Yeah, excellent. So to start with our kind of foundations and gcse stuff, I imagine you're very comfortable with the Soka toa stuff. The more exciting part in gcse is the sign in the cosine rule. So we'll start with a little recap of that and then we'll move on to the a level stuff. So Yeah, how do you feel with sign ine and cosine rule? Do you feel comfortable with those? Can you remember how they go? Yeah, Yeah, I can remember. Excellent. I think one of the more exciting elements of it is knowing which one to use when. So just as a starter, I'm going to give you two questions. One of each, and I'm largely interested in seeing, can you identify which rule to use? And as ever with me, if you can't remember exact formula, I'm never particularly fussed about that. I think it's far more important. You know what's in your toolkit, even if you might need a reexactly how it goes. So let's start with this. 16 and probably worth having a calculator on you today. Yeah, I have one here. Excellent. Right. Let's start with this. Let's. Yeah, excellent. What makes you think cyal is a good candidate? What feature of this triangle when there is well a side and its opposite angle are known and then another side or an angle is unknown? Well, spot on. Just a little starter. I'll give you permission. Finish this one off for me. Yeah, excellent. And I'll take it as rich. You can plug that into a calculator. Little bit of a spoiler alert. I told ywas gonna to do one of each. But even so, have a think about how youspot that this one needs the cosine rule. Twis it such as you have three sides or three angles, including the unknown, I suppose. Yeah. So what actually I'm not sure why these numbers are so silly, but so let's go with it. Or if you don't have a matching like angle inside, think about it. Yeah, I think about in terms of the pairs, I think is probably the best way. And then let's go for finding X, Z. So could you show me how the cosine wall would go here? Wait, no. It's a zero, and then with A A B cosine zero A B, C. To bc. Cause so a then big a Yeah. So it's always worth thinking about the relationships between the letters. So I would think of it as it's Pythagoras and then we're subtracting off the two ones we have here and the anvil corresponding to this. It's all it's all patterns rather than these letters that we choose don't mean anything. Okay, excellent. See if you can finish that off for me. I'll take it as red. You can plug that into a calculator. And actually, no, let's finish this one off because simrule executing the last bit is softener and a little fiddly. Let's let's finish this one. I mean, okay, so also. I mean, you've got the right sum. There might be an error in the answers. I no, no, I wrote one. I wrote one. And that calculates with there. So be 9.56. Love it. Excellent. Nicely done. I don't think we need to spend little time on that. So the big transition for a level is thinking about sine, cos and tan now as functions, rather than something we're doing in a particular triangle. We're thinking of this functions anyi'll done in, get something out. And I think the best place to start with that is is the graphs of these functions. So just so I'm not lecturing you, if you already know all about this, tell me where you're at with that. Do you know what any of the graphs look like for sine, cos and tan? Well, sign on. Sign should wait. Sine of one. Wait, sine of zero is one, right? No, it's zero. So it should start from here. Then a cosine should then start from here. Nice. So let's start with just those two. So you're up doing right. We've got the same wobbley shape and we just need to get it the right way around, which starts at zero, which starts at one. The other important thing we need, I'm just gonna to get a fresh one, is to know where our key values are coming. So if we start with sign, this is the main chunk that keeps repeating, right? Do you know what these there are four important angles. And I suppose knowing what's on mas and I'm in, can you fill in any of those? Max is one minimum is negative one. Lovely. That is. A great sign of 90 and 182, 70 and b 60. Excellent. And just while I'm in quizzing you mode, can we put those in terms of high as well? Certain in radians, however. Too is that's 180 right? So that's 90Yeah okay, so 180Yeah. Three, two I and two I. Excellent. So there's a bunch of symmetries and periodicities going on here. The most basic one is that this unit now repeats. So the dotted line. So we say that sign has periodicity of that. 360 or two pi, which means that whatever angle I have, so let's say have some angle here, thatgive me some sign ine value. If I add 360 to that angle, then I'm give you the same value. Exactly. So we can just read off all our symmetries and periodices from knowing this graph, which I think the most efficient way of doing that, all sorts and symmetries going on. So could you comment on if we have an angle theta and 180 minus that angle theater? Sorry, I don't understand. So have a think about if we have an angle theta, and then we think about 180 minus that angle, can you have a look at the graph and see what you think might happen there? Gonna go somewhere smaller than so again, let's pick a random. What if we have some angle? Then can you mark on where's 180 minus that purple angle? So could you hazard a guess what special thing will happen when we have sine of theta and sine of 180 minus theta? They're the same. Exactly. So we have a liassymmetry down here as well. What else could we comment on? We could have a think about what happens with the the negative values. So thinking about the shape, thinking about the fact that we're carrying on like this, what do you think will happen if we compare sign ine of an angle and negative of that angle? Equal, equal. Wait, no, it should be the inverse value. And what exactly do you mean by inverse? Times negative one. Yeah so that's not quite an inverse inverse what means different things in different contexts, but it's just the negative of it would be the simplest way of saying that. Yeah, excellent. So I'm not necessarily saying that's an exhaustive list of properties, but basically my point is no, your graph and you can read everything off and just use what you need in a specific question. That would be my attitude towards that. Okay, let's now have a think about pause. So you told me that this is the one just starting a bit, shifted along, etc. So again, could you mark on the important values for me? So I'm thinking about these ones in particular. 