 A couple of announcements before I jump in. This is a bit of a weird one. I'm going to be saying no calculators today because there's no calculators on this test. This is the last non-calc section of any test, which means any trigger I'm going to be asking you is going to be exact value questions and special triangles and all that stuff. Also, if you go to the website pitmath.com and you click here where it says Math 12 and you click here to access Math 12 Provincial Exam Reviews and Video Tutorials. I sent these links out to you as well, I think, on the weekend or just before the weekend, like a while ago, a couple of days ago. Anyhow, Trig Part 1 Review and Trig Part 1 Review, you can find two summaries. It'll be fairly similar to what I'm doing with you today. If you just want to hear it in a slightly different way a second time or whatever, or I didn't answer the question you specifically wanted me to answer. Also, if you want, nope, never mind. These are all both Unit 1 Trig Parts 1 and 2 combined. I did give out a second half of the first review, the even-numbered pages, which for some reason didn't photocopy, so if you were away this morning, thank me later. Can you turn in the booklets to page 20? What I want to do first of all is do straight Q&A. In other words, if you glance at a question and you say, I have no idea how on earth to do this question, then I'll do it with you. And I would like to spend a bit of time probably talking about graphing, because I suspect there's still a few questions about graphing. I'm going to do that. And then secant, cosecant, cotangent, and tangent, the ugly cousins. I'm hopeful that the cast rule and all that stuff is pretty good. And all I need to talk about with those ones, Amanda, are the curveball type questions. So you all have this handy dandy thing in front of you. On the first page here, are there any questions? And you need to be honest, because if you're the only one that doesn't understand, that's still worth asking. Are there any of these that you would be going, I would have no idea how on earth to even start this question or what to do. Yes, number three, absolutely. So this is what I'll cut and paste in the tutorial. Trig, part one, tutorial. I won't copy the whole little test bank, because some of it will be redundant. I want to save paper. Okay, I think I like this question, or I might ask you a range, which is similar, or a minimum value, but you're going to see something like this on your test. Okay, it says, find the maximum value of this. What they're really talking about is vertically how high and how low. Dominique, I said vertical, and here's how I visualize this. What's the vertical displacement? I think I'm down here somewhere. What's my amplitude? Which means I would go five up, how high would that be? Yeah, and five down, how low would that be? There's your maximum value. There's your minimum value if I ask that, or there's the range between negative eight and positive two. So if they wanted the range, negative eight, less than or equal to y, less than or equal to two. And you'll notice the three doesn't actually show up directly. It's indirectly in here. Is that okay? Any others on page one? Yeah, okay. Five, I have no idea. Which trig function is it? It's cotangent. There's a reason I have no idea. I really haven't memorized cotangent. Ah, what does cotangent go with? Here's what I've memorized about tan, besides the fact that it happens on a beach. Did I mention I was in California? Okay, shut up. And you know what? I'll even do the, bring in the graph paper, Mr. Dewey. There it is. Heck, let's even bring in some graph paper. Heck, Mr. Dewey, let's even insert a little graph. Here's what I know about tangent. This is what I've memorized. First of all, it's shape. Okay, I got that. Sort of looks like a letter T if you stretch the T out a little bit. I kind of remember it that way. I know for a fact that tangent goes through zero, zero. And I know it has an asymptote right there and right there. Is that okay, Ellen? Where is this asymptote? This is what else I've memorized. At 90 degrees or pi by two and at negative pi by two. You see, this little part of memorization tells me an awful lot. Can you tell me the period? How long is one wave? Well, from negative pi by two to positive pi by two. Oh, period's pi. That's one tan function, not two pi, like cosine and secant and secant and cosecant. Co-tangent and tangent are a period of pi. Okay, also it has asymptotes. Where would co-tangent have asymptotes? Here and every pi because that's a period. Now, if tangent has asymptotes right there. Oh, they want the value of pi. I'm going to have to extend my graph a little bit further, aren't I? Okay, pi by two, three pi by two. Sorry, pi by two, two pi by two. Here's three pi by two. I'm pretty sure there's an asymptote like this. I'm pretty sure tangent goes through zero there again. I'm pretty sure it looks like that. Is that fairly reasonable to do? It takes a little bit of time, like about five seconds. But, you know, the five second trade-off, Sabrina, for not having memorized yet another graph, I think I'll live with