 Page 345. I want to talk about using identities to solve equations. Now, I don't think I put one like this on your test. At least I know it's not on the written. There might be one on the multiple choice, but I think I threw one on one of your quizzes, and this does show up on the provincial. So we're going to cover it here. Using identities to solve equations. And it says solve the following equations where 0 is less than or equal to x less than or equal to 2 pi. And the first one you'll notice has cos squared and sine. Now, if this was a multiple choice question, I would glance at the answers and if they were decimals, I would say, hi graph and calculator. But they've given me between 0 and 2 pi and I think they're suggesting I can solve this algebraically. But I've never given you something with two trig functions in it. What the heck can I do here? I can target that cos squared. See it? How can I write cos squared as a sine? And your hint is, look at the top row of your formula sheet. What is cos squared if I want it as a sine function? Kyle. Okay, if you said 1 minus sine, 1 minus sine squared. What I can actually do to solve this is rewrite the 2, but instead of a cos squared, put a 1 minus sine squared and then minus 3 sine x equals 0. What do you think we would do now using your, because you've got some good equation solving skills, what would you do now? Can you get rid of brackets? Yeah, totally. 2 times 1 is 2 minus 2 sine squared x minus 3 sine x equals 0. What kind of an equation is this? It's a quadratic. How do I know? It's got a squared. I don't like this because the quadratic term, the squared term is negative. We've never really taught you to deal with squared terms that are negative. Do you know why? How can you make it positive Troy? Just throw everything the other side. I'm going to plus this over, plus this over, minus that over. It's going to be plus, plus, minus. We're going to run out of room here if we're not careful, but that's okay. Now this is a lovely quadratic, just like the quiz that you wrote recently. It's going to factor because there's no GCF. I checked for that already. You know what? I'm almost positive there's going to be a 2 sine x and a sine x. Again, if you don't know how to factor these, I'm around after school. I will happily teach you, but I need to find out what method you were taught first and then I can reinforce that method. There are so many teachers teaching different methods. I generally don't give one in class in grade 12. For a negative 2 there, I think I want to go plus 2, minus 1. Does that give me my 3? Yeah, it does. Wish I knew how my brain did that? No idea. What are the roots of this second bracket? Sine x equals negative 2. Why can I do that? Oh, yeah, lowest sine gets as negative 1. Oh, thank you. They gave me one factor with no roots. Save me some time. What are the roots of this first bracket? Sine x equals 1 half. Have I got a triangle with a 1 and a 2 in it? Yeah. Cast rule. I'm running out of room here, but we'll see if I can fit this all in. C A S T. Sine is positive here and here. Which triangle is this? Oh, this is the 1, 2, root 3 triangle. Thank you. Glad you made it. They talked to me ahead of time, so no dice for them. Hey, which angle has a sine of 1 half? The bottom one or the top one? 5 by 6. In fact, I'm pretty sure I'll get x1 equals pi by 6 and x2 equals 5 pi by 6. That's a 5 if you can't read my writing. So there's an example of where you can use a trig identity. How would you know that you were supposed to use a trig identity? Well, if it was multiple choice, you would glance at, first of all, you'd see there was more than one trig function. And you would glance at your answers. It was multiple choice and they wouldn't be decimals. Although I'll be honest, if it was on the calculator section, I'd probably still just choose to graph left side, graph right side, find where they cross and change my answers to decimals. I think it'd still be faster. But doable. Oh, and if it was on the written section, I guess you'd have more than one trig function. It was on the written section. I've told you, though, on your test, not going to happen. On the provincial, I don't think I've seen one on the written. I have seen it every couple of years on the multiple choice. Let's try b. Sine x minus root 3 cos x equals 0. Hmm. Suggestions? Crashing calculator? No, I'm going to try and do this algebraically here. First of all, is this a quadratic? No, it's not. I'm not going to be so attached to making this equal to 0. In fact, I might, because there's only two terms, I might say, why don't I write this like sine x equals root 3 cos x? Why don't I try fussing that over? Looks nicer. Anybody see it? I'll pause. You need to go tan equals root 3. Where'd you get that? Oh, divide by cos. Divide by cos. What happens to the coses over here? Can they cancel? It is factored. And Troy, what is sine over cosine? In fact, you get tan x equals root 3, and I'll make it obvious I'll put the root 3 over 1, because it is the 1, 2, root 3 triangle. There was a tangent hidden in there. That's kind of a clever trick. That one does show up just once in a while. The odd time. But did you hear how I recognized it? I said it's equal to 0, but it's not a quadratic. Is that why you're always, what kind of an equation is this? It's not a quadratic. So I said, I don't need to be so attached to keeping it equal to 0. We can solve this. Do I have a triangle? Oh, first of all, Castro. C A S T. Tangent is positive here and here. Do I have a triangle with the root 3 and the 1 in it? Yeah. Which angle, the bottom one or the top one, has a tangent of root 3 over 1? Opposite over adjacent. Pi by 3. Pi by 3. And I get x1 equals pi by 3, and x2 equals 4 pi by 3. Ta-da. So I'm going to add a couple of questions for you to practice. Let me try 7. Number 7. Now I've got a choice. If you want to, I can do a few more identity examples from that big hints for trig identity sheet. By the way, that hints for trig identity sheet, that's not homework. I just always fold a copy of their side because it's very hard for me to make these up in my head. That way I got a bunch of extra practice identities I can do with you if you want me to. Or