A Level Maths Tuition Lesson 9 Sine Rule, Cosine Rule and Areas of non-right angled triangles

hmm foreign hmm okay let's get started hey guys welcome back to another one of the a-level free tuition lessons today we're going to be doing all of the cosine and sign Rule and area of a triangle questions that can come up so I've got another video on this but for the GCSE side but to be honest the a level and GCSE content are very very similar for this topic however they can it's going to be obviously on the higher end right so it's going to be a bit trickier so without further Ado let's get into it so we have a triangle ABC so what I like to do in all of these questions is to draw out exactly what we're talking about so we can just label them from left to right A B C side a b is length 15 centimeters AC is 12 centimeters and BAC so that's the angle between them is 60 degrees and for a it just says find the length b c it's looking for this one here so all you need to do here is if we label this X for example is we can use the cosine rule because we have it's pretty much the same however there are some differences so I would say it's probably on the higher end of the GCSE Spectrum but there are some kind of quadratic equations of stuff that come up a bit later but it is fairly similar okay and as you can see we have two sides and the angle between them so to work out the opposite side to the angle we're going to be using the cosine rule as a reminder yeah sure so so far all I've done is I've literally just started this is the first question okay so we've literally just drawn out the diagram and labeled the sides see cos a now if the letter is kind of confuse you just remember that the side a is always opposite the angle they give you and sides B and C are the ones that make up the angle okay so you have these three pieces and the opposite side so if we decide what we have here that means x squared is equal to B squared it doesn't really matter which way around to do it so if we just say 18 squared plus 12 squared minus 2 15 12. then cos 60. so again shouldn't be too bad just do the whole thing on a calculator so if I just get my calculator up so we have 15 squared Plus 12 squared minus two B is again 15. again it doesn't matter which way around you actually put them because you're just adding them all timesing them so this calculator you can get it from the Casio website you can get a free trial for a few months but I actually paid for mine because I use it in so many different things so yeah equals so it doesn't actually equal that equals the square root of that because don't forget it's a squared not just a three week 21 but doesn't tell if this is uh if they want it to any kind of significant figures so we're just going to write 13.7 that's it and that's three marks so hopefully not too bad if you have any questions please stop me and let me know and I'll continue bear in mind there is a delay on the stream though so circles and absolutely that will come up uh shortly let me actually see exactly what I plan to do that so the equation of a circle we did two weeks ago and we did the transforming graphs four weeks ago so you can go back and have a look at those so two weeks ago there was the um Circle lesson which is the equation of a circle and all that stuff and the puppy just has calculated the area so remember the area of a non-rectangle triangle is just half a b sine C where again A and B are just the two sides that make up C so if you have like an angle like this that'd be angle C and A and B are the ones that make up that angle so we only have one angle it's 60. so let's just choose these two sides and that will give me the area yeah so a half so I'm just going to call it 0.5 times 15. it's 12 and then sine 60. 45 root 3 which is 77.9 centimeters squared and again don't forget the units because they are very particular about the units so you're not too bad to begin with but it's going to get tricky and trickier as we go along so please let me know if it gets too tricky okay so again uh this question again it's fairly similar but it's slightly more tricky we once again have a triangle ABC so let's just draw this out a B has length eight centimeters BC has seven centimeters AC is six centimeters and we need to find the size of the angle ABC so a b c would be this angle here let's call it X degrees so for this question once again we use the cosine rule because we have three sides and we want an angle okay so we have however we don't actually want the side we want the angle okay so if we label our sides up this is going to be a is going to be capital A and then B and C can be the two others it doesn't actually matter which way round is okay so if I want to rearrange this equation for cos a I would add move it over to the other side so I have 2bc cos a plus a squared equals B squared plus c squared subtract the a squared to move that over to the other side and we get 2 BC cos a equals B squared plus c squared minus a squared and it does matter what you call a squared but the other two it doesn't actually matter and then lastly divide by 2 BC and we get cos a equals B squared plus c squared minus a squared over 2bc nice now you can actually just go straight for the angle by inverse causing everything and putting that all in a bracket so B squared plus c squared minus a squared so again B squared plus c squared that's 8 squared plus 7 squared a squared is 6 squared in this case and minus six squared over 2 BC so 2 8 and 7. now give your answer foreign squared minus 6 squared over 2 8 and 7. not too bad hopefully again let me know if you are stuck or if that doesn't make any sense now we just write and that's the three okay and lastly all you need to do is find the area of the triangle