 72 then I would have drawn this instead okay so now we need to think of our angles so first things first what I'm going to do is find the other positive angle so that means I'm going to go counterclockwise so that is going to be because this angle here is also 72.5 remember when you draw the lines the angles always the same then that angle is going to be 36 60 minus answer which was just by c2.5 which is 2875 let's draw that in green okay now what I'm going to do is find my negative answers so I'm going to get rid of this line here Boop my negative answers are going to be - 72.5 before I continue if I have a look at the range it's 15 to 375 so in other words I'm not considering any negative values right so if you're comfortable you can just ignore this part if you're comfortable because you can see I'm not going to need any negative answers so why bother even writing them because all I'm going to do when I write them is immediately cross them out because we're not even considering any negatives so if you want you're more than welcome to just ignore this part but for those of you that want to you know you need a little bit more that's fine and then this next value would be 360 - 72.5 again which we've already done is 287.4 and I am aware that I did the negative afterwards which again is not in our range cool so we're practically done can I just okay I did write that in green cool so if I'm now working out x what would I do well I would subtract 15 from both sides then I'll divide by two so let's do that 72.5 again think of it in terms of algebra you wouldn't divide by two then minus 15 you would subtract 15 then divide by two those give you two different answers so subtract 15 and then divide the answer by two and that gives me 28.75 but it says 1 decimal place so 29° uh sorry not 29 what am I talking about 28.8 de and for the next one we're doing the exact same thing so 287.4 minus those 15° and then divide by two 13625 so 136.34 five marks man like I get that this isn't the easiest topic but I'm hoping that even if you're a bit confused you can see that once you get the hang of this it's so many marks for something that isn't really that bad and I want to point out one more time there are two other methods which I would like you to explore if you're really uncomfortable with the cast diagram if you want to more than welcome to every year I've had students that either use the rules if you look up trig equation rules you'll find it online and the graphs and they do well they get the answer but I will say that they hate the topic far more than people that use the cast diagram because with the cast diagram there's less to remember sure learning how to use it takes a bit longer but once you know how to use it I'm going to be real with you sometimes there was a point in my life where I had gone a year and a half without tutoring any a level maths and then I had to to a level maths and the first topic was solving trigger equations and with the cast diagram I was able to do it still you know it's it's not that bad it's a very useful tool okay so for this question there is another slightly new thing however you're actually going to figure out that it's not anything bad so we have this solve between 0 and 360 5 cos x - 40 = 2 now you might think what does the five do in front of the C well that's a really good question what if we just divide both sides by five so the reason why we have to change things for stuff like this is because we can't get rid of like for example the minus 40 we can't get rid of it before we do the inverse CA but if you can get rid of it before you do the inverse cause for example this five then get rid of it divide both sides by five first that'll be my number one piece of advice and then it just looks like honestly a bit easier than the last question we did so 2 over 5 which is of course 0.4 now what I'll do is I'll quickly apply whatever this thing is inside the bracket to my inequality so I'm going to subtract 40 from all of it so that's minus 40 less than equals to x - 40 320 okay cool now we do inverse CA blah blah blah all that fun stuff easy peasy lemon squeezy so let's do the inverse C real quick inverse CA of 0.4 gives me 6642 it says two decimal places so we're going to keep it to two decimal places I know shocking equals 66.4 2° like so now let's draw out our cast diagram c a s again our actual answer bit here the thing on the right right hand side is positive so we're going to draw a line in both positive quadrants for cos which is of course A and C my angle is positive so we're going to draw it counterclockwise so 66.4 2 to work out the other angle well we can work out the negative ones if we want we do technically have a little bit of a negative here but if you notice so negative means we go clockwise that's going to be 66.4 2 cuz remember the angle between the line and the horizontal axis is always going to be the same for these two so we can see that's netive - 66.4 to CU again it's clock uh clockwise the other negative value is going to be all the way around like that and that's going to be 360 - 6642 so we do 360 - 66 6.42 which is -29 3.58 - 29358 de both of those so far are not suitable so let's find the other positive value so if we get rid of the negatives actually just this big negative one if we're going around this way we're once again going to have 360 minus 6642 and that's going to give us positive 29358 which is indeed inside this range so if you look at this these two are not inside the range but this first uh this first one here and the last one R so now all we need to do is add 40 to all of these values so that would be 106.4 and also this one will be 333 .58 de it should also be noted by the way and this is actually something that's a bit smaller is sometimes in very rare situations but it does happen they might say two three significant figures and if you notice when I added 40 I gained a significant figure in which case I would have to trim off one of the answers so let's say if it was for Sig fig here when I add 40 for fig is actually only these first four digits that's another thing they can do to really screw you over annoyingly however in most