I can give you the rest of the class. You want me to do a couple more examples or you want to just take, I see people nodding. Okay, give me one second, then I'm going to pause. So I just gave you your take-home quiz, and give me a second. I need to go find hints for proving trig identities. Bring that puppy up. And let's try, what, 22? Okay, sure, let's try 22. Tan n all over tan n plus sine n equals 1 minus cos n all over sine squared n. All right, suggestions. Okay, I think I would do that. By the way, starting with the most complex side, although is the right-hand side already in terms of sine and cos? I probably am going to come back to that later, I suspect. Let's see. So you're saying right the left side in terms of sine and cos because there is more than two trig functions. So tan is going to be sine n over cos, not s, Mr. Dewick. Good gosh. That n is throwing me off. I'm so used to Greek letters. All over sine n over cos n. Did it again. Plus sine n, Mr. Dewick is my math teacher. He says don't you dare not write that as a fraction. If we have one fraction, we want all fractions. How many levels does this fraction have? Now, if it was just one fraction over one fraction, it would be, how do I divide by fraction? It's not going to work here because I got one fraction over two fractions. This is a complex fraction. I'm going to show you that same trick. It's a great trick. If you master it, if you learn it, it'll almost always save you two or three lines. What's the trick? Look at each mini fraction. What's my common denominator? Cos. I'm going to go like this. Add a bracket. Add a bracket. I'm going to multiply by cos over cos. You can put it here, here, or you can put it here, here. I don't care. I usually have more room on the right side, so I'm going to go cos n over cos n. But to help me keep it all organized, Stacey, over one, over one. By the way, what am I really multiplying by when I multiply by cos over cos? The whole key here is if you ever multiply by something, it has to be a one because you want to keep these two things equal. If you multiply by something other than a one, now this will no longer be equal to that. You can't prove it. You'll end up doing 15 lines of nonsense. Here, you see it? Cos cancels. And you get all over. Here, you see it? Cos cancels. And you get sin n plus. Here, by the way, I encourage you, draw the little loops I learned a long time ago by putting that little chunk, chunk, chunk thing. It gets rid of a lot of my dumbest tricks, keeps me systematic. Here, is anything going to cancel? Nope, sin cos. Victoria, can I cancel? Have I factored? Don't you dare! And again, why am I so big on avoiding it? Cos as soon as you do that, now the rest of your work is wrong. You can't make it look like the right side. You can't do it. One of the reasons these are tough is as soon as you make an algebra mistake, you're out of luck. Suggestions now. Where, factor, where? Oh, the bottom? Really? By the way, what is the GCF in the bottom? What do I have on the bottom here? That's a little bread crumb, I think, telling me I'm on the right track. I think, I think, I think, I think, I think, maybe? So I would go like this. It's going to be sin n all over sin n bracket 1 plus cos n. Victoria, can I cancel? Have I factored? Then I can. Oh, but what would be left on top? Yeah, let's do that. I don't want to leave it totally blank. That would be silly and confusing. So here's what I have. 1 plus 1 over cos equals 1 minus cos over sin squared. Now what? Now you've got two options. I've heard Troy here say sin squared equals 1 minus cos squared. And that would work. And I'll show you how you could do it that way in just a second. Here is where I would say conjugate. Because on the left-hand side, do I have a binomial denominator? Yes. With no squares. Yes. This is the conjugate. This is where I think I would say, by the way, what is the conjugate of this Troy? What's on the top over here? That's also how I knew to do the conjugate. Because when I go 1 times 1 minus cos, I'm going to get the top. The breadcrumbs along the way. So I would go like this. 1 minus cos n over 1 minus cos n. And on the top, woohoo, I get 1 minus cos n, which means that bottom must work out to sin squared when all is said and done. Let's see. When I multiply this out, I'll get 1. And then Kyle, can you see I'll get a minus cos and a plus cos. What's a minus cos plus cos? Nothing. And then I'll get a minus cos squared n. What is 1 minus cos, oh, can I cancel? By factor? No. What is 1 minus cos squared? Okay. 1 minus cos m all over sin squared m q e d. Now Troy had an interesting approach. So once you've written that, put your pencils down and look at what I had written in the blue here. Sorry, actually look at what I had, sure, written in the blue here where we had 1 all over 1 plus cos. Troy, here's another approach you could have done. What did you say sin squared was? You could have gone 1 minus cos n all over 1 minus cos squared n. Now the reason I don't really like this approach is because kids are so tempted to cancel. Because if you're going to rush 1 minus cos and 1 minus cos squared, it looks like they cancel. Especially because if you cancel a 1 minus cos on a 1, by the way, can I cancel? Have I? Okay, you can't. But if you cancel that and that, what would you get on top if you canceled out a 1 minus cos? A 1. And what's on the top over here? What, like, kids are really, really tempted to do that. However, this factors. How many terms are there in the bottom? 2 minus sin is the first term of perfect square? Is the second term of perfect square? This is actually different to squares. This is actually 1 minus cos n all over 1 minus cos, 1 plus cos. Can I cancel? Have I factored? Then I could argue that those two brackets are the same. There's a 1 left behind, and lo and behold, I do have 1 plus 1 over cos. Sorry, 1 over 1 plus cos. Either of those approaches is totally valid. We're at the stage now where for a lot of these alex, we've got two or three ways to get there. In fact, I can think of at least one other way that we could have got there too, but I won't go into it. The key, Justine, is don't make algebra mistakes. Don't cancel when you cancel. Victