 So, a little bit about the tests for those of you that haven't watched the video or didn't come in for the tutorial, if you haven't watched the video you probably should because I gave a bunch of information about the test but I'm going to re-give it right now. This is what you're going to see on Friday. The test is, I can't remember 14 or 16 multiple choice questions, I can look it up around 14 multiple choice questions and then around 25 marks or so worth of written. I think the test was out of 46, if I'm remembering properly. Here's exactly what's on the written section. The very first question, number one is going to be solved with a period change, example the sign of 3x equals something. Question number two from quizzes 4 and 4.5, hey that's worth adding right here. Question two from quizzes 4 and 4.5. Then I wrote down, there's going to be two or three quadratic trig equations, actually I think this is going to be question number two, I think there's some kind of a word problem. It will be tides or tire rolling or ferris wheel and I think I gave you a little hint in the way that I wrote this, if you're paying attention, if you're not I can't help you anyways. Then there's going to be two or three quadratic trig equations, one is an exact value. Then there's going to be a one where you need to use the calculator to find the reference angle. So you'll get for example, I'm making this up, sine theta equals one quarter. You don't have a triangle with a one and a four in it, oh it's a second function sine of one quarter, get the decimal and it's pi plus or pi minus or two pi plus or what and you'll get it as a decimal. So one exact value quadratic, we have to factor, find the roots carefully. And one exact, one non-exact value quadratic, maybe three quadratics I can't remember. And then the written is going to have several three or four t table style identities where you have the line down the middle QED and all that good stuff. Victoria, you can switch back now. What about on the multiple choice? Okay, absolutely, now this is not an order, I can't remember the order that I made the multiple choice, but absolutely there is going to be a solve by graphing somewhere. How will you recognize your supposed to use your graphing calculator? Desperate answers, okay. That technically is going to be more than one because I'm probably going to give you an application problem where I give you the equation and we're going to use our graphing calculator for that. Although the odds are those will be decimal answers anyway. So you'll recognize that the answers probably want to use my graphic calculator. I am going to ask you to find a decimal version of a reciprocal trig function. So on your previous test, I said, for example, find the cosy cant of five pi by six or seven pi by three or it would have to be some kind of a reference angle of pi by three, pi by six or pi by four as an exact value. Because you had no calculator. I am dealing on this test going to say, for example, find the cosy cant of 2.9. You know what, just for practice right now, what is the cosy cant of 2.9? In other words, make sure you know how to do it. I don't know, it should be actually, it's meant to be free marks. The one that I've highlighted, that question, that pink chair matches Tyler's attention span right now it seems. What'd you get? Sorry? Louder Dylan? 4.18, is that right? Are we getting that? How do you get it? One divided by the sign of 2.9. What if I ask for secant of 2.9? One divided by cosine of 2.9. What if I ask for cotangent of 2.9? One divided by tangent of 2.9. I doubt it'll be a 2.9 although apparently frequently when I make up random numbers, they overlap. Maybe it is a 2.9 on the test, that would be ironic but I hope not. There's gonna be a word problem where I give you the equation and then ask you to do something with it. And a little reminder for times of the day, if it's 330 you can't enter 3.3 on your calculator, it's 3.5. Or if it's 345 p.m., that would be 15.75 on your calculator. I'm either going to give you an X and say find a Y or I'm going to give you a Y, that's where you, if I give you an X value, it was trace button. If I give you a Y value, it was send it across as Y2 and find where they cross. There's going to be some kind of a question about restrictions. I'm going to give you some kind of a trig fractional thing and I'm going to say, tell me the restrictions. What can't I divide by? Can't divide by, a cosine can't be easier. But remember, many of your reciprocal trig functions are fractions in disguise. And if you don't know quite what that means, ask me about it in a few minutes when I start taking questions. There's going to be an addition identity where alpha or beta is one of the four corners of the circle. In other words, I'm going to give you sine alpha plus beta, sine alpha minus. One of those, one of the alphas or betas is going to be an X or theta. And one of them is going to be either pi by 2, pi, 3 pi by 2, or 2 pi. And I'll even tell you what the answers are. The answer is either going to be negative, positive, positive, negative sign or positive sign. Because those are what they always work out to be. There's going to be something like quiz five, questions two and questions three. So make sure you looked at quiz five, questions two and questions three. Although I'm putting some quadratic trig equations on the written, I'm also going to put a couple on the multiple choice, probably one quadratic that just has a squared but doesn't have a regular term. So you don't have to factor it. You can just get the squared by itself, square root both sides plus or minus. And I may put one on there where you need to use an identity. Remember, we did one day