With a few questions, just to recap what we did last time. And then I'll get back to those harder, more multipart questions we were looking at towards the end. So just by way of a recap, let's have a go. This one, here's an equation for you two times, ten theta, -15 degrees, because 3.7, I'd like you to find all the solutions between -180. And 180, I say recap, there is a subtly new thing we're doing here, but see how you get on solving this equation. There's the permissions. Have a go. Could you work that out for me short? So you want to know invest hand of 3.7 over two, very nice. That's going 61.6. Okay. Okay. Describe to me what you're doing in that last step you just did. Shouldn't I be subtracting the 15 degrees on adding the lithere? You go, if you're always just making sure to yourself you'll clear what the step is, you'll catch these little slips. Oh, nice. Good. Yes. Little tip, I think being dotted lines for the asymptotes helps withdrawing it. A good drawing. I like that. Okay. Tell me about that calculation and see if you're not quite sure. Wanted to find that, which I would add to here to find that. Yeah. So I think the important property to remember here is the periodicity. So they're the same chunks repeated with 180 gaps. So that the key facts I would remember about ten is that when we add on 180, we just get the same. That's the pattern here. Once we add in a whole other 180, we're back to where we started and that's minus as well. 180-72.6. The other way around. Yeah so basically whatever your first one is for tan, add or subtract on lots of 180 will give you something that tenans to the right the same amount. Yeah, excellent. Just one comment there. I would say this step here is the best place to find all your solutions. It doesn't matter in this case because we're just adding or subtracting, but doing it at the final step could get you in trouble if you had something like a two in there. So just to show you what would be even one step better at this stage here, I would list all my solutions here. Doesn't matter in this case, but if we look at something like maybe the next question. We could do sine two. Theta equals 0.3. Let's now do this for comparison and bear in mind why we need to do it at that first stage. Have a go at that. And Yeah, I can just do any calto calculations. Youlike, no, I've got the online calculator. Uncle, excellent. Nice. That's our core answer. So again, let's do everything between -180 and 180. So bear in mind where we need to get our extra steps in for it to be correct and yet very good thinking. Let's start with a sketch. Yeah, so well, maybe if we know that the top is one, maybe you could do a horizontal line approximately where point three will be. Yeah. Yeah, I think that gives a feeling. So where's that other mystery angle? Not quite. So there's two things to consider here. One is what the symmetry or periodicity we're using is, but the other is where to do it. Now as I said, I would recommend you do it before you divide by a half, because when we're doing the inverse sign here, we're saying, what are all the things that sign to give? Point three, that's where all the multiple answers come up, and that's where we're using the function of sign. So given withdrawn sign, I would recommend at the point where you take the inverse sign, that's where you should list all your possible answers. So basically, there needs to be another step in between here because basically if we do it at the end, we'll be halving it in the wrong time. Yeah, so we're liwhat are all the possible values of two theta. What's everything that when I take the sign of it given point three? Yeah. 13 in 180 minus. Excellent. Yes, we're doing a sort of symmetry around 90. Excellent. Brilliant. And then we divide it through. That would give us two solutions. There are actually two more. It's a bit of a sneaky point, and I'll tell you what's going on, but I'll give you a moment to think about. Not quite because thatwill give us the negative, wouldn't it? Yeah. The thing to think about is we've been told the range we're looking at is -180 to 180. But if theta has to be within that range, what does two theta have to be within then? 360? Yeah. So did you see that we're going to Yeah, because we're going to half it. Our original answers can be a little bigger. Excellent. Yeah, brilliant. So can you list our other our other answers then? Very good. Excellent. You've really got the hang of the symmetries in the periodicity. Very good. Excellent. That gives us four. Yeah, I think it's really helpful to draw the graph. Oh, I so agree. Even now, I'll still get a quick diagram going. Excellent. And then we'll go through with the halving. So two points to emphasize here. One is the sneaky business of if we've done something to a theater inside the trig function, we might have to broaden our brackets originally. And the second thing is we we list all our possible values right when we do the inverse sign and then we do the rarranging. Excellent. That seems solid. Yeah. Happy with that. Yeah, brilliant. So that's just a slight extension of what we're were doing last time. So what we finished with last time was looking at our trig identities and doing some more sort of algebraic stuff with it. So I think I'll just send you that sheet again. Give me 1s. None those this one. Okay. So we were looking at lausheet Yeah so we were looking at these kind of more algebraic ways of looking at it, looking at a few sort of identities that we have and definitions and an awareness that as we go on through the a level, we'll accumulate more identities to play around with. So looking at this sheet, do you remember we do you remember we had to go one. And we started to so I think we should dive back in. So do you remember vaguely how you did two a or if youlike we can just go through it again, but I definitely want to dive in it. Yeah. Remember even the be. I don't think we got onto B1, but let's pick up