90 then however two excellent 18. It's wait, is it 180? Yep, nicely. Yeah. So I took three. Three over two pine, yes, I think he knew that. Yeah, three over a two pi a 270. I think you've got that one. Excellent. So a little think about some symmetries. Again, even if we start with the last one, now, if we compare theta and minus theta, what's going on now? Cause of theta and coof? Minus theta, what's the relationship between them? They're equal. Yeah. So this time, nice and symmetric. Now what about again, let's think periis t if we take theta and we compare it with theta plus 360, what's going on there? Still equal. Equal absolutely wraps around Yeah period. And we could think about other ones, things like what could we do? We could do sort of theta and 360 minus theta. But all I'm doing is I'm looking at the graph and knowing that we have lines of symmetry in all of these one. So there's a whole bunch of symmetries you can read off. But I think let's just do case by case when we get to some questions. So then the last one and theta. So there's little few things to say about that. So in gcse, we're thinking of this as toa opposite over jent. More common at a level is to think about it in terms of sine and cosine. Do you know what the equation is linking Todes, coine, cosine, theta divided by sine? I'm not sure about which is which. So let's opts it way around, sign thedoes the way around. Yeah sign ine, of course. But you got the idea. Absolutely. So having you come across the graph of ten, we might try and figure it out from scratch if not. But Yeah, show me what you know about it. Something sort of like that. It might not be the best, but you got the gits. We've got sort of a repeating squiggly line and we got some asymptotes here. So let me do a drawing and then we'll talk about it. So it's a sort of, and I would advise drawing in the dotted lines. That does help. And then that's at 91 hundred and et why does that? All right. That's 118. And that unit is just repeating. Now looking at our previous graphs, if necessary, can you tell me or have a think about why are there these asymptotes at 90, 270-90, etc. As sign approaches these values, it becomes one. And when cosine approaches these values, it approaches in zero, becomes infinity. So the important thing of that is that the bottom is becoming zero. What happens on the top doesn't really matter why it's not zero space, but we're dividing by a quantity approaching zero. That's what's convenient in terms of efficient memorizing. Just no way or zero as a of cause are and then you know, we your asytare similarly, why do we have zeros at zero, 180, etc. Can you analyze that in terms of sign? Sign up 180s. I think it's zero. So it's always going to be zero. Yeah. So in terms of not having to do loads of memorizing, if we know where the zeros of coare, we know where the asytotes are, we know where the zeros of sign are, we know where the zeros are, and then we just have to remember the squiggly shape and we're done. Liany questions. So far, no excellent. Let's do some questions at this one. Things. Okay. So as a sort of illustrative question, I think there might come across a point we need to discuss. But let's just think about this. For starters, can we find all the values of x in the interval? Zero to 316 such that sine x is a half. So before we dive into this, do you have a feeling why we're being asked for multiple solutions here? And do you have a vefor why we're being clearly alone? Yep. So the Maryline, and so if we don't have a domain, we would get infinite values exactly. If we have sx is equal to what? Something between one and minus one? Yeah we're going to get infinitely many values. So we're always given a range and we always need to be very careful. We've caught them all. So let's have a think about this one. So fill your way along, see how far you can get. How would you go about finding all the values for sine x equal to half? Calculator is allowed as well. Which would be equal to. So it's we find 30 and then it's going to wrap around many times. So and then we find the corresponding values excellent. So 30 are kind of base one. Now we need to find with the other ones. Now what I would do and do to this day is draw the graph. There's there's other techniques you might have seen with a unit circle, for example. But personally, I think you can't miss any if you draw the graph is my take. So Yeah, whichever technique you favor. 