in where we had, there's a cos squared and a sine. Wait a minute, cos squared is one minus sine squared. I can plug that in and now everything's sine and I'm happy. We did a couple like that. I think one of those might be fair game for multiple choice. And then there will be lots of identity kind of questions. Right, the following is a single trig function or especially going backwards for double angle and addition identities. In other words, I'll give you the right side of cosine two theta but in disguise or the right side of the sine of alpha minus theta but in disguise. And I'll ask you to kind of go backwards and put it together. There it was again, so fighting my cold. That's the test. So I'm going to start out by asking on the review, either identities review or big trig review, are there any of these that you would like me to go over? Now is your chance to ask. Yes, from the review or identities review. You don't need to tell, sorry, trig identities review, I didn't open that one. Oh, you know what, I think I have that one. Do I have that one here, Mr. Duick, have I printed that here? I don't, open it up. Number four, this one, this one, okay, yeah, whatever, I'll deal with it. This is an older one and I seem to recall, although it's only worth two marks, I found this one kind of yuckier than a two marker in my mind. I'm guessing it became a two marker that year because they just had to make it less to make the numbers work so the provincial is out of the correct total. Ready for my plan of attack? I am, I always glance at these ahead of time and kind of try and predict a bit of a strategy. I notice a binomial denominator with no squareds. I'm already thinking probably conjugate later on, but I definitely see more than two trig functions. I think the first thing I'm going to do is rewrite everything in terms of sine and cos, secant squared minus sine theta over cos theta, that's what tangent is. All divided by cos theta. But never write this. Did I write a fraction on the top? Well, then write a fraction on the bottom. I mean, let's keep our brains from having to strain too hard. Because now I do have, although this is a four level fraction, it's not a complex one where I had fractions inside of fractions and plus. So you know what, how do I divide by fractions? Put it in multiply. That's what I'm going to do next. So I kind of know where I'm, I don't know where I'm going, but I know what I need to do next because that looks ugly. So I would go like this, one over cos squared minus sine theta over cos theta times one over cos theta, I'm multiplying by the reciprocal. Good, good, good. Oh, top times top, bottom times bottom. I get one over cos squared minus sine over cos squared. Now what? Yes, because I got one fraction here and conveniently I have a common denominator already. And even if I'd had like a cos instead of a cos squared, I would still probably be smiling because it's a fairly nice, easy common denominator to get. The fact that I have a common denominator, I'm kind of going, hey, okay. So I'm going to get this one minus sine all over cos squared and I'm trying very hard to keep this on one screen. So sorry for not scrolling down so much. And I've already had my nerdy adrenaline rush because do you remember what I said I had noticed earlier, the fact that I had a binomial denominator with no squares conjugate? What is the conjugate of this? So what would I multiply the top by? What do I have on the top over here? That can't be, okay, now I know where I'm going. See how I figured that out? Now, I'm going to do the conjugate and then I will show you a second alternate approach because now we have enough that I can show you more than one way to do it. So I'm going to choose to go one minus sine and one minus sine. Same thing, top and bottom, always, always, always, always, always. And when I do that, I'll get one minus sine on top. And what do I get when I foil out one minus sine times one plus sine? I think I get one, and then I get a minus sine plus sine, right? And then a minus sine squared. And you are allowed to do that shortcut in your head, they're okay with that. You know what one minus sine squared, how much you want to bet one minus sine squared has to be cos squared because I'm pretty sure it is, isn't it? We're done. That's one way to, let's finish it off, Mr. Dewey. One minus sine theta all over cos squared. That was turning this using the conjugate into that. There's a second way and it involves continuing to work on this side. What is cos squared? One minus sine squared, right? So you could have done this. Now the most common mistake kids make now is they start cancelling because on the left hand side, what's on the very, very top? A one, so they want to go, oh, I'll go, I'll cancel that, and a negative, and a that, and there's a one that, no, can I cancel? Have I, this factors? Perfect square, yes. Minus sine, yes. Perfect square, yes. This is also difference of squares. This factor is into, well, I'll leave the top. One minus sine, one plus sine. Can I cancel? Have I factored now? And I have one completely matching term. Although, if I cross everything out on top, what's really left on top? Which is kind of what I wanted. So there's a second way to get there as well. I would then write the final answer of one plus sine theta. There's a second way to get there as well. I take either of those. Is there a third way? Probably, I haven't even looked. What's secant squared the same as? It is something, what is it? You could replace this with one minus tan squared, and probably work some magic there as well. I think, though, that would take about four more lines if I'm trying to do some math. I'd give you four marks. The key is for