that. Now, here's the key then. It looks like the tangent of pi is zero. So what would the co-tangent be? Undefined because zero has become undefined. That's one way to do it. The second way to do this would have been to simply say, you know what, tangent is defined as y over x. And if I go to my lovely little unit circle, pi is right there. What's my y-coordinate right there? Zero over, what's my x-coordinate? Negative one. What is zero divided by a negative one? When zero is on the top, that works out to zero, and then you would say co-tangent of pi was the reciprocal, one over zero, undefined. So I can get there in a couple of ways. Which way is faster? Well, probably this way. This way is more flexible in that it can handle angles that aren't necessarily on the edges here. I could probably make an intelligent guess. Any others on this page? I don't know, but number four is a weird one. I was kind of wondering if nobody would ask. Let's find out here. This is a way that I can ask you if you understand graphs without blatantly asking you to graph something. So it says, which of these values would result in it having no x-intercepts? You know what? Let's look at it with D being zero, with no vertical displacement, and then we'll ask, what happens if D changes? I would do a real quick sketch. What's my amplitude here? Which trig function? Any phase shift? That's why I started right there. Okay, it's going to look like this. Except this one with a two there, it's a horizontal compression. That's going to be a pi right there. I really don't care about horizontal, because they're saying which would not have any x-intercepts. The domain is all real. So no matter how much I shrink or stretch it horizontally, there's going to be x-intercepts. The only way I could not get any x-intercepts is to move up either so high or so low that my graph wasn't touching the x-axis anymore. If I moved two up, you know what? I got negative four to play with. I'd still have a couple of x-intercepts. If I moved three up, I'd be kind of right there. I'd still have a couple of x-intercepts. Heck, if I moved four up, I'd still be touching once right there. I'd still have a couple of x-intercepts. You know what, Brett? You're right. It is D. Five up would have no more x-intercepts. Example of a mini curveball that I would hope you wouldn't panic. I hope you would say, well, why don't I do a sketch and kind of think about things here? Turn the page. Or next page over. Any on number six or seven, whoop, or eight or, hello, try that again. Any on six or seven or eight or nine that you want me to go over. So you'll absolutely see number six, except it's going to be radians. Okay, I'll ask you to find a reference angle or a principal angle or both, or a coterminal angle. It's going to be radians though. Range. Yeah, here's a nice algebraic one. I don't know what you're thinking about. You can always ask me to do if you want to. Number eight, here's a graph. Tell me if it's the correct equation. Sure, that's fair game. Any of these? Thank you for finally asking. It's kind of pausing there going really. By the way, good news is, seven is a little curveball. Okay, and it really ties into the one that Ellen asked earlier about how range works. You see, I know that this value here, which is the lowest, came from the displacement minus the amplitude. I know that this value here, which is the highest, came from the displacement plus the amplitude. Let's try that, maybe. Oh, wait a minute. I'm not going to be able to get a whole number. I'm just going to get an algebraic expression. I'll plug that in and I'll get the a by itself and see if that helps me. If I plug that into there, I get negative k plus a plus a equals 2k. Carly, what's a plus a? I think I'd write that next. I'd plus the k to this side. You know how many k's there are as it turns out? Three of them. Get the a by itself. Apparently the a value... That's wrong. That's wrong. Over 2. Thank you. Oh, let's go back and try the same thing. By the way, I'm kind of having a feeling that probably this is the wrong answer because it'd be a fluke if they worked out to the same thing. See, you kind of think like a test. It might be. Let's do the same trick, but this time let's get the a by itself over here and plug it in to here and see what d ends up being. So if I take this guy and I rewrite it as a equals 2k minus d, and now I plug that in there, I'll get 2k minus d plus d. Sorry, let's try that again, Mr. Duc. Substituting it improperly. I'll get d minus 2k minus d equals negative k, right? I'll get a minus 2k. I'll get a plus d equals negative k. What's d plus d? 2d or not 2d? That is the question. And I have a minus 2k. If I plus it over here, I'll get that. Ah, you know what? Turns out d is k over 2. A is 3k over 2. Now I would consider that like an a minus level question. This is a curveball, but do you see how I haven't done a single thing new here? I just haven't panicked. I said, apparently they think I can figure this out. Let's pull up some grade 11 algebra. Now, honestly though, what I would have done is I would have glanced at it. I would have said, what the