30 is. So 153. So this technidrawing the line across at the level you want means you can see very clearly. If you've missed any. Excellent. Let's try a few more. What about cos x equals root three over two? Again, let's do zero up to 360. Could I have some more space if and Yeah, thank you. That should be the only value. So can you tell me how far your graph is going now? How long have how much have you drawn? How wait. Oh, wait. No, that's there's another cycle. Yeah, so you're in danger of missing one. So have a think about where 360 falls for cause. I'll. Point this one here is 360. The way we about it is it's when where we get back to where we started because that's the periodicity. So can you see the bonus one then? In here, absolutely. So now let's think about these symmetries we're using. So 180 is falling here. This is going to be 270 as of 300. 300. Tell me how you're getting 300. Two 70 plus three. So have a think about your symmetries there. What symmetry are you using? Which line of symmetry are you using? Or periodicity. What property are you using there? This Yeah. So that doesn't mean that this is going to be the same as this, does it? But that something useful tells us that this is going to be the same. Is this so Yeah, exactly. So if we wanted to learn a set rule, wehave 360 minus theta gives this the same there. But I think just draw it into a case, the case as my personal take. And yes, very good. 390 as well. If we were going on. Excellent. Let's just do a tan one, then I'll move on. Let's do ten. One of route three again zero up to 360. Let's do. I would recommend drawing in your asymptotes as well. I find that really useful. Excellent. Very nice. So we said that the periodicity of sign and cause is 360. Can you see what the periodicity of ten will be then? 180. Excellent. And could you phrase that as an identity with tters? Like please in you to see the past 180, right? Any tweak that. Much better. And Yeah icing on the cake, some brackets. Excellent. Very nicely done. That's a really kind of foundational skill, being able to find all those solutions. Very nice. Does that all make sense? Yeah, excellent. Next little sub point is there's some special values that kind of need to be memorized that come up in non calculator contexts. So I don't know if this is something you've seen, but obviously with our zeros, 90s and 180s, we've already covered in the graph zero, one minus one, etc. Some also useful values that are unfortunately need to be memorized are 30, 45 and 60. Is this something you've come across at all? Yeah, but I haven't memorized any yet. Fair enough. They don't exactly stick in the mind, which is why I would not memorize a table. I would memorize which triangles to use, because there are two triangles that we kind of get for free, because we know all the lengths and angles of an equilateral triangle, right? We know it's 60, 16 and 16 11. So what we do is we chop it in half and then we have a right tangle triangle where we know side length, ths and the angles. So could you use this triangle for me to read off all of the following? Have a go. Wait, sorry, what do you want me to do? Could you repeat, please? I'd like you to use this right angle triangle we've constructed by chopping an ecoateral triangle in half to work out all of these values I've written down. All you need to do is do a bunch of different Soka tower gcc trigs, and then you get all of these for free. And some pythgers, I should say. Nice. Some of these you might need to work out the third side. Think about that. One squared plus plus that squared think about that plus minus. Wouldn't that make it now think about the order negative. Have a think about the order. If in doubt, give it a name, call it something. Yep, absolutely. And think about the order. Which of those? The hypotenuse? Yeah, so the two other lengths squared ads to give the hypotenuse squared now please. Yeah obviously I understand what you mean. And this is fine as a shorthand, but just to be aware, this here isn't technically is it just to get really good habits in there? I see what you're thinking, but just always good habit to not write equals when you have mean equals. Yeah, excellent. And we could tidy that up by putting just two on the bottom, couldn't we? So rothree over two is the other side of this triangle. And then we have everything we need to just read off and the other angles. Yeah, we do tend to start writing fractions rather than decimal at this point. Yeah, it's a sort of general style thing. Have a think about that. Let's think about that one. Nice. So that's two thirds of our table for free. All we need to remember is that it just comes from chopping an equalateral triangle in half. That makes sense. Yeah, excellent. The final bit comes from considering another triangle where we can work everything out for free. We're going to have right tangle with two equal lengths. So for convenience, I'm going to go 11. So then can you tell me what are these angles and what's the third side? Those angles are. Wait, sorry, could you repeat the question? So our next construction is we're going to have an isoocsceles rectangle triangle. So I'm going to have two sides of the same length with a right angle in between. So can you tell me the size of the other two angles and can you tell me the length of the other side? On the other side, the hypotenuse. No, I just can't remembering it. I'm not looking for remembering. I'm looking for working out. So go back to basics. It's a right angle triangle, you know, two of the length, how you work out the third. So again, we've got all the angles, all the sides. We are now in a position to just read off all of these. Have a go. Very nice final p we some questions. So there's lots of trig identities, things that we know hold about these functions because we've already seen some actually. So sine theta is a sine of 180 minus theta. Tan is sine over cause cause theta is cause 360 plus theta. These are all trig identities. And there's loads of nomore in the second year available. But in the name of taking things one step