an identity, don't cancel unless you factored. Or if you can cancel on the same line, if it's a plus minus, plus sine, minus sine, five. Cancel those. In terms of vertically, don't cancel unless you factored. And don't multiply the top by anything if you're not multiplying the bottom by the same thing. If you do that, then you can muddle your way through carefully, through all of these as long as you're careful. Second key is never, ever leave an identity blank. I'm going to assume all of you can rewrite things in terms of sine and cos. And that'll give you one or two marks every time. Not full, but some. Is that OK? Next? Yep. On the identity sheet still? OK. By the way, have you guys looked at my online answer key that I put online there? Because that's OK. Because when you're at home and frustrated, let me just show you what I put up there. It's called Identity Review Answers. It looks an awful lot like that. Just letting you know so that when you're studying, just letting you know, oh, look at that. The reason I'm so big on this, by the way, is I find you learn it better by figuring it out, by looking at this, than me explaining it. I think sometimes I explain things too well, which is kind of weird. But I think sometimes I make it look too easy. I will do number 10, though. Numbered same? Yeah. Where was I, Mr. Dewick? Identity is review 10. Ooh, double angle. Ooh, double double angle. That's cool. Yeah, definitely I would do this. Even though this is the uglier side. Hey, this is 1 over sine, and I'm done that side so fine. Even though I do that one first, yeah? Well, sine 2 theta, I only have one choice. That, to me, is almost my favorite identity, because I only have one option from my formula sheet. What is sine 2 theta? Yeah. So this is 2 sine theta, cos theta over cos theta, plus sine theta. Now, cos 2 theta, I have several options. I'm just going to don't write this down. But in my mind, I'm saying, I'm not sure which one I'm going to use there yet. I'm going to try and do a little bit of math in my head and see which one works better. Because I am noticing this. I am noticing I end up with a over here. What trig function did I end up with over here? Sine. What number is in front of this? I'm going to use the one from cos 2 theta that has a 2 sine in it. That's a total guess, but I bet you that'll help. It's got a 2 sine squared. Which one is it? 1 minus 2 sine squared? Now, take a look. This is, again, me being a nerd. What do I have on the top over here by itself? Hey, I got a 1 to show up. And I like the fact that this is a minus 2, even though this is a sine and this is a sine squared. I'll bet you, somewhere along the line, I'm going to end up going 2 take away 2 and stuff is going to cancel. And there's my 1. I bet you. Got to try something. How many fractions do I have here, Leah? How many here? Let's write him as 1. Common denominator. What would my common denominator be here? Ah, what do I have on the bottom over here? Ah, okay, okay, this is good, this is good, this is good. Let's do this, Mr. Duke. So I'm going to write this all over sine theta. What would I multiply a 1 by to change it into a sine theta? Sine theta, same to the top. And I'll get 2 sine squared and the plus sine dropping down. This is already over sine theta, so this whole term just drops down and I am getting my nerdy adrenaline rush. Leah, can you see 2 sine squared minus 2 sine squared? Aha, there, I don't need to worry about factoring. Those do cancel horizontally. And I'm left with final answer, 1 over sine theta. That's probably the easiest way to do that one. There's other ways, but if you picked a different double angle identity here, you'd probably add two or three lines. If you use cos squared minus sine squared, you'd end up replacing the cos squared with a 1 minus sine squared and you'd still end up with a minus 2 sine squared. If you use the 2 cos squared minus 1, it'd be kind of silly, but you could also still get there. It'd be a little work though. Next. Yes, love to. 95, someone's done some homework. I like 95, ah, hang on, yeah, I think I do. Sorry, I think I do. And it's from the August 2000, sorry, the August 2003 exam. I think it was question five from the August 2003 exam if you're looking at the high tech numbering system here. Okay. If you've learned one thing from me this year, hopefully you've learned that you never look at a multiple choice question without also glancing at the answers. Because when I glance at the answers, I see 12s and 6s. I am thinking double angle, because there's something in front of the X. Okay. So I'm already heading that way. And I'm thinking cosine double angle. So I would be staring at my formula sheet at the cosine double angle. When I look at the cosine of two theta, I do have one with a sine squared in it. What is the cosine of two theta, the one that has a sine squared in it? Suggestions. GCF is a four I can pull out. When I pull the four out, what will this become? A one. What will that become? Okay, that can't be a coincidence. Four bracket. One minus two sine squared, six X. So what's sitting where the theta is? So what would two theta be? What's sitting where the theta is? Six X. What would two theta be? 12 X. Na-ah, na-ah. Fact, I think, isn't it? I know it seems easy when I do them. But when I said there's going to be several multiple choice questions where I give you the right hand side of a double angle or an addition identity and I'm making you go backward. Yeah, hello, there it is. That's a fairly tough one, by the way. Not an easy one, but that's B plus. Troy's going, what? Yeah, next. No more. Oh, there should have been more, or people haven't, there should be lots. 33, love two. Oh yes, this is