cases they do decimal places because they're not Pooh heads another new style of question so we have a tan S = 3 and aha I hear you say probably not you just square root both sides right you are right yeah we don't need to deal with this this is not part of our jurisdiction don't worry about it because we can get rid of it before we do the tan thing we can just square root it but you can probably tell by the sound of my voice things are not as it seems remember remember when you square root you get a positive and a negative so this is where we're going to have to split our page in Twain so split it in half down the I mean approximately down the middle we have Tan x = positive < tk3 and we have Tan x = < tk3 now this doesn't change the limits or anything so we're still looking for only positive values so you might rightly think there's probably going to be no values on this side maybe maybe but we're going to check anyway because that's what we do so we're going to do both of these nice and easy so get our Tan x out we're going to do shift tan we're going to start with the positive ones just because it kind of makes sense to me of course root three I pressed F3 instead of uh the number three 60° okay fairly easy so 60° let's draw our little cast diagram it down here c a s and t now we have a positive answer so let's draw our lines where tan is positive which is in a and t and we have 60° which is positive so we're going to start from here now here's where I'm going to start showing you some shortcuts because we're not looking for any negative values I'm not going to bother reading the clockwise version hope that makes sense right it saves you a bit of time but I want you to kind of get comfortable with these small shortcuts that you can take and then our other positive answer again as whoopsies as we know this is still going to be 60° right because it's always the same so that's 180 a flat line plus 60° which is going to be 240° and that's it right there there's no other values within 360 cool now let's do the same thing but with < tk3 so change that to a minus sign -60 probably could have worked that out ourselves but it's always nice to do check so let's go through it if you remember whenever you have a quadratics or a squared It generally speaking increases the number of answers that you have generally actually doubles them so let's see if that holds true for trig equations so first of all you notice that we're in we've got a negative answer which means we need to put a line where tan is negative which is in the S quadrant and the cosine quadrant now it's a negative value so we drawing this clockwise now do I care about negative values the answer to that is no so I'm not going to bother doing the other clockwise value I'm not going to bother right because there's no point so I'm not going to do any clockwise values because we don't care about negative angles so what am I going to do instead what I'm going to do is I'm going to go counterclockwise remember you can always read clockwise and counterclockwise always if I wanted to I could read negative values from this one on the left I could read negative values from the one on the right likewise I can read positive values from the one on the left and positive values from the one on the right there's no limit okay so you can always do both so this would be 180 minus 60 so that would be 120° which does fit inside this boundary right so that works and likewise if I continue this line going off that would be 360 minus 60 which is 300° which also works so in other words it does actually have multiple more values it has 1 2 34 has double the number of values so at the end I'd literally just write in a box x equal to 60 120 I like to put them in order of size I don't know why I'm just like like built that way you know those are our four answers for our five marks again for a five marker it's not that bad there are a far worse questions that are worth fewer marks in a level maths believe you me but hopefully that makes a bit of sense and hopefully you can see what kind of happens again the aim of these videos is to expose you to all different types of questions again I want to preface this one more time I've left this kind of L into the video but this isn't covering every exam question that exists we're doing that a bit later we're going to go through whole exam page papers so you will find other questions but you should be able to use what you know from this video to answer them but if you're ever unsure you can always email me my emails in the description you can also leave comments and here is the final type of question and as you can see it's a big fat nine marker so you might be slightly intimidated but do not worry in as not too bad there are two equations in as that you need to be able to use to rearrange triger equations so if you look here it says show this equation cos^2 x = 4 - sin x can be written like this and there are two equations you can use so I'm just going to write out this initial question first and then I'm going to show you the two equations first of all that you might be expected to use is the fact that Tan x can be written this and the next one is this in A2 so in year 13 you you will be expected to use a few more but those are the only ones you need for as okay only these are the only ones you need to use for year 12 for your year 12 marks and all the exams in year 12 which is really handy because it's actually quite simple if I look at this and I look at the answer what is the difference well first things first is T here at all no okay so I'm not using this one I have to use this one that's how easy it is if you have tan you're using the tan one if you don't have tan you're not using the tan one easy easy now let's actually look at the difference so we have sinx sinx that's the same four and two okay they're still numbers so that's still the same same in air quotes but if you notice there's cos squ here and no cos squ here so all they're doing is they're rearranging COs squ in this equation and subbing it in that's it done that's how easy it is and that's why I've only put one question in here just as a recap if there is a