at time, just looking at the one that's in the first year available level, which is very much the most important, I would say, comes up all the time, absolutely crucial. And I'll show you how we derive that. This is where I would get the unit circle involved. Ves, in general, I think sticking to the graphs makes more sense, but proving this, so this is a circle of radius one. So if I take point. That's defined by this angle here. Yeah how far I've gone around the circle. That's so. Can you tell me what the x and y coordinate of that point is in terms of theta? If this is a circle of radius one, see if you can work out this is going to be one. And so that's. Is that your coordinate cost data? Or you just thinking. Yeah so Yeah, that's the excellent so and I've got to get in the right way around for Yeah Yeah excellent. So our x coordinate is called why coordinate is snow? To be fair, we haven't explicitly covered this, but you know what the equation of a circle is? Equation of a circle centered at the origin with the radius one. Do you know what that would be? You mean x squared plus y squred equals? I was hoping you would have seen that. Now we have a nice, simple proof of this identity. So all the points on this circle, we know that the x coordinate squared plus the y coordinate squared equals one. We know that the x and y coordinate of this general mo point are cotheta plus n sine theta. Therefore, we know this must be the case. Z equals to one happy with Yeah that came from and that is the case for any beta value. So this is our first really important triidentity. More to come, but good to absorb than one at a time. American reckon, okay, what makes sense? Yeah, excellent. So let's do some questions. And just to mention something now more looking toward the future. Trig identity questions tends to be one of the a level topics people really struggle on, not to be negative about things, because once you get to a level, there's lots of them. And people think it's a bit of a magic trick. How do I know which one to use? So I think the thing is to do is to take them on one at a time and kind of really consciously put them in a toolbox so you're really aware of all the strategies you have for approaching them so that once you've learned them all and you're looking at a question you're not overwhelmed by, but I don't know which ones to use, you can just go through your toolbox and see which of these is appropriate to the question. Just to sort of mention that now let's try some questions. Actually, I'm just going to send you the sheet. That's probably easiest 1s. This is identity here. And you can stop practicing that kind of attitude even with these ones. If you can't see immediately how to tackle one of these questions, just think about what's in my toolkit. So far, not a lot because we haven't done them all yet. But we've got this one we just mentioned. We've already said tan is sign over cause and we've got a couple of things like this one here. We've got our periodicity equations, so bear that in mind. Let's start at the beginning. So one a. You scroll it down, please. I think I can interpret that pause as you realzing. We've got a zero equals zero situation, that old chestnut. Let's analyze how we got there. Why did we get to a zero equals zero? I mean, you've scrubbed it out now, but can you remember what you did and analyze why did we get into that unhelpful situation? Because then if we know why we got there, we won't do it next time. I don't know. So what you did is you've rearranged an equation into a new form and then substituted that new form back into the equation. That's not going to give us any fresh information, is it? That's the kind of thing that's going to come out with a zero zero, because if we have a new piece of information, like in simultaneous equations, then we can substitute that in and get something useful. But if we just substitute something into itself, basically, that's when it kind of breaks down and we get zero, zero. Now this is really good practice for intentional algebraic manipulation. So have a think about where you're going, think about what you're aiming for, and let that inform what you do, because there's so many options, right? There's lots of algebraic moves we could do. So think about what we're aiming for. We're aiming for ten is minus a quarter. So if something that might pop out at you right away is, well, we don't currently have a tan in our equations. So somewhere along the way, we're going to need to introduce a tan. So focusing on what you're aiming for, give it another go. 未呢认为。Okay, technically, because zero is equal to ten, zero to sine of a. And the other thing to bear in mind is this isn't going to be done purely on algebra. We'll need one of those glonometry identities we've looked at. So we're going to need to use some fact about sine, cos and tan to show this. It's not going to be done on shear algebra. We need to use a relationship between sine coand ten. So that the way. Around here. But think about that move. Has that got you any closer to having a tan? No, no, when we've now got on x -90, we just want x's. What's the simplest relationship you can think of that links sign and cause to tan? Exactly. This is a really good candidate to use because it starts with the two things we have and it ends up with the thing we're after. So see if you can run with that. That is the. No coin. Think about your careful well, no, if you're thinking of just times in both sides of that equation by four, then Yeah, absolutely. So a way of catching any mistakes is to always think what the process is. So how have you arrived at this first line here? What was the process to get here? By