exactly what I'm talking about. I like this question. I like this question. I like this question. Doesn't mean this specific, but this kind of question. Ian, I glance at this. I see coast, coast minus sine, sine. That's an addition identity somewhere, right? Um, what is coast, coast minus sine, sine? Okay, so here's what I would do to solve this because this is a lot to keep track of. I would say, I'm pretty sure that's alpha. I'm pretty sure that's beta. And you just told me it's cosine of what? What's alpha? No, what's alpha? Ah, what's beta? So what's cosine of three x plus two x? There you go. This is what I mean by, I'm gonna give you the right hand side of an identity, and you better be able to go backwards, okay? Yeah, and by the way, for the addition ones, they're recognizable because it's two terms with sines and coses and sines and coses. I always label the alpha and the beta. That there alone usually gets me to the answer pretty quickly. And then which one is it from the sheet? Carefully do my substitutions. Okay, next. What's that? Sorry? 13. Lucky number 13. Number 13. The length of your attention span in seconds. Here we go. I told this thing to you because I care. Yeah, no, not gonna panic. First, things that I notice. What's sitting right here? What don't you see in any of the answers? A plus one. What's sitting right here? So I'm heading towards a cosine double angle identity because it's got an 8x, two times 4x is really what I'm thinking. Is there a cosine double angle identity that has a minus one that might cancel this plus one, which somehow vanished over here? See, I put all that together using clues here. What is the cosine double angle identity that has a minus one in it that would wonderfully cancel this plus one here? Read it to me. Okay. Accept, accept, accept. If this is that, if this is supposed to work out to 8x, what's theta? Say this again. If this whole thing works out to 8x, what's theta? Say it one more time. If this whole thing is 8x and I want this to become 8x, what would have been there so that when I multiplied it by two, I got an 8x? This is what they must have started out with. Theta is 4x. So here's what I'm gonna say. I can rewrite this whole thing as this is a double angle in disguise. It's the cosine of 4x, cosine of 4x, which is two cos squared 4x minus one. Two cos squared 4x minus one. Then a plus one is still sitting there all over two. I think I've lost a few people. I'm gonna pause here. Look and think and ponder. This fair game, totally fair game. Becoming clear, some of you haven't taken a close look at this review yet. I didn't give you any homework on that. Anyway, yes, this would be non-calc on the provincial. It's gonna be, if it's on your test, it's gonna be calculator. So here is your cheat. I have no idea what to do. Leave it, finish the rest of the test, come back, muck around with it some more. Oh, there's five minutes left. I'm gonna graph bracket, cos 8x, close bracket plus one, all divided by two, with a window up. You know what? Zoom, trig. I'm not gonna get fancy. I'm gonna just use the built-in trig window. And am I in degrees? I bet you I am in radiance, okay? When I graph it, I get that. When you graph one of these four, you'll get the same thing. And only one of these four. There's your cheat. How would I graph cos squared 4x? I would then turn this graph off temporarily to turn a graph off without having to retype it. You just hover over the equal sign and press enter. Now it's turned on again. See the equal sign is now black. Now it's turned off again. So here I would go, how do I type in cos squared 4x? I would just go cos 4x cos 4x. Not the quickest way, but it certainly works. Graph, I think it's the same. Let's find out. Turn this one back on, graph. Oh, am I getting the same graph twice? That's a cheat on multiple choice identities, which is why they went to a non-calculator section. But you can certainly exploit that loophole on this test if you really want to, if you're panicking. And yes, that would have worked for the previous one. It works for any multiple choice identity. You graph the question that they gave you. You graph the four answers. It's just not very fast or efficient because graphing four things takes a while. I would probably actually, for the view window, go from zero to two pi. I think that looks a little easier to see what's going on. There you go. Victoria, you had a question. Identity is number three. Ooh, ooh, cool. This one's a beast. Let's find out. Well, the right-hand side's easy. I mean, I'm, you know, cozy camp. That's one over sign I'd do that first just because then I have that feeling of satisfaction of finishing off an entire side. Victoria, what are we gonna try on the left-hand side? Sorry? No, no, what strategy am I gonna try? There's something screaming out to me right now already. Uh-oh. What am I gonna do first thing? Absolutely. How many I get? I got coast, sign, tan, secant. I'm rewriting it all in terms of sign and coast. And I think there's gonna be fractions, certainly tangents. I'm gonna put everything over one that isn't a fraction. I'm gonna write this as coast theta over one plus sign theta over one, sign theta over coast theta, all over sign theta over one, one over coast theta. I would totally do that and you'll get marks for that. Now what? I think I tidy it up. Top times top, bottom times bottom, top times top, but I'd write these as single solitary fractions. I would write this as coast theta over one plus, what's sign sign? What's the bottom work out to? You're correct. And now my decision is made for me. This is a complex fraction. It is four levels with plus signs. So I can't just go right from the multiply. I am going to