heck? I would have put a star next to it. I would have moved on to the next thing. Are you all a bit of a kid? I would have worked on it for no longer than 60 seconds. I like, I found if I get my brain working on something like this, and then I move on, what happens subconsciously? Your brains keep working on it. You've all had the experience of suddenly 10 minutes later, an answer pops into your head. Really? Okay. Thanks for coming for the 10. Okay. No problem. I'll throw this online. I may be available Thursday after school. I'll find out, sorry, Thursday, ask me. How did I get that? So this goes right back to the question that you asked me earlier. I said, you know what? The range is the lowest point. Is this minus that? And the highest point is this plus that, right? Five down and five up from the amplitude. A down and A up from the, sorry, I said five down and five up from the amplitude. Five down and five up from the vertical displacement. A down and A up from the vertical displacement. Oh, and they told me what the answer was. Apparently when I went A down, I got that. And when I went A up, I got that. Is that right? This is a tough question. Don't kid yourselves. Pardon me? I like them better than them. I make far less dumb mistakes with them. Letters than numbers. Alex, I sure can. Are we okay at number nine? I hope finding the quadrant, you can sketch it and find the right quadrant, right? Number eight. Now I'll probably ask you to recognize the graph. I'll definitely ask you to graph something like in the most recent quiz. So let's see. Vertical displacement and amplitude I try and find first. Alex, how high? How low? Careful. Negative three. Total distance? Amplitude is nine, right? Nine up, nine down, nine up, nine down. Dominic, what's the amplitude? So if you go nine down, you'll be in the middle. Or if you go nine up, you'll be in the middle. The vertical displacement is six. Okay. And right away, Emily, I would go like this. No, no. What's the amplitude? What's the highest point? 15. Go nine down, right? You must have gone nine up from the middle and you must have gone nine down from the middle to get to negative three. It's got to be six, all right? And amplitude goes there. Vertical displacement goes there. So I crossed out B and C, or to B and D. Now I need to find B. Now remember, B is not the period. B is two pi over the period. How long is the period? Well, one wave goes from here to here from zero to 10 pi by three. I think the period of this graph is 10 pi by three. So I think B is going to be two pi over 10 pi by three. Vlad, how do I divide by a fraction? So this is going to be two pi times three over 10 pi. Okay, the pi's cancel and I get six over 10, which in lowest terms is what? Three over five. Is that okay? That's how I'd handle it. Although I'll be honest, I would have canceled the twos out on this line and got straight to the three over five, but that scares some of you. Some of you just like to multiply and then reduce. Okay, easy. Number nine is okay, yes. Emily, we're good. Smiling for some strange reason. Oh, your book? No, leadership? Sorry, I'll write you a note. Turn the page! Any here on 10, 11, 12, or 13? 10, how would I find coterminal angles? Add or subtract? What? Two pi, although I'd use 18 pi by nine. Here, I can do it in my head. 11 plus 18, 29 pi by nine, and 11 minus negative seven pi by nine. The answer looks like it's B. Number 11, this looks like a job for? The arc length. I can't remember it. Oh, please do, I hope. Really look for the word arc, sorry. Sure can. Okay, I want to ask you a multiple choice cast rule question, but I can't just say, hey, where is sign positive? To try and come up with four wrong answers is tough. So this is how we will ask you if you understand the cast rule. That's my way of saying I like this question. I like this question. There's going to be one something like this. Except I'm going to, well, I'll show you. The only thing I don't like about this is it's too much typing. Amanda, I would never write secant M is negative. I would write secant M is less than zero. And tan M, I would write tan M less than zero. How would I write is positive? Greater than zero, okay? So I'm not going to write out those words. Is negative, is negative, forget it. Okay, C-A-S-T. I don't have a rule for secant. Spence, what does secant go with? Cosine. Cosine is negative according to this question. That's going to be here and here. And I do have a rule for tangent. Tangent is negative. Tangent is going to be negative here and here. Where is the overlap between which two angles? What angle is this up here? Pi by 2, how about here? Between pi by 2 and pi, has to be. That's the question that I'll give you and it's probably going to be on the first page where that's how I'll test you. Do you understand the cast rule? And to make it interesting, I'll probably have one of them be a reciprocal or both of them be a reciprocal function. Is that okay? 13, sure can. Glad you're asking. So number 13, what do they want me to find? Did they give me the sine and the cosine as fractions? As soon as I see that, I'm going, oh, they gave me x and