both sides, by four. Exactly. This from here to here is Oh no, no, not quite. Not quite. Wait, then I would get that. No. Yeah, exactly. So first we've divided by cause that's legitimate. We might want to think about or do we want to make sure we're not divided by zero but basically fine. And then here divided by four, you see how just thinking what the process was, you caught your mistake. So it be really comps of that. And then you're very nearly there. And then we're basically, Yeah excellent. Something to bear in mind is a show that means we need to show every step really carefully because just arriving at the answer won't overly impress anyone because they gave you the answer. So it's an invitation to be super clear in know you're working. But Yeah, nice. So this is the kind of thing we do with our trig identities. First job, identify what are potentially useful identities for us. Think about what you have and what you want to get to. And in this case, that's looking like a good candidate. And then we do very careful, accurate algebra manipulation. At some point, we'll have to use the identity that we identified. Lovely. Okay, have a go at part b. And first of all, can you tell me what hence means using the results from above? Excellent. So have a go at that. It might just be my your handwriting, but make sure you've got that minus in there because thatmake a difference. So inverse ten of minus water. Yep, good start. Nice. And you know I don't mind if stuff isn't memorized for yet, so I'm going to scroll back and show you the tan graph. But do we have one fourth, though? So well, this is a reference. You might see that, right? No, no, I was thinking about a cosine or something. So just have a take that in and then we'll go back to a working so that's what tan looks like that Yeah may prove useful, but absolutely, we've got our sort of base value. I think about where you've put that line. All right, it's negative 14. So unfortunately, we don't get any for free. We're going to have to think about the patterns and the symmetries and the periodicities. So can you remember what we said? The periodicity of turn is or work out for that matter, 180? Excellent. That's going to be useful here. And then b 60 minus that. So very nicely done. Perfect. Lovely. Let's have a look at question two now. Could you scroll a little bit down, please? So again, we've got to show that. So we're thinking about really careful workings. And again, I think the first job to do is think about what which our identities are potentially useful here. Yeah sign cosign ine exactly. Yeah, that fits the bill. We've got stuff squared knocking around. Yeah looking at what we have and what we're aiming for. Nicely done. Okay. Can you explain where that first line is coming from? Since they're like four cosine squared and five sine squared. Okay. So you're so you're taking the four of them both out and there's one science script left over. Yes. So are you just missing a five that think about your sign? All right, there's five Sighere. I know you're writing on an awkward board now, but just for reference, in an exam, don't make the exam think, Oh, where's that first line coming from? Really? In an exam, the first thing you write should be the left hand side or the right hand side, one of the sides. And then you start doing some stuff to it. And each step it should be clear what you're doing. But for now, I'm happy to do a bit of mind reading. Okay, where next? So b well, that isn't quite done, is it? Yeah, okay. Yeah. I can see all the right stuff is going on when you're doing it formally, when you're trying to do a show that with a identity in the middle, you need to pick a side, write down that side, apply a series of steps where it's clear what you're doing at each step, and end up with exactly the other side, not something that you've realized is the same as you need to start at one side and end at the other. I'm happy to see that you just got one idea now. Okay, so another hence how good it be. So maybe a little more going on to this hence part this time. What it looks like you've already clocked is that the hints part is that we can transform that five sine squared plus five sine x plus four cosine squared x into what we found in part a. Yeah, it's the same as that sine squared plus five sine plus four. I can see that you've already clocked that. That's really good. That's really listening to the hence thing. What that suggests is that it's easier to think about this equation. Than the one that we're given. It suggests that this is an easier one to solve. So really, the trick is, can we spot why this is a nicer equation to solve? No, sorry, could you describe this equation and what kind of equation is it? I don't know, is it linear? The only algebra that appears is sinx, right? Yeah. So it's an equation in sinx. You could think about that in isolation or we could do substitution. We just call sx something else. We have this and what kind of quadso. It's a quadratic ensign. That's why this is a much nicer form. Now in just about out of time. We will pick up there next time just in case communications are a little slow from the agency. They didn't actually check with me. I was free in January and I'm actually traveling for the second half of January. So I'll be away for a couple of weeks. So have a think about whether youlike to do to a week in the first half of January or something like that. But just in case the agency is slow to communicate, I'm away from the fifteenth for almost three weeks. So just have a think about whether youlike to back an extra few lessons in before that, just to let you know. But great stuff today. I will see you next week. Bye.