so again all you need for the area of a triangle a half a b sine C so all you need is two sides and the angle between them which we now have because we worked at the angle between eight and seven so the area is half A and B are the two sides that make up the angle so eight and seven and sine 46 . five mm-hmm yeah we get 20.3 centimeters squared so again hopefully this is just a recap of what you've done in GCSE okay so hopefully that seems kind of familiar maybe some of the harder questions in GCSE but there any questions about that at all because it's going to get a bit more tricky now perfect good here again hopefully that's just a bit of a recap so here is where something gets a bit more interesting okay so they give us a bunch of letters and numbers if you notice they don't tend to give you the actual triangle so in Triangle d e f so let's just draw out relatively big triangle and just call it def and I don't actually know what it looks like but it's a good model Ed is five centimeters so let's do that EF is six centimeters given that sine of angle def so that would be this one is two-thirds so let's call that angle X so sine X is apparently two-thirds they also tell us that d e f is acute so in other words it's less than 90 degrees it now says find the exact value of cos okay so here's where things can get a bit more um tricky or a bit interesting if we come up with a triangle for sine X right I'm going to make a right angle triangle out of what this angle is here now if you remember from your sohcahtoa sine Theta is equal to opposite over hypotenuse so in this triangle here that I'm creating and again whenever they ask you to find the exact value of cos and they give you sine or tan or vice versa you're going to create a right angle triangle with this okay opposite of hypotenuse but we know that our opposite of hypotenuse is two-thirds so opposite is two hypotenuse is three okay and if you remember that cos Theta is equal to adjacent over hypotenuse which will be this side here so I already know that the hypotenuse is three I just need to work out the adjacent so if I have a right angle triangle I know two sides I'm going to work out the third I'm going to be using Pythagoras and if I have the hypotenuse and I want to work out one of the other sides all that means is I do the hypotenuse squared subtract the other side squared now that's pretty tricky but this comes up in mechanics and it comes up a lot in your paper one and paper two okay so it comes up in as and A2 content so if you're not sure please let me know it's a bit tricky ah okay really good question so I'm just gonna write this out and I'll get to it so the question is how comes we can use this when it's not a right angled Triangle Well what they've given me is that sine of this angle in this not right angle triangle right so this triangle here isn't d e f it's not this triangle but if they tell me that sine of an angle is two-thirds I can create a new right angle triangle from that information okay so can you see that okay because sine of an angle is always equal to opposite over hypotenuse so I draw the second triangle because now I know what a is it's root 5 I can work out what cos is absolutely sure so let's start from the beginning okay they tell me about this triangle here okay this triangle is not actually needed for part a okay it's not needed so if you want we can move this slightly further up it's not needed for part a the only information they give me is that sine X is two-thirds now if I just told you that sine X of an angle is two-thirds okay so if we ignore that and I just said sine X equals two-thirds what's cos x then would this working out make sense so ignore the above triangle if I just gave you this information would it make sense that we can do this because what you'll find with a level questions is that they're quite long they give you like a b c and d and they're not always fully linked okay so if I give you that sine X is equal to some fraction I can always say because that's what the answer is that's what the question says find cause so what I did is I just called def X so all they're asking me to do is find cos x that's all that the question's asking you to do so again because of sohcahtoa because that's what they've given us they gave me sine X and they want me to find out what cos x is these are all really good questions by the way it's good you're asking this because this is a brand new skill for a level it can be a bit tricky so in general if anyone says sine X equals this what is cos x you can always write it as a right angle triangle because sine X is always equal to opposite over hypotenuse of a right angle triangle mm-hmm does that make sense ah perfect it's a really confusing new skill right but this is what I mean by there's some new stuff in the cosine rule and symbol when it comes to this it is quite tricky so that means in this case cos x which as we said over here is adjacent over hypotenuse is just going to be root 5 over 3 and that's it that all okay so I think quite a few people just joined if you're unsure of what we just did let me know and now is probably where we have to actually deal with the um other triangle so to be honest this information isn't needed for the first part so what we can do is I can take this and just drag it down to where I need it and now from Part B we're actually using the information that we need okay now sine X equals two-thirds then I can immediately work out what the answer is right I've seen your comment just give me two seconds of writing something um so basically use the sign rule you sign to label the sides of the triangle and then from there you use Pythagorean cos then use the yeah so in part to B and C we're actually going to be