at b. Maybe we had a slight discussion about it. Have a go at that. Picking up at two b, how would you go about that? I would substitute it in. Excellent. And Yeah, we identified what kind of equation is that? I don't know it to me, or how would you go about solving that? Can you see a strategy? It said we do a. A place placeholder for sinx. Yeah, that's absolutely one way we could go about it. Yeah. Absolutely. So it's going to be a quratic. Just check for and just for a bit practice, go just for practice. We won't do all of them once we've got to quadratics, but let's just solve this one right to the end. So is there something interesting about those two solutions? Does anything jump out at you? Of y is equal to sine x then. Must be good to four for minus one. And is there anything interesting about either of those? So anything leap out to you? No, sorry, that's not a problem. We'll see what happens when you try to finish solving it, when you try to find x, given these two possible values for sine x, and I think you'll find something interesting happens, it's undefined able for which one. Negative four, absolutely. And why is that? Can you give me an interpretation behind that? What's going on there? Maybe thinking about the graph. Negative, should I say that again? There's somewhere here at negative four though. Well, think about what the negative four is doing. If this is the graph of y, all right, it's no, no, it's y. So then it's absolutely sine x is always somewhere between one and minus one. So whenever we see something outside of that range, we know instantly we're not going to be any solutions there. Excellent. So we just left with minus one. And now the only other thing is to look at the range. We're doing zero to 360 now. So what's the appropriate negative. We were doing zero to 360Yeah. It is net negative. Nike, yes. Well, thatbe outside of the range, wouldn't it? If you look on two b defining the values of x in the interval zero to one, zero to 360. So 90 is not in -90 is not in that range, but there is a value in that range. But sine of minus one is negative nine. Well, I mean, it depends one of the values that signs to give minus one is -90. Yeah but that's not the appropriate one for our range. You can use that as a stepping stone to a seven. Absolutely. Yeah. And you've highlighted on the graph. Excellent. Very nicely done. So I'm going to pick out a few more of the, I think the more interesting ones here. What about three a? Can you see what's going on there? So we're solving for zero to 360. I'll give you some more room three a. I can move this along a bit. Could you remind me the identity? Is it sine squared x plus cosine squared x? Or is it minus? Yep. Excellent is one that's a key one in our repertoire at this point. I'd say there's one other main one. If I give it to you, it slightly gives away those questions. So I'll give you a moment to think. Or maybe I'll give you a Yeah. It's sine over cosine. Excellent. That's kind of all we have at this point. We've also got our periodities and symmetries, and you can state them in terms of identities, but these are the two biggies, I'd say, at this point, very good. Wait, it's two sic, so. Bewof over complicating? No, I'm just not sure what I would recommend you do is move. That cause over to the other side first, see what happens when you do that. Yep, excellent. Then what does that suggest? Tell me what the process is that got you from between those two steps, dividing both sides by a oscience. So does that directly. You're doing multiple steps once, aren't you? Two sign ine x divided by does not immediately give you that. So how about you write out each step at a time rather than trying to jump ahead? Oh, so it's two depenabsolutely. We've just got two sine of coso. That's ten. Absolutely. And that's nice because now we've put it down to just one trade function. Is both tangent or the inverse tangent. The inverse tangent, we'll get on to code tangent later. That's what we don't to not mix up. So inverse tangent or arc tangent, we call that absolutely. See what capculates this check we've got. And then we're in the right range. 5.67 nice. That sounds about all and have a check. Six. Yeah nice. Any other values within the given range that we might want to double check for? So we're doing zero up to 360. We're still positive since you Yewe're going up to 360. Excellent. That's a really nice clear diagram there. 180 plus 26, five, seven. Excellent, very nice. Okay. Next up, have a look at. See. E3e. Yeah, could you throw a bit please, thank you. It wants us to make a tangent again. We could try that. Yeah, we've got signs and causes. We might be able to reduce it down to just a tan. See what happens if you try to do that or imagine what might happen. Can you see what the problem might be? You're going to have a 31 over cosine x Yeah or Yeah we're going to have a that sort of three is a bit awkward, isn't it? When we divide through by a cause, that's just gonna to be three of a cause. But also we've got an excess of sine, don't we? That sine squared. When we divide it through by cause, we'll have a sign of a cause, which is lovely, but we'll also have another sign knocking around. So it looks like that won't quite work this time. What's the other strategy we've come across? What's the other tool? We have one sorry, I say that again, using the square. Yeah. So can you tell me why that looks like a good candidate here? Practicing using our skills of attaching the right identity to the right question. Why is that a good candidate? We already have a side squared. Yeah we have a side squared. If we could translate those into cos, squared thatbe a good situation. Yeah, okay. See how far I can get with that. Turn the lights on. Good back. Youlike me to quote the identity for you. No, I you know remember the density, but not very neat in. I can't just multiply both sides by cosine squared, right? Wait, you can. The question is, is