y and r. Or they could give it to me as points. We've done that on a couple of quizzes where they actually gave me something comma something, x and y and I could figure out r. But here, sine is what over what? Sine is what over what in terms of x and y and r. So I think it's y over r, by the way, Kido. I think that. Cotangent is what over what in terms of x and y and r. So I'm pretty sure that one of these, there's a negative, one of these guys is negative. So let's think here. According to this question, sine is positive. Okay, that's here and here. Cotangent and tangent is negative. Where is tangent negative here and here? Where's the overlap? I have to be in this quadrant. In this quadrant is my x-coordinate or my y-coordinate negative. Which one? Convince me. Y, how come? You're to the left, right? Because I'm to the left and I'm above, right? I'm above the x-axis. I've got to be positive y. I'm to the left of the y-axis. It's got to be negative x. Which trig function do you want me to find? Coast, Tyler, which is what? x over r. There's a couple like that in the review. There is one particularly cheap, nasty, yucky, I think slimy one. So tricky that when he did the answer key. I actually saw the stats on it because it came out on the provincials back when they used to release stats. And something like 90% of the kids got it wrong. And of those 90%, 80% picked the same answer because it had a trap within a trap within a trap. And 80% of the kids found the trap within the trap and they were so proud of themselves. And they missed the third one. Which one am I talking about, you ask? Let me go hunt for it. It's another one similar to this, but it's cheap. Bear with me for a moment, please. Oh, I hated this question. I'll recognize it right when I see it. Oh, yeah. Because it's a great discussion question. Heck, Mr. Duke, go pit page. I even remember it's at the top of the page, I think. Tells me how much I hate the question, but I can remember what it was. Got to be getting close, Mr. Duke. Yeah, this one here. This was mean. Very similar to the question Carly, you just asked me, which is why I don't mind doing it right now. It's kind of like, hey, let's review this. So they did mention the unit circle. Remember, as soon as they say that, I know r is 1. Nice. And they told me that cosine theta is m. Really, what they've told me is that's my x-coordinate. Right? Because it's x over r, but r is 1. Y is going to be the square root of r squared minus x squared. Oh, what quadrant did they tell me that we were in? What can you tell me about the y-coordinate in that quadrant? What can you tell me about the x-coordinate in that quadrant? So a whole bunch of kids chose this one, which is wrong. And here's why. The x-coordinate is negative. But what can you tell me about cosine in this quadrant? Positive or negative? Did they put a negative here at all? Then that m must have a negative hidden inside of it. It has to because cosine has to be negative. When I wrote down x equals m, that there has a hidden negative inside the variable that we can't see because we don't know what the value is. And so, yes, I agree with you. The x-coordinate is negative. But it already is because they said that statement there. The correct answer here was that one. That's a cheap one. Because everyone was so proud, they caught the negative. There's the first trap I told you about. And then they said, oh, and of course, x is negative. It's the second trap I talked to you. I just put a negative there. And they said, OK, it's that one. The third trap is actually if that was the case, they would have had to have had a negative here, too. Right? Do they? That must mean whatever number is sitting in there that's hidden must have a negative inside there already. I don't need to add an extra one. Cheap. Cheap. I will not give you one that's stupid on your test. What I do like, though, I think it's totally fair game to give you something like this where you have to use x and y and r and come up with an algebraic expression, either to find the points or give you the points and find the trig function. Yes. I'm going to wash my hands after that one. I hate that question. Turn the page. You going to make it Dominique, you sure? OK. You want me to go over, I will, if you want me to. By the way, this test has lots of curveballs. Don't freak out because this practice test you're looking at, there are going to be some nice easy ones, too. In other words, if you're going, no, I like this because it is actually a bit of a higher level. Vlad. Sure thing. What do they want me to find, Vlad? Of? Now, exact values is a trigger phrase. It's either special triangles like one, two, root three. But look at your answers. Are we using one, two, root three? Or you're going to be using x and y and r. And I think that's what it wants us to do here. Because I think what they've told us is that my x-coordinate is that. Yeah. My y-coordinate is that. And if they want me to find cosy-cant, well, that's going to be y over, sorry, no, Mr. DeWitt, r over y. Right? Yes? Look up. Yeah. I mean, let's become good test writers, please. I'm ready without knowing