using the sine and cosine rule but you don't need it for the first part which is annoying but yeah well done and we're going to be working the side DF out uh actually let's not use eggs for this question really well done no problem yeah it's it's not too bad once you understand it right but it's the understanding like the the leap it's it's quite difficult but once you get it it's not too bad and now you know it you're going to be able to do it really easily uh in future yeah I I think it's a bit of a tricky jump to make okay so that angle is actually 48 point so 41.81 even if I can learn how to read and perfect okay so Part B literally just says find the length DF so I've drawn the triangle out I'm just looking for the length DF okay honestly I wouldn't mind but there's not really much points I don't think there's many other people here I think it's just you so I don't think it really means anything um yeah just work out uh DF so we have two sides and the angle between them we want to work out the third side so that means we're going to be using again the cosine rule again classic so in this case my a squared is y squared and B squared doesn't matter which way around we do it five squared plus 6 squared minus 2 5 6 cos 41.81 and then all we need to do square root it and get our value nice and easy so 5 squared plus 6 squared minus two five six cos we'll see one and square the answer 4.03 let's call it centimeters three significant figures so again that was another four marks four marks just for that it's not too bad most of your papers are probably out of 80 marks depending on what paper you're doing so I don't personally find that tricky and then lastly part C going to be using find the angle e f d Okay cool so e f d that's going to be this angle over here and let's call that said okay so what we can do is actually replace these letters now so why we've worked as 4.03 we've worked out X to be a fairly long number but 41.81 okay so if we want to work out a an angle and we have its opposite side and another angle side pair so can you see that this angle is opposite this side and we have this side which is opposite the angle we want we can use the sine rule you can also use the cosine rule again but that's quite long okay so I'm just going to use the sine Rule and the sign rule just simply says so again I've seen your message there's going to be two seconds to write out the sign Rule and I'll read it was a a level math I say do really about the BL section can be good so yeah so how about after the lesson I'll go through all of that stuff is that okay I'll definitely go through all of that so yeah cool perfect um so we're going to do here or I'll just talk about it after this question um whichever one you prefer so all we have here is this is the angle we want we have its opposite side we have another angle and another opposite side so sine of the angle I want which again I've just called Z the opposite side is five centimeters equals sine of the other angle which was well actually we already know what it is it's two-thirds so we don't have to write this out but I'm going to write it out anyway so instead of sine 41.81 I could have just put two thirds because that's what it told me it was over 4.03 and now this is a simple rearrange um rearranging so if I just times both sides by five I get that sine Z equals 5 sine 41.81 over 4.03 so if I go back up here so this is going to be the inverse sine 5 sine 41.81 over 4.03 and we're done a bit long but not too bad inverse sine bracket five sine 41 point a one close bracket down 55.8 degrees to three significant figures Okay so the question just got asked which is really good really important um is essentially if you do all of us virtually all exam boards you have three papers you have paper one again if they're called units it doesn't matter right but you have three papers you have paper two and then you have something called your applied paper right so again if they're called something different in your um exam board it's still the same thing right um You have one paper two paper and the applied section which is the stats and mechanics right so the question was if I do really bad with one of the pure sections can I make up with it with stats and mechanics so because there's three papers in total they're all going to make 33 percent of your grade right whoops so that will give you a third of the grade so if you think about it that way let's say you get zero percent on paper one and then a hundred percent on the next two papers you still get sixty six percent it'll actually be 67 because of how that works and what that will be is usually it's equivalent to a grade A or B so you can make up for it but it's best to kind of aim for like an average it I haven't really seen most of the time people actually do worse in applied than they do in pure most of the time so but you can technically make up for it right but that's the way I would um work it out I'd work out what percentage I got from this so let's say I got 20 in total from here another 20 that's 40 I'd have to gain 30 here to get 70 you know so the next question basically said uh how many papers is there so it's three papers how many marks all of them are worth the same and it's usually 80 marks but they're all worth the same amount and lastly what's the grade boundaries great boundaries for maths change all the time all the time every year it changes by quite a bit sometimes it's been as low as 50 for an a sometimes it's been as high as 80 for an A so it varies so much you can look up last year's grade bandages if you want yeah so with applied it's all of mechanics and stats so every single bit of mechanics and stats only appears in the applied paper for paper one roughly it breaks