that useful? You're at liberty to multiply both. It's not useful in any way. Where's the only place that you could substitute in this identity? Into this equation. We can't relate it to the cause part, can we? Because there's no cause that appeds. We have to relate it to a sine squared or a cause squared. Or I suppose if wehave just a number by itself, we could relate that in. So it's, we can't substitute in here. There isn't a cause. So our options are we could use this sine squared and relate that to this sine squared, or I suppose we could use this three and say it's three times this one. Would you reckon which of those sounds promising? I think both sts are okay. Okay, well, pick your Favand, give it a. Don't know how to do it. Okay, well, which one should we try? Substituin for the three. Okay. So we're saying it's three, lots of one. We say one is equal to sine squared to x plus cosquared x. So it's got of just a direct substitution, isn't it? So what would that look like? And then substitute for the one the equation we have. Yeah it's basically simultaneous equations. It's how do we combine these two bits of information? Now since we're sort of focusing in on it quite detail, I'm going to say that actually I would be going for the other one. I'll show you why afterwards. But just so we're focusing the right thing, I would actually be trying to substitute in here. So what we want is to have sine squared is equal to some stuff, and then we're going to substitute it in to allow two times that stuff plus three Cox equals three. So can you use our identity here to fill in the box? What is sine squared x equal to. No, so you're happy that we have an identity called sine squared x plus, cos squax equals one. Yeah. So all we're doing is some really basic algebra. I'd like you to rearrange that to make sine squared x the subject. That's using the identity but rearranging it to give us an identity for sine squared x. And just think about what you did there. Just the boxes. The box which is substituting in. Yeah, I mean, know, just multiplied it. You've lost some things along the way. Just try doing things one step at a time. So we've got the sine squared is equal to one minus cosquared. So Yep, Oh asi was doing that as a multiplication, but Yep, absolutely. Excellent. So thinking through the rest of the equation, why is this a better situation now? Excellent. We've got a quadratic. Yeah, I'm going to just plug in the formula, but I can I just not write it on the board for now? Yep, absolutely. I trust you to do this final step. Yeah. X goes minus one or minus. I think you don't want the minus in there, so just check what you've plugged in. I think we want a half or one to just check what you've popped in your calculator. Right, right. I think I don't know. I probably plugged in three or some yes, a husband. So once we have a quadrastic, we're on home turf, aren't we? We know what we're doing, get our values for the trade function and then we're doing the stuff we did last time. So really, with all of these, once you've got to a quadratic, you're basically home and dry. So can you see that for any question of that kind of form? So that was e, but think also about c, for example, I or J or l, you either all of these that have a ssquared or a cause squared, possibly a sign or a cause, it's at least worth trying to use this identity to get us down to a quadratic in ner trig function. Yeah, Yeah. So what about someone where it's not as obvious what's going on? How about, for example, what about g that doesn't follow that pattern? So what do you think you might do for g? Need to use both of the identities. Yeah, I reckon so. Yeah, let's see how that goes. Wait, no, it's apple divided by person. Look. We're not plus, it's minus. Yep, so we could move it all to one side. That's. Is there any advantage to that step? Is there any advantage to having it all on one side? So then I could apply the identity. You're just as well placed to apply it before would be my hint. So we've got a sine squared and we've got a cause. Which of those two is going to be useful for applying that identity? Absolutely. And how do we express sine squared using that identity? Absolutely. If that's answer. I mean, if that's in, Yep. And then we're home and dry onto quadratic excellent. I'm just seeing whether there's any other more interesting ones in three, but I think you've got the hang of that. So what I'd like to look at next is question five. So we've been doing solve that. And the nice thing about that is if you get to the right answer, it's pretty clear you've done something right. But prove our final result isn't gonna to be anything new. So it's all about how we communicate it. So think of it more as like a bit more like your essay subjects at school. You're convincing ingly. You're showing me every step of why it's definitely the case that these two sides of the equations are equal. A really clear way of doing that. Whenever we have proved that left hand side equals right hand side, pick one of the left and right hand side, write it out and then try to get it into that form. Exactly. Yeah, nice. So let's start with a why not five a. And. Sine x words plus cosine x plus. Two sx sx. Should you go to one plus two? Yeah and making it clear what you're quoting at that point you're using about identity. Excellent. What about let's do I think they're all worth doing actually, let's move on to b. Never think about that. Yep, that's better. So sine squared x is equal to one minus sign. Look, seven. Yep, that's looking useful here. Yeah, very nice. Just to say, there's sometimes not a itable wrong answer about which side to start with. I recon they're about even here. Cool. Let's move on to thoughts. I have more ideas about that side. Absolutely. And that is the right way to do it. Which one is giving me ideas of what to do next? Absolutely. Great. Well, that slowdown a bit, please. I don't think that's useful there, right? Good