anything else. I've got a 50-50 chance of guessing right now. In fact, I've turned it into a true or false question. Same odds. Oh, and tan is y over x. Oh, what would happen to the ks? What happened to the negatives? Technically, I'm done. It's got to be a, let's actually go further and convince ourselves that we can find the root 29, however it appears. r would be the square root of x squared plus y squared. Right? By the way, if you guys are all academic kids, Joel, if you're not in the habit on a multiple-choice test of crossing out wrong answers every time you do a step, you're wasting precious time. You'll notice I finished this question. Actually, no, I didn't even, I don't think I did a third of this question, and I have the right answer. It is going to be a. Get in that habit. Of lab, the negative is going to cancel. 2k squared is 4k squared plus 25k squared, and I'm going to get this, the square root of 29k squared. Oh, what's the square root of k squared? I think I can write this as a k in front and a root 29 right there, which explains why in this one, of course, the k did vanish too. Yay, it is a. Is that okay, Katie? Did I do anything new there? No, stubborn and clever. And, you know, relaxing, not panicking, looking for certain trigger words, exact values, at special triangles, unit circle, or, oh no, look at my answers. There's no root threes and root twos. Ah, this is going to be that stupid x, y, and r stuff. I've done a bunch of these. Why do they have to say where k is positive to avoid that stupid this here, where m could be positive or negative and actually had to be negative to satisfy this question? I hate that previous question. Any others on this page? The real question is not where you're supposed to be. The real question is, are you even here right now? The alarm bell goes off. You know where it goes off? Right here. Is that one third factored out of the x? Three pi is wrong. I'd cross out a and b right now. In fact, you know what? I'd cross out a and b and then I would say, when I factor, well, let's become negative or positive when I factor it out. Positive, so it's got to be to the left. I'd circle d and I wouldn't even do the math. Right? Can we start learning to write tests that way a little bit? Well, Mr. Dewick, are they all like really? You'd be amazed how often you can quit on a question. Let me give you a great example that I assigned to you guys earlier today. It was a quadratic trig and I just did it today. Let me go find it, Mr. Dewick. Might have gone too far. I was at the top of the page. Well, even something like this one here. So this would be totally fair game on your test. But if it's multiple choice, I'm not going to finish this question at all. First thing I'm going to do is this. Cosy cant, so it's going to be re-write it. Cosy cant x equals negative 2, because I get the cosy cant by itself. Cosy cant goes with which trig function, Tyler? Sin negative. So I know the answer is here and here. Yes, that can't be right because that's not in that quadrant. That can't be right because that's pi by 3 is in that quadrant. Right? And in fact, you know what? All I'm going to do now is find the reference angle and quit. If the reference angle is pi by 3, this is the right answer. If the reference angle is pi by 6, this is the right answer. I'm not going to do anything more than that. Why would I? You guys are looking at me stunned. You guys don't write tests this way. You need to. This is how you write multiple choice tests. And you'd be amazed, especially in trig, how often you only need to do about one-third of the question. And you're done. So sin is negative there and there. Let's see. This is negative 2 over 1. Do I have a triangle with a 2 and a 1 in it? Yeah, that one right there. Cosy cant. Cosy cant goes with what, Tyler? Sin is opposite over hypotenuse. So I'm looking for a hypotenuse over opposite. That one right there. My reference angle is pi by 6. I'm done the question. I'm not going to waste my time. Why would I bother doing more? Yeah? Isn't 2 pi by 3 what? I don't care. It's still not legal. Oh, I could have crossed that out too, you're saying. Apparently I didn't even need to do that much. Thank you, Ellen, for pointing out a short cut that I missed. Trig in particular lends itself to that because there's only four quadrants to play with and because half of them are negative and half of them are positive, almost any question can turn into a true-false question, if anything else. Let's keep going back to here. So phase shift. If you did factor out the 1 third, the same as dividing by 1 third, same as timesing by 3. Which pair of functions below have the same period? Okay. What's the period of cosy cant? Don't know. Oh, Amanda, what's cosy cant go with? Have they mucked around with the x at all? Say no. Then the period here is 2 pi. What's the period of tangent? Ugly cousin. What is the period of tangent? Pi. Have they mucked around with the x at all? Say yes. Period is pi over b, not 2 pi over b, like in sine and cosine, which would be pi over 1 half, which, oh, only smokes. I'm pretty sure the answer is that. But let's keep going