down to the stuff you did in as and this stuff is the stuff you didn't Aid to so generally speaking it's a lot of the integration differentiation there's a lot of proofs whereas in as you have things like the cast diagram you have algebraic division yes yes that's correct paper one and two is only pure paper three or the third paper is only stats and mechanics okay hopefully that answers your question um and that should be okay for that yeah in general people people want to do a pure so you might hear them called Pure one and pure2 it doesn't matter two of the papers are pure only third paper is stats and mechanics only yeah good question to be honest paper one tends to be the easier one you get one grade you get an overall grade they'll tell you what you've got in each paper like a percentage or as marks so they might say you've got 60 in paper one 70 in paper two 65 and paper three but you get one grade which is your maths grade okay hopefully that's all excuse me sorry I'm drinking some water and then we go into that question foreign again it depends on the grade boundaries but 86 is very good to be an A or an A star in most years okay in fact I'm pretty sure in almost every year that's been an a star so again just look up last year's grade bound which is the best way to do it I mean tomorrow's the best way to do it is just aim for 100 in every paper because the thing is if everyone does really badly in the country the great boundaries go down if everyone does really well the great boundaries go up so it depends on what happens in your year cool anyways let's do the next question so triangle pqr so let's draw and call it P okay uh PQ length nine PR length 10. and the overall area is 30 centimeters cubed that's where it even okay yeah so what it will do is without average it so the way it would work is they take the mean right so let's say you take you get 30 percent in one paper and then 80 and 80 in the next two they'll add them all up and then divide them by three so that gives you 190 divided by three you would get 63 percent so all they do is they take the average they take the mean okay find that set light uh length of side QR so the thing is we can't actually go straight to working out what the length of the side is so if you're ever unsure in an exam of what you need to do just try and work some stuff out okay so if they give me the area let's think about what the area is where the area is half a b sine C so they give me the area I can work out the angle which would be this angle here okay so the area is 30 that equals a half a B's nine and ten sign y so that would be 30 equals 45 sine y divide it over you get 30 over 45 equals sine y which means y is just inverse sine 30 over 45. 41.81 degrees okay that does like bring me closer to working out the length of side QR well I have an angle now and two sides which means I can use the cosine rule to work out the last side so again in an exam also keep in mind everything you work out in an exam will give you marks okay so even if you weren't sure what to do with this after you've done it you've actually already gotten three of the five marks available right so even though you haven't got an answer you've got most of the marks so never leave a question blank okay if there's one thing you take away from this lesson just make sure you always fill in the answer like fill something in okay because I will give you marks so we have 9 squared whoops messed that one up it's 10 squared minus 2 9 10 and then cos 41.81 all you have to do now is square root it and you get 6.84 centimeters 3sf okay puppy just says find the angle of p q r p q r would be this angle over here which I'm going to call Z a bunch of different ways you can work this out um but probably the easiest one is the sign rule so sine Z over the opposite side which is 10 centimeters equals the other angle side pair we have which is sine of the angle we just worked out 41.81 over the opposite side to that which is side x which we worked on the previous part is 6.84 so therefore sine Z if you times both sides by 10 you just get 10 psalm 41.81 and again please stop me if you have any questions at all it's the best way to learn and then you get Z equals something so Z is just going to be the inverse sine of that there we are well inverse sine 10 sine 41.81 over 6.84 77.1 degrees another answer okay how was that I don't think it was too bad again just try and work out as much as you can generally speaking you get marks for it I have two more questions and again the last one is actually really difficult well I say really difficult it's it's advanced so be sure to stick around to that I can skip ahead to the more advanced one or we can just go through the next one as well in fact okay this question is also um quite interesting because it combines what we did last week which was the cast diagram but this is the extreme one okay so let's have a look at this one so let's try our triangle and you're probably going to notice something but okay I don't know what happened there no idea what happened there so a b c a b is X plus three a c is X an angle between them is 60 degrees work out the value of x okay so again I said this is pretty tricky uh it's not it's not that bad but the problem I find with students sometimes is they look at something like this and they draw something like this and they panic and they shut down okay now think about it just use the cosine rule well done that's exactly what you do um oh I actually forgot one of the other sides all you need to do is use the cosine rule easy right so the cosine rule says to the opposite side squared so let's do that X plus 2 squared we'll equal the other side so this case X plus 3 Plus um a squared it's x squared minus two X plus