clear fractions by multiplying the top and multiplying the bottom by a common denominator. What will my common denominator be? Oh, pray tell. Let's see. What's on the bottom here? What's on the bottom here? What's on the bottom here? What's my common denominator? Coast. I don't need sign. Sign's not in the denominator anywhere, right? We'll get the mini fractions. I know there's a sign in the denominator, the big fraction I'm talking about. All the mini denominators. My common denominator here is gonna be coast. I'm gonna multiply the top by coast and the bottom by coast. But I'm gonna do four levels so I can clearly see what's on the top on the top and what's on the top on the bottom and what's on the anyways. So I can clearly see what's going on and I'm gonna remind myself we're gonna do that. Good, good, good. Behold my angel. I don't think anything will cancel. I think I end up with coast, coast. Except I'm not gonna write coast, coast. How do I write coast, coast? I totally agree with you. Plus, behold my angel. What happens here? And what's left behind? I think you're right. Behold my child. And so what's left behind on the denominator? Oh, what do I have on the bottom over here? Ah! What do I have on the top over here? What do I have on the top over here? Is sine squared plus coast squared one? Ah! Oh, that was so much fun. I'm telling you right now, this complex fraction trick is a great trick. Now you could have, Victoria said, since I only have one fraction in the bottom, you could have put this in brackets and said, how do I divide by a fraction? Flip it and multiply and multiply this whole thing by coast over, that still works, but I'm telling you, as soon as you have a complex fraction, if you multiply by the mini common denominators, in one line, you'll go to a two level, nice fraction with good stuff happening. Okay? Next! Yes, 13? Yeesh. You know, the less there is, the more nervous I get. Like the one Victoria read to me looked so ugly. I knew it would probably fall apart in about three lines and there'd be lots of clues along the way as to what I should try and do next. By the way, Victoria, as soon as I wrote this, and I said my common denominator was coast, I already noticed I was gonna end up with a sign on the bottom. I knew I was on the right track, about two lines earlier. This one here, I gotta say, Justine, not giving me much to, I'm not sure. Not really giving me a lot to work with here, are they? I think there's exactly two trig functions. I may rewrite everything in terms of sign and coast, I don't know. The first thing I'm gonna ask though, because I see a cosecant squared by itself, and a secant squared by itself, I'm fairly sure there's identities at the top of the page that have cosecant squareds, or secant squareds by themselves, but I can't remember what they are. What is cosecant squared the same as? One plus cotangent? What is secant squared the same as? One plus tan. I'm doing some math in my head right now. I'm trying to visualize a one plus cotangent there, then I get three terms, here I only have one. I may come back to it, you know what? I'm gonna try this side, and I'm gonna try rewriting it in terms of sign and coast. I got nothing else to go with here. Yeah, let's try that. What is cosecant squared? What's secant squared? So when I tidy this up, top times top, bottom times bottom, I get that. Since I've rewritten this in terms of sign and coast, how much do you think I should re, I think I should rewrite this in terms of sign and coast. So this is going to be, and I just, I haven't had my nerdy adrenaline rush yet, but I'm starting to smile, because here's what I would, well, I'll let you maybe figure it out. Now what? What would my common denominator be, by the way? What do I have on the bottom over here? How much do you want to bet I'm gonna end up with? Now, you ready? What would I multiply top and bottom by here to get sign squared, coast squared? And I'd have a coast squared on the top. What would I multiply here? Sign squared, I'd have a sign squared on the top. Are you saying I'd have a coast squared plus sign squared, which is, hey, that's what we're gonna have. Do you wanna try the rest of it on your own? But that's where that one's gonna end up. I have to say, once I started that one fell apart, but I'm also gonna say, I wasn't quite sure what the best approach was. Could you get there by replacing these with one plus tan in one place? Yeah, probably in about a few extra lines. The other thing I considered was replacing these with one plus tan and one plus coast and foiling, or one plus tan and one plus coast, you can't in foiling, would have worked, I imagine, but there would have been extra lines. I think that'll get you there the cleanest, like that. Now, questions that have not been asked of me, and maybe it's because you guys really understand these, and that would be awesome if that's the case. I'm just gonna kind of go on a temporary limb here and I'm gonna say, I wanna make sure you guys know how to solve something like number 14 on the written. You guys are good solving a quadratic trig equation like that, factoring it, finding the roots and all that good stuff. No one's saying anything, so do I need to do, I will do this if you want me to, but if you're okay, yeah, okay, because this is the quiz that we wrote a 20% on in the top left corner. That was my way of saying, hello, there's a big chunk of your written, and no one has asked me about this. Maybe I nailed the lesson and everyone got it, but it'd be the first time I've done that. Hey, what kind of an equation is this? How do I know? Got a squared. I need to factor it, and you do this, got a number in front. Okay, so for part A, E and how