just for giggles. What's the period of secant? Don't know. What's secant go with, Brett? So secant would have a period of 2 pi over b. James Coase. Pi. What's the period of cotangent? Don't know. Oh, what's the period of tangent? What did we just say? Ugly cousin. So cotangent will have a period of pi over b, in this case, pi over 2. There's your period questions for all those ones, right? And having said that, turn the page. Now, this is not a great example of how I will phrase questions on your test. If you want good example of how I'll phrase questions on your test, look at review number one, part one, and then part two that you got today. I took the questions phrased similarly to the provincial exams. This is from Alberta, but this has some nice higher level questions, Alex. I think it's a good review. Any on this page that you're going to, uh-huh, don't know. Or all of them. I'm good for that too. Number 19, great question. Ugly cousin, right? And I need to make sure I read the question very carefully. It's asking which does not have the same asymptotes. Okay. Remember the tangent graph? I sketched it a little while ago. I'm just going to visualize it, because I'm trying to get you better. If you slide that two up, because that's what that is, well, that move the asymptotes at all. So A does have the same asymptotes. It's not the answer we're looking for. If you slide it Pi left, will that move the asymptotes? Well, it will. What was the period of tangent? Pi. If you slide it exactly one period, won't it now overlap with itself perfectly again like you started out with? In that one. If you compress it by a half, would that move the asymptotes? If they were here, and now we compress them to here. See, it's half you were shaking your hand. I think it does. Like I think instead of being at 90 and Pi by 2, Pi by 2, I think it'd be Pi by 4, Pi by 4. So I'm leaning towards C, but let's go process of elimination because we're good test writers. D, if you stretched tangent vertically by a factor of 2, would that move the asymptotes at all? That'd be that. Now what I'll ask you is simply, hey, what's the equation of the asymptotes? I don't like these ones. We have to do all four answers to try and find the right answer. I hate questions like that. I've tried to get those kicked off of the provincial 10 exam as much as I could. I failed, but I tried. Any others on this page? I learned some sprinkler systems are just being tested. It's just a test. Thank you. The sprinkler systems? No, that's it. If the water comes on, we're suing they're sorry behind. So yeah, there you go. We should all get soap and things, right? You know, little shampoo, right? Honestly, that water in there would be so disgusting by now. By the way, do you know how the sprinkler systems work, though? It's a very neat system. My understanding, and this might not be with this model, but the water in there is stopped off by a piece of wax, and as soon as the temperature gets hot enough to melt the wax, the water will flow. Clever. Pardon me? In theory. Meanwhile, any others on this page? Like what? Oh, number 10? 20, you mean? No. Radiance, this is technically fair game. That's a little overkill, though. What it's saying is, so instead of 10 degrees, let's say I gave you a nice angle in radiance. What would happen if we did a vertical stretch? That wouldn't change the X intercepts. And the slide. Oh, the X intercepts have moved 10 degrees left. Pi, whatever radians left, that I'd expect you to handle. Okay. Oh, instead of 3pi by 4, we moved pi by 4 radians left. Now we're 2pi by 4. Okay. So yeah, I don't think I threw one like that at you, though. Like that fair game, and that would be a fair game as a curveball. Okay. Number 18, hopefully you could answer the, you know, period. Range. Phase shift. Which ones are true? I'm guessing one of these two is false, but maybe not. Oh, I look at the answers. Only one of them is true. Should check that. Okay. Written. This would be, find the arc length in terms of pi. So I would give you this angle in radians, and it'll be multiple choice. On your written section, I've already told you. Two graphs, and then four or five trig equations to solve, which we haven't done much of yet. We'll talk about that in a second. No. 22, no. 23. This is applications of trig functions. We're going to do that next unit. So no. 24. I'm not going to ask you to find the equation from points. I would ask you to find the equation if I give you the full graph. Ferris wheel, not yet. Number one, that you guys have this on page. Okay. So I broke the course, the units up a little bit differently than they did on this particular test. So number one would be fair game. Finding, being able to solve that minus eight from both sides, divide by four, and you'll have a negative two. Hey, have I got a triangle with a two and a one in it? Yes, it's the one, two, root three triangle. Whereas C can't negative, knock yourselves out. I bet you this one as well, you don't have to do part of the question and cross them out. But this I would also put on your written so you had to do the full thing the hard way, the long way. Do