three X cos 60. and then all you need to do is find out what the value of um X's so as you may know cos 60 is one half so that means that this 2 times a half actually goes away because you have 2 times a half which is one so this actually become a lot easier we have X plus 2 squared equals X plus 3 squared plus x squared minus x times X Plus 3. so all we need to do now is it's a quadratic equation so we just need to expand simplify and then use the quadratic formula or whatever else we want to do to work it out let's expand this so I become x squared plus 4X plus 4 equals x squared plus 6x plus 9. if you're unsure of how I got to any of this please let me know so I'm gonna leave it there for a few seconds make sure you can expand that in your head make sure that's okay or if you want me to go over in more detail let me know perfect so these two x squares cancel out and now we're going to simplify both sides we have x squared plus 4X Plus 4. equals x squared 6X minus 3x gives me 3x Plus 9. so now if I I can actually get rid of both of these X squareds because I can just minus x squared from both sides so that gives me 4X plus 4 equals 3x it's 9. 3x is smaller than 4X I'm going to subtract 3x from both sides we get X Plus 4. equals nine minus 4 so we get x equals 5. so that was one of the um I've mentioned this in quite a few of my videos but 10 of your marks have to be unseen questions okay which is where they give you something you've done before like the cosine rule but they give it in a situation you've never seen before this was one of those in um unseen questions and most students didn't even attempt it and the ones that did uh got it wrong so now I've gone through it you might see how easy it is but in an exam it's very easy to panic when you see something you don't recognize yeah which hopefully makes sense so again in an exam if you come across something you don't haven't seen before which is guaranteed Don't Panic just try and use what you know because they have to give you something you're able to do okay so we have some time if you want we can go through this question which is uh they're also squared squared squared and squared right so in the diagram all the lengths aren't squared but in the cosine rule we need to square all of them right so the cosine rule is a squared so all of them are squared it might just be my handwriting is a bit bad oh good question Okay so in the cosine rule this side has to be opposite that angle okay so these two things have to be opposite if I have a look at my diagram the angle 60 degrees is opposite X plus 2 squared because X plus 2 is the side BC and the angle is BAC so the only restriction on the cosine rule is that the A and the capital A have to be opposite each other for it to work so in this case X plus 2 has to be squared equals to the rest of it uh does that answer your question does that make sense well done Alchemists that's exactly right okay so the last question that we'll go through um again this is one of those unseen ones because they've combine two topics okay so let's draw it out again we have triangle ABC so this question and the question we just did are some of the hardest questions you'll find with the cosine and sign rule so if you're okay with it brilliant if not that's again fine and just go over it one more time 10 centimeters BAC is 30 degrees and it says find two possible sizes of the angles ABC if you see here giving your answers to two decimal place is so I think this is quite important but if I asked you sine X equals a half what does x equal I'm going to assume that most of you would go to your calculator and do inverse sine of a half right and you'll say that it equals 30 degrees and you'll probably say that that's the only value there's actually an infinite number of values so if we look at the sine graph it's probably the best I've ever drawn it actually it goes on forever right now if that's plus one a half is here look how many times the graph touches a half it will touch it an infinite number of times so this is a recap for those of you that missed my last lesson on the cast diagram but essentially for any value of sine causal 10 there are an infinite number of answers okay so when they're saying find two possible sides of the angle well the first things first this is the sign rule right because we have a side and it's opposite angle and we want another side in its opposite angle wait I think I've labeled this BAC that's fine okay slightly annoying but we're going to be using the um cosine ball sorry the sign rule because we have a side and it's opposite aside in its opposite angle and another side and we want to find its opposite angle that'll give us what the value for Z is okay now for y so arguments that's 100 correct um but there are other values of X as well right because you have 180 minus X but you can also do 360 minus X um however we have a restriction on this okay so we have this value and I'm going to work out what Y is yeah so you are 100 correct though well done now value of y 40.5 degrees let's call it actually it's two decimal places the way I'm going to work out Zed right because that's what they're asking me for right they're actually asking me for Zed this angle here the way I'm going to work it out is I'm going to do 180 because all of the angles in a triangle add up to 180 minus 30 minus what Y is okay but they're asking me for all of the possible values of Z because y has multiple different values okay so what are the restrictions we have on angles in a triangle well they can't be negative right they have to be greater than zero right I think that's a fair statement to say is there an upper limit well they have to be less than 180 degrees because if one angle is 180 degrees it