does this factor, my friend? I agree there's gonna be a two cos and a cos, totally. And then with a minus one here, and I want a minus one in the middle, I think I wanna put a minus one here and a plus one here, because that would give me positive one, take away two, minus one. Oh, let's get rid of all that. Then I would say to myself, what are my roots? What are my roots? Can you solve from here, cast rule? This is gonna be exact value. Let's see, C, A, S, T. Cos sign is negative here, and here I think it's the one, two, root, three triangle. I think it's gonna be pi by three. It is, so that's pi by three, that's pi by three. So X one is two pi by three. Oh, we've gotten so much better at radians. X two is four pi by three. Here I would do my little alarm bell if I wasn't coughing and sick right now and my voice wasn't so sore, but I would yell, bellow, pretend I did, jump. Okay, because one, you know what? That's not gonna work for my, you know, I'm just gonna sketch it. You know where cos sign is one high? Right there. Which is what angle? Zero. And technically right there, two pi. Ah, nope, they restricted two pi. So there's my three answers. And then if they want the general solution, is there a period change at all in here? Cos I will give you one with a period change. I told you that. Otherwise it's just gonna be this plus, and this, and that all plus two pi, and we're in as integer. Yes, could I do one like two from quiz five? Sure, maybe. Which quiz five version did I give you? Version two or version one? Well, how about I open up version two? Question number two, like this baby here? Sure, I'd be happy to do that. I'll make one up. Example, and it would go something like this. If, pick a trig function, cos theta equals negative three over five and tan theta is less than zero. Pick another trig function. Actually, let me do this. Instead of theta, I'll use alpha because that's what we're gonna end up using here anyways. Pick another trig function. Secant, okay. And secant beta equals secant goes with which trig function? It goes with cosine, so I want hypotenuse and we're opposite. I'm gonna use the other one here. 13 over 12 and sine beta is positive. Find an exact value for sine or cosine. Value for sine or cosine. Sine of alpha plus beta. I think I tried to make this one work out evenly. Sort of. I have no idea what is sine of alpha plus beta. Sine alpha plus beta, sorry. Plus alpha plus beta is sine plus plus cosine? Oh, okay, I thought it was minus, my bad. I really don't have them memorized. Let's fill in what we know here. Do I know the sine of alpha? No. Oh, I know the cosine of alpha, negative three over five. I know the secant of beta. Secant goes with which trig function. So this is actually, you know what the cosine of beta is? 12 over 13, I can fill that in. 12 over 13. I'll put a plus sign there. What I need to find is that and that. Well, let's go tackle alpha. When they gave me this, what they really told me is that x is negative three and r is five because cosine is x over r. Seems to me I can find y. Although I have to be a little bit careful. They've told me that cosine is negative, which is here and here. And they've told me that tangent is negative, which is here and here. Sorry, not, that's wrong, Mr. Duk. Cosine is negative there and there. Tangent is negative there and there. Right out the cast, real, Mr. Duk. Don't be so lazy. Anyways, I think this is telling me that I'm definitely in that quadrant. So sine is positive. And I think if I go y squared equals the square root of five squared minus negative three squared, I think I end up with y equaling four. I think sine ends up being four y over r, five. I'm gonna do the same thing for beta. I know that r is 13 and x is 12. What is y? Oh, and I know one more thing. They told me that secant and therefore cosine is positive, which is here and right out the cast, real, Mr. Duk. Don't be so lazy, which is there and there. And they told me that sine is positive, which is there and there. You know what? I must be in this quadrant. So y is also positive. And I think when you do the Pythagoras, when you go y equals the square root of this squared minus this squared, I think you end up with y equaling five, I think, which means sine is gonna be y over r, five over 13. I'll end up with 48 over 65 plus negative 15 over 65. I'll end up with 48 minus 15, because I have a common denominator already. How lovely, 48 take away 15 is 38 take away. 33 over 65, is that right? Any others? Yo, do an example of number one. Oh, period change. Ah, that, sure. Now I know where we're, sorry. My brain went, huh, wait, one. That was a while ago. So take a trig function. Cos of, and they'll put a period change in there. Now, it's gonna be a two or a three, because anything more, you get so many answers, it's just a waste of time. Because remember, if you put a two in front, you get four answers, well, first of all, if you put a one in front of the x, if you never write two answers, usually. You put a two in front, four answers, put a three in front, six answers. If I put a four in front, how many answers are you looking for? Forget it, okay? So let's max it out, let's go three x equals, and you want this to be an exact value question, or do you want this to be decimals? I don't care which. Okay, so let's find, let's see, how about, oh, we'll go negative root three over two, zero less than or equal to x, less than two pi. Why don't you type some in hand writes? I don't know. Do you see the period change? Can you, in other words, can you notice there's something in front of the, you have to spot that, okay? And you can't go, well, I'm gonna divide by three, and that's gonna become us, and you can't do