you want me to do it? Yeah, cast rule, right. Exact values. Uh, no no no no no, no no, no, no, no, no. No, no, no, no, no, no, no, no, no, no. also something like this I'll do about three more you can write these down or you can watch I'll print this up when you're done if you want copies I think I'll ask you to do something like solve the risk when I make these up and it might be exactly the same one as on your test because because what well okay whatever it happens it happens there's only so many Pi by 3 ratios and Pi by 6 is angles I can come up with right let's go secant squared theta equals negative 4 over 3 for 0 less than or equal to theta less than or equal to 2 Pi Sabrina would it be okay if I had given you the question originally like this in other words could you get minus 4 from both sides if I'm gonna get secant squared by itself I don't think I will on this test I will on your next test though we'll start saying I don't have to get the trig function by itself for you all the time this is a quadratic trig how do I know it's got a squared it's gonna be four answers this time Alex how to get rid of squared what I have to remember when I square root plus or minus so this is gonna be secant theta equals problem here mr. do it get rid of the negative can't take the square root of a negative see what happens much I make these up myself plus or minus the square root of 4 over 3 and Alex what is the square root of 4 2 and the square root of 3 is root 3 I use my cast rule if I didn't have the plus or minus here Spencer I'd have to do a bit more thinking but here there's gonna be it could be positive there's gonna be four answers what are the answers oh I need a triangle with a 2 and a root 3 in it do I have one this is the 1 2 root 3 triangle which of these angles has a secant of 2 over root 3 okay secant goes with cosine adjacent over hypotenuse so hypotenuse over adjacent of 2 over root 3 the bottom one or the top one which is how big Amanda which means that's pi by 6 that's pi by 6 that's pi by 6 and that's pi by 6 can you rattle them off to me Amanda isn't that common denominator trick nice what if for giggles I said this instead of between 0 and 2 pi how about between negative 4 pi and negative 2 pi sorry to do a domain shift if I'd given this to you right away Tyler I would have ignored it temporarily I would have solved it as is and now I'm gonna subtract or add I think in this case subtract because these are too big multiples of 2 pi although I'm gonna use instead of 2 pi what's my denominator here Tyler so I'm gonna subtract and I'm even gonna get more clever than that Emily underneath here really really small I'm gonna write that this here is actually negative 24 pi by 6 and this here is actually negative 12 pi by 6 because that'll tell me when to stop why not make my life as easy as possible so 1 pi by 6 take away 12 pi by 6 what do we get is that between negative 24 negative 12 well then do it again negative 11 pi by 6 take away 12 pi by 6 by the way Katie sometimes you may find some of your angles are already in the correct domain okay good happy joy 5 pi by 6 take away 12 pi by 6 is negative 7 pi by 6 is that between negative 24 you know what I think from all of these I'm gonna have to subtract twice aren't I so 5 take away 12 is negative 7 take away 12 negative 19 pi by 6 okay Carly what's the next one then 7 take away 12 take away 12 that's right it's negative 17 pi by 6 gotta keep going till I get to the domain what and what I'm saying is yeah keep going till you get to the domain I can give you whatever domain I throw at you now I'm not gonna throw something in the negative millions or positive millions in fact even this is a bit far but yeah and I could go you know bigger in the positive direction right so Carly now you figure that out can you get the last one 11 take away 12 you're at negative 1 pi by 6 not small enough yet take away 12 more there you go the last thing I'll throw at you on your written by the way your written will have a quadratic where am I sorry hang on here your it will have a quadratic it will have a reciprocal and it will have a domain change I did all of those in one question just to save time I'll probably separate those as three separate little skills though for me there will be probably a reciprocal but I doubt it'll be a quadratic reciprocal there will probably be a domain change but I doubt it'll be a quadratic or a reciprocal and there will be a normal standard boring one with a domain change problem I can't remember I've looked at it for you and then the last thing is the alarm bells the undefined or the ones is or the zeros is for example solve 0 less than or equal to theta less than not or equal to mister do it 2 pi cosine theta equals 0 or cotangent theta equals 0 cos theta equals 0 see the first issue is is 0 positive or negative the castrel doesn't work at all I'll put a copy and send it out instead this is where we fell on our unit circle we said temporarily let R be one we called this the unit circle and if we visualize a radius slowly draw there slowly sweeping around like that cosine is what over what in terms of x and y and