tan then you use this first one there's not you're using the second one then you see what disappears either cos squ becomes sin squ or sin squ becomes cos squ that's it that's the only thing they can give you in as so all you need to be able to do is rearrange this but come on man you just minus sin squ or you minus cos squ that's it so we want to rearrange for cos squ so I'm just going to minus Sin squ from both sides so cos squ x is just 1 - sin 2 x and that's how easy it is sub that in rearrange Bish bash BOS you're done so 6 1 - sin^2 x = 4 - sin x expand we have 6 - 6 sin^2 x = 4 - sin x and now you have to decide which do you want to move everything to the right side or to the left side we have a look at the answer the sign squ X is positive so I'm going to be moving everything to the right hand side okay because again here it's negative so I need to move it over to make it positive yeah that's pretty much it I was going to say some more stuff but there's no real need to so add 6 sin squ x to both sides so we have 6 sin^2 x then I'm going to put the minus sin x because that's what it tells me to do in this equation here and then 4 - 6 is - 2 = 0 which as you see from the answer is exactly what we've done easy as Sin it then says hence solve and you guys know how I feel about the word hence hence literally just means use part A so I'm not going to write this equation I'm going to write this equation minus sin x and how do we solve that well if I have a look this is a quadratic equation right if I replace if I write the general form form for a quadratic equation instead of X we just have sin x that's it so in other words x equal to sin x okay so all I'm going to do is solve my this equation using the quadr the uh calculator the quadratic formula on the calculator except it's not the quadratic formula I keep calling it that but it's not so menu down twice one to the right equation funk polinomial degree since it's squared it's going to be two type it in as if it was this so I'm going to get rid of this for now because we don't technically need it so the number in front of X or in front of the squared term is six number in front of the nonsquare term the linear term is minus one and the constant term is min-2 equals twice so we have sinx = 2/3 and we also have sinx equals something else uh minus a half we're getting a load of marks already this is two marks already I haven't really done anything yet now what do we need to do well it's the exact same as our previous question we need to do two separate um solvings so what I'm going to do is I'm actually going to move this I'm going to move it now that we've done with the hence bit I'm going to move it under here because we've done the hence part so what we're going to do is I'm going to spread this out a little bit just so we can have a nice bit of room straight line right and then all we're going to do is solve it as per usual what is our bounds it's 0 to 360 so let's write this down over here as well 0 less than equal to X less than equal to 360 so let's go back to our original thing so up twice to the left we're going to do inverse sign 2/3 that was close 41.8 does it say how many decimal places one decimal place 41.8 by the way where it says where appropriate what that means is if you got a whole number let's say 40 you don't need to add the point Z it's only where you need to actually round that's a fat thing that's 10 St okay so let's do the good old fashioned cast diagram c a s t again I'd recommend by pausing the video now and attempting these two ones yourself just really quick on a piece of scrap piece of paper just draw it out quickly and since it's a positive answer we're going to draw our lines where sign is positive so that's in A and S and I can see that my angle itself is positive so that is here and then because I don't care about negative angles I'm only going to draw the positive angle here so as we know this is still 41.8 of course it is so we're going to do 180 minus 41.8 so8 - 41.8 gives me an answer of 138.236.128.7 with the negatives because we don't care about negative angles so here let's do the same thing s minus I'm going to do 0.5 because I can't to use the fraction button you find I'm dreadfully lazy okay again cast diagram good old fashion by the way I know I get some I get this question quite frequently is can't I just use the same one the answer is absolutely if you want to the reason why I do two separate on is because when I'm teaching you guys if I start drawing an extra two lines and doing all that stuff it can get a bit complicated and confusing so for you go ahead but bear in mind you might mix yourself up for me I'm not going to do that so minus a half means we're going to do it where sign is not positive which is in the CT areas regions quadrants whatever you want to call them and it's -3 so indeed we're going to go this way around now I don't care about negative values do I so I'm not going to go clockwise anymore I'm going to go anticlockwise in fact I'm going to put a nice neat line through there so if I go anticlockwise or counterclockwise I have this value first so that's going to be 180 + 30 which is 210° and I'm also going to have 360 minus 30 which is 330° both of which are inside this region so once again it's a quadratic so I get four answers makes sense right so my final final answer I put at the bottom and I write it all together just to make it super super clear to The Examiner to my teacher to whoever is reading this that I've got enough answers again they only check your working out well okay not always but they check your working out after theyve checked your answers if you they can clearly see you got all four answers right just tick it tick it and that's a six marker that just tick for free so that was a really long video so well done for making uh through making it through the whole thing I'd recommend finding other questions and also attempting these questions again by yourself in a day or two again links in the description uh feel free to share this like it subscribe Etc if you do enjoy this kind of content just to keep me motivated in making it um but yeah hope you have a wonderful day