that. But you do have to spot that there's a period change. Why are you going on? Because every year I have some kids who just do this as though there's an x there, and I'm going, no, there's a three x there. Do you see the period change? Say yes. And I would make a big note. The period is actually that, two pi over b, which is three in this case. But then I would say, even though I've noticed this, I'm going to ignore it temporarily. I'm gonna replace it with the letter A. Y-A-ary, because I like that letter. I'm gonna solve, remember doing this now, it's all coming back to you. Solve this as normal. Castrol, cosine is negative here and here. One, two, root three triangle. I think pi by six is my reference angle. Pi by six is my reference angle. Pi by six is my reference angle. I think my first A value is five pi by six, yes. And my second A value is seven pi by six, except I wasn't asked to solve for A. I was asked to solve for, not three x, just look. No, I wasn't asked to solve for three. The answer to three is three. I was asked to solve, I wasn't asked to solve for A. What's the variable they gave me in my question? X, that's okay, ready? Scene one, act one, take two. I wasn't asked to solve for A. What was I asked to solve for? How would I get the x by itself if I knew A? How will I get this by, you know, divide by three? If I wanna turn this into an x value, divide by three. Now dividing by three, same as putting an extra three down there, it's gonna be five pi over 18. Dividing by three, same as putting an extra three down there, it's gonna be seven pi over 18, but there may be more. How do I know? I'm gonna add the period to each of these values, except what's my denominator here? Times by six, times by six. Can I do that, please? And that way I can do the math in my head. I think there's possibly a third value if I add the period to this. What's five pi by 18 plus 12 pi by 18? I'm pretty sure it's 17 pi by 18. Oh, and what's seven pi by 18 plus 12 pi by 18? I think there's a fourth value of 19 pi by 18. There may be more, by the way, when am I gonna stop when I hit 36 pi by 18? Yeah, two pi, but I'm doing the common denominator in my head. Let's see, now we've added to those two to get these two. Let's add to these two and see if there's more. Add the period to this, what do we get, my friend? Yes, is there a sixth one? More specific? What's the answer? 31 pi by 18? Is there a seventh one? There shouldn't be, but I'll double check. If I add the period to that, I went bigger than 36, uh-uh. Just to show you what we really did, Ian. Clear, clear. Here's cosine of three x, three x, Mr. Duk. Three x, let's try that again. Here's negative three, sorry, negative root three, good gosh, can I watch this more over two? If we normally, if we didn't have a three there, if we just had a plain old x going from zero to two pi, we would have one cosine wave and a line root three over two high, negative root three over two high. Let me just change the view into a tiny bit. I'm gonna go from negative two to two, so you can see it a bit easier. There's my cosine graph. As soon as we put the three, as soon as we add the period change, we get this. What we are doing is finding these first two. How do I find the next one? If I add the period, I'll end up right there, that gives me that one. If I add the period, I'll end up right there, that gives me that one. If I add the period, I'll end up right there, that gives me that one. If I add the period, I'll end up right there, that gives me that one. That's what we're doing with this little trick right here with my little clever, don't deal with the period change right away, find what it would be with no period change, then do the period change, because it's a lot easier, then add multiples of the period. I think there's a quicker way to get there, but it's more complicated and not much quicker to make it worth the more complicated. Okay? Yeah, sure can. Oh yeah, I like number, I think I like number two. Although I suspect I've put this as the multiple choice and you're right, it looks so simple and I'm even gonna contradict you a tiny bit. It looks so simple because it is. But it's so simple that our brain's gonna trip on itself. First of all, did they add the domain restriction? So when they say this, that's the magic phrase for general solution. Okay? I'm gonna find the first two and then I'm gonna add multiple of the period. What kind of an equation is this quadratic? How do I know without having to think for very long? Got it squared. How do I solve a quadratic? What's the first thing that I always do? I'm gonna go like this. Troy's seen it yet. What? GCF. Which, oh, what's the first thing that we always, always? Oh, did Mr. Dewick have this question in mind when he went on that big rant? Because it is so easy. We're looking for the complicated, complicated. Wait a minute, two brackets and actually note as it turns out, this one's got a GCF. This one's got a GCF of cosine and it's got some nice roots but some bad roots because I get cos x equals zero, alarm bell, cast rule's not gonna work and I get cos x equals one, alarm bell, cast rule's not gonna work. In fact, what I would do to find the actual angles is simply sketch the cosine graph and I would say, well, where is cosine one high? My third solution is gonna be zero. Where is cosine zero high? Let's see, that's two pi, that's pi, pi by two and three pi by two. Here is my x one, pi by two. Here is my x two, three pi by two. That's the answers between zero and two pi. Did they give me zero and two pi as a domain restriction? No, no, I say to you, they actually want me to give them the general solution so it's going to be something like this. Solution, x one equals, oh