r x over r and since r is one it's just plain old x what this is really saying is cosine is 0 when x is 0 is x 0 here nope it's one what about up here what about here this would be of cosine theta equal negative one what about right here sure not close mr. do it theta equals you could also have arrived there by sketching you could have gone like this well here's my cosine graph where is it 0 right there and right there hey that's pi that's 2 pi that's pi by 2 and 3 pi by 2 what about a reciprocal okay where will cotangent be 0 when tangent is what it's undefined now the other way to think about it is this cotangent is x over y what would make this fraction work out to 0 when x is 0 strands enough it's going to be the same two answers this one isn't it x is 0 right there and right there so we would say theta 1 that has to be a 0 theta 1 is pi by 2 theta 2 is 3 pi by 2 and that's also where tangent is undefined the last thing I did want to talk about if you want me to is graphing trig functions but if you're okay on that then I'm also good on pausing you guys want me to do some graphing trig functions or not sorry yeah okay I'll have to import graph paper here which you guys won't have in front of you unfortunately so we'll just do it together when I hit print unfortunately the graph paper doesn't print but I'll label my points when I make these up they sometimes go so let's find out there y equals negative 3 cos 3 bracket x minus 4 close bracket plus 1 I really don't like this question because there's a 3 there and a 3 there and that would mean some kids might get the wrong answer for the wrong reason anyways I'll leave it we always started out making the list what's the amplitude here Carly I agree what's the period don't say 3 because it's not 2 pi over 3 what's the phase shift is there an alarm bell oh excellent 5 by 4 to the right what's the vertical displacement and then here's what I also added I said take the period and divide it by 4 which is going to end up being 2 pi over 12 sticking an extra 4 down there or in lowest terms pi by 6 and then I said pick a good x scale a common denominator for this this and this Amanda when I graph this I'm gonna let each square be pi by 12 and what I'm really thinking is every dot is two squares apart that okay oh and one a whole wave is gonna be times by 4 times by 4 8 squares long yes and my phase shift is going to be 3 squares right so let's see what this looks like try to bring it in some graph paper here hopefully that'll be enough and it's kind of funny I really hardly even look at the equation anymore I'm all interested in this the only thing I'm telling the time I'm gonna go back to the equations when I ask where my graphing sine cos negative sine or negative cos I start out what's my x scale so I'm not gonna label each one that's gonna be really crowded how about one two three four five one two three four five six pi by 12 one two three four five 12 pi by 12 good enough one two three four five negative six pi by 12 and I'll add more if I need to what's my vertical displacement and that is going to end up being the middle of my graph my amplitude is going to be going up and down from there in fact I could have lanced at this and told you the range was go up one and then three up three down from there and they get up to the positive four because that's my amplitude one two three one two three one two three oh that's up here you mean that's a four you're right learn to count mr. do it do it right I had the right place though okay Carly phase shift what how many squares my first point's gonna be somewhere on this line here where my graphing positive sine negative sine positive cos or negative cos starts where which one okay got sign starts in the middle positive goes up negative goes down cos starts up high negative cos out starts down low so you know what my first point is there where's my next one how many two squares right one of those middle one of those top one of those middle one of those bottom right counting in Spanish just cuz hey let's go to left where will I be here yep how about one to here how about here yep so instead of starting I try to think of oh sure I can do this I don't want to gum this up copy three pasted down here same equation negative three sign I'll do it in highlighter yellow here here here here here here here here here this is why we said any cosine graph is actually a sine graph if you phase shift it right and every sine graph is actually a cosine graph if you phase shift it right you there's always an infinite number of answers what they give you a graph and say tell me the answer that's why I'll probably make it a multiple choice for that section because you'll have only one right answer to pick from having said that I'm gonna nuke that because that highlighter won't pick properly and they're gonna freak out don't need another one or is that alright well I ask you to graph tangent you know how to find pure good and domain well I ask you to graph cosecant or secant no you don't have to find period asymptotes domain any other folks okay let me press stop here if you want to hand in the books that would be great