assessment. We don't really want to get a tan involved. There isn't a tan in what we're aiming for. Maybe I start with that. They're say, Yeah, good idea. Sometimes it does just take trying one and thinking, actually the other looks better. So what can I do now, though? So there's something really nice about that expression. It's kind of a stred at this spotted situation. You spotted it, you can convert it again. How do you mean what we could do that bring us back to where we started? It's actually more it's more of an algebraic property. There's something, there's a way you could rewrite the top. Just using algebra doesn't bring any trig in. I'll give you a moment to stare at it. Difference of two squares. Spot on. Absolutely well spotted. So see what happens when you do that. Yeah, excellent. And indeed, to finish things off because this one looks a little more interesting. There's potentially a little more challenging this one. No. I don't think it helps. Not immediately obvious. Yeah because we don't really want to bring a turn in Yeah but we definitely just want to play around. So we've tried to rewrite it with tan. That doesn't really seem useful. There's no obvious way to use our squared expression, is there? Because there aren't any sine, squared or cosine squares. So that suggests that we've kind of used our two tritools. We might just need to resort to some algebra. It's quite a neat on this. Which might require a bit of a clue, but I use both of the sides though. Well, I would say yes for getting some ideas, but there is a risky move. Once you start playing with both at once, you can end up with something that looks right and is logically nothing. E. So always for your final answer, start at one to go to the other. If you're stuck, then you can kind of work on both in tandem and see if you can get them down to the same place. So kind of start with the left hand side, try a few transformations. Start with the right, try a few. And if you can get them to the same place, then you can kind of reverse engineer something Yeah but be very careful of combining them wrong or mixing the logic. So I could give you a hint if youlike, but it's kind of an interesting question. So take this as an opportunity to engage the kind of creative maths side of your brain. And have a play around, see what you find. So there's no actual trigonometry involved. It is. But the first step I would do is just algebraic and then you're in a position to use one of the trig identities. So you've started with one side and you've divided top and bottom. That's absolutely always valid. Disavantage of this is we haven't come across any ways of dealing with one oversign. That's the next bit of material. Can't you just multiply there on top? Yes. But that's not quite right then, because if you're going to times top and bottom by sign, then the bottom will become one minus sign. So it's kind of looking a bit complicated. Actually that can't be right. Well, gonna to miss and then all you've effectively done is rewritten cause ses sine times ten I think something might have gone a bit wrong there actually. There is Oh, that's the wrong way around, isn't it? That's not ten. That's one over ten. But good playing around. What happens if we divide top and bottom by something that seems to not quite work out? So a natural next step. What a few times top and bottom by something. When you come up with some good candidates, it doesn't really matter which one you start with. But what could be timed top and bottom by that might be useful. So you're doing the the right hand side and your times tip by times it by cause. Yeah. So advantages of that. We now however, cause squared, that's something we can play with. Yeah, absolutely. So we could see where that goes. And there is definitely an animal of playing around with this just because we're almost out of time. If we start with the cause over one minus sign, we eventually want to relate it to something over cause. Actually, sorry, you've done exactly what I was thinking. I was just thinking of it in a different way. For that reason, I would time top and bottom by coyeah. Absolutely. So see what happens pursuing that. Excellent. Yep, this is looking good. Excellent. And then there's one last move we need. Very good. Another difference of two squares. Excellent. So I think the way to tackle these, to be clear what your tools are at this stage, we have the cosquared plus sine squared. We have ten and the definition, but we also have all our usual algebraa tricks. Excellent. So this sheet could be good for homework. I'm away for a few weeks, maybe just to mention something quickly because you mentioned it. And it's kind of the next thing we will look at is cotangent versus arc tangent because we want to not mix them up. And that's kind of the next thing welook at. So in our last 14s, arc tangent is the inverse. Yeah how tangent is the reciprocal? So one over ten. This is also sometimes written as there a separate function for that though because it's sometimes useful, it doesn't mean anything beyond just one over ten. So therefore, cause over sign, it's sometimes convenient to speak in those terms. There are also corresponding ones for one over cause is Seck. And one over sign it's c Yeah. If we're dealing all in reciprocals, it's sometimes more useful to talk about cosec and cosec. And we have equations like one plus cosquared is cosec squared and things like that. So very similar to our sine squared plus cos squared. We have similar types of equation linking these the these are three other ones. Just as a sort of sneak preview of that thatbe. I think the next thing to look at Yeah getting a really nice deep understanding there. I think I'm away for a few weeks. Highly recommend the rest of that sheet. And just any of those worksheets on physics and matths tutor are very good. And just yes, store up any questions you have and I will see you in three weeks time. Great bye.