heck, Mr. Dewick, let's get really, really clever. Solution, nope, nope, nope, that's so cool. And that was easy, plus period change. Ian, by the way, the one that you asked me here, I'm probably not gonna do this, I'm gonna ask for the general solution so you would find these first two and then you would add the period. Most common mistake is kids wanna add two pi and what's the period? Two pi over three times n as an integer. And that way, if I want to put a bigger number than three here, like a four or five, I'll always ask for the general solution because then you're still defining the first two and quitting instead of then adding the period, adding the period, adding the period. So I can put a bigger coefficient in front if I want to. Any others? Six from the trig review, written or multiple choice. Identities review, from the identities review. Have you looked at my online answer key? So this one I went kinda weird, you ready? How many different trig functions are there? Only two. So you could do this by rewriting this in terms of sign and post and that will get you there. I started out and then I actually ended up erasing a bunch of stuff and I said, wait a minute. I do notice I have a binomial denominator with no squares. And now we're gonna add another trick to your bag of tricks. What would the conjugate of this be? What's on top over here? That's how I knew I bet you I want to use conjugate. The fact that the conjugate appeared on the top on the other side, that's why I said this time it's not gonna be a last resort. Now it was my last resort aside from rewriting this in terms of sign and cost, which is why in my hint sheet I said if you have more than two trig functions, then almost always write it in terms of sign and cost. If you have exactly two, sit and pause, think, do some math in your head. So I went conjugate, cosy can't plus one, cosy can't plus one, specifically because when I thought that in my head I realized it was sitting over there I got this, and then I ended up with cosy can't squared minus one, and you know what cosy can't squared minus one is from your formula sheet? It's on there, cosy can't, cosy can't squared minus one. What's cosy can't squared minus one? And then I said can I cancel, it is factored. Now I factored out a cotangent squared and I ended up with that. That's probably the best way to do that one. If you try rewriting it in terms of sign and cost on both sides, you'll get there, you'll get complex fractions on both sides, you'll have to clear the complex fractions, you will get there. I think mine though, well, one, two, three, four lines. I think if you do it rewriting in terms of sign and cost, eight lines I think in my head if I'm counting correctly. Is that okay? One of the few times that I started with conjugate right away, but do you see why I did? I noticed it's sitting on the top over there on the right, I noticed binomial denominator with no square, that's always my trigger. Oh, and it's sitting on the top over there on the right. That can't be a bad thing to try, gotta be good. Yes, I am actually pretty good at these. No, there's me doing the math in my head and counting. You can hear a track of eight lines? Well, okay, try. What I really said was it's four lines to do the conjugate no matter what, how many extra lines to rewrite it in terms of sign and cost and clear fractions, four more lines, that's where I came up with the idea. I didn't do the whole thing in my head. I asked how long would it take me to do the step to get to the conjugate? So it wasn't quite as impressive as you think. Or maybe that was more impressive because I knew enough to know that I didn't have to do the whole thing in my head. You're my hero, Mr. Do, oh, stop it, I'm blushing, please, no more compliments, I'm all embarrassed, please. Any others? So today after school is gonna be very similar as this class. I'm not gonna, like if you ask me a topic or an example, I'll do it, but I'm not going to spend 45 minutes straight just talking, whipping through the hole, I did that Monday. And I also did that last year and put that video online as well. Both of those will help. Folks, really, I think for those of you that are struggling with identities, look at my online answer key, I put it under blog me, it's sitting there, that helps. Trust me, I think 99% of you, by looking at this, for example, Amy, if you had seen this, I think you would have figured it out, right? Or Justine, if you had seen, I can't remember the one you asked me, but I think you would have figured it out. And I think you'll still have the aha that comes along with it, which I think is beneficial, okay? I've made up to about number 80 on the trig review. I haven't got all the way through, I haven't even got close to the written. I'm gonna save that as a PDF and I'll throw that on there tonight. It's better than nothing. But I was doing every single question, not just the ones that I assigned you, which is why it's taken me so long. But I also know that tonight, I'm not gonna have to free out, well, I could take my tablet with me and while I'm watching the game, I can try. Forget it, I'm gonna have fun. I like you guys, I really do, but I'm also trying to occasionally have an evening off. So I would have got it done this weekend, but I got dropped kicked by a cold slash flu thing. So I was on the couch chicken souping on Sunday. Wasn't really interested in doing that. Were you here on Saturday for the BC? Like I was miserable, eh? I was staying away from everybody. People were coming to shake my parents. I'm contagious, don't, you don't really, don't want, I have the bubonic plague or whatever, it was disgusting. All right, I will post this online as a video.