means the other two angles are zero which means we have a flat line it's not a triangle so let's look for other values of Y that are within this range so I believe Alchemist you use the um basically the graph but you use kind of 118 minus the value will give you the other value it's 100 fine I use the cast diagram because it gives me some other values as well although in this case it will only give me that one value so we found out that 40.54 the other place where so where the cast diagram works is you draw out your angle and then you think of where else is it positive because again signs positive tear if you're unsure about this I would recommend going to the previous video which is where we did the cast diagram I explain it in a lot more detail but as Alchemist said it'd be 180 minus the answer but the idea is you can keep going around this diagram infinitely right unless we have boundaries okay so if anyone ever asks you if sine X is a half what's X you should say what's the what's the boundaries because there's an infinite number of answers which of course isn't particularly useful and so the other value would be I did use a calculator just being completely honest because I don't really want to do mental maths to that extent yet 139.46 so the values of Z there are two values either Z equals 180 minus 30 minus 40.54 or Z equals 180 minus 30 minus 139 0.46 okay and those would be our two values there are no other values because if we go any higher we'll actually get a negative answer for Zed right because if you go around again it's going to be too big so 180 minus 30 minus 40.54 we get a value of 109.46 that looks about right cool and if we just change the last digit it's 139 point six again I'm being a bit lazy 10.54 that's it so those last two questions we did they were kind of unseen uh supposed to kind of trip you up kind of questions and again if you notice it combines topics right so this one it combines cosine rule with solving quadratics and this one combines um sine Rule and cast diagram or solving trig equations so what you can expect in your exam is that they'll try and combine two topics that they've never combined before so you can kind of maybe predict roughly what they're going to do okay but I'd recommend to doing as many questions as possible I hope that helps do any of you have any questions um Alchemist I think you've been here before but welcome back if not or welcome for the first time but I feel like I recognize your name yeah let me know if any if you want me to go over anything or double check anything again I'm very proud of you guys actually messaging you know using the chat function uh if you don't understand it's the most important thing you can do again treat this like tuition um any questions at all oh wow perfect um oh it is okay fair enough first time I thought I recognized your name but okay welcome for the first time then in year 13 sing for the mouse cool by any chance do you do physics as well Saturday never matter already are you going to redo it to try and get a better grade or this is just for fun if there are no other questions I think that went pretty well pretty happy with an a star I mean you can't really do much better than that really well done do physics as well nice I actually have a master's degree in physics so I did maths and further maths and physics at a level as well generally people who do math and further maths generally speaking do physics as well but yeah nice one but really well done on a star and maths it's really good best combo exactly yeah standard I assume you replied for maybe physics or maybe computer science no you don't you don't need to do further math for engineering you don't actually need further Maths for physics but it will help you in University of physics but you don't need further maths to do engineering no my man absolutely it's a good degree good fun well yeah yeah congratulations to a star math so really well done and definitely well done on doing physics so I actually want to do physics lessons as well I'm just trying to find the time to do it because at the moment obviously I work as well you know I'm not exactly making big bucks on YouTube or any money at all but I do want to do physics videos as well quite desperately because uh that's definitely where my passion is so again I would like to do further maths at the moment uh because I'm working It's tricky to find the time uh because I have like two I also tutor as well so I have tuition classes most days and I have I work in a school so I'm trying to find time because again further maths and physics content is my that's definitely where I want to do stuff so at the moment I'm just trying to go through asna to Pure then well sorry as A2 regular maths then as and A2 further maths so honestly probably be after you finish your um of course unless I managed to stop working before then if my tutoring picks up a bit more then I'll be able to do more kind of live lessons for further math as well so fingers crossed yeah um she's normally run until seven but I think we're all okay with this it's been great talking to you guys um but if there are no other questions have a really good evening have a good weekend stop doing maths do physics I appreciate that Alchemist thank you um stop doing maths I don't know a lot of people have requested maths but I do understand needing physics because it's very tricky I'm gonna try and find some time okay I'll try and find some time maybe in the weekends although I don't know if people really want to watch physics videos in the weekends but yeah also yeah I appreciate that Alchemist thank you very much but yeah I'll let you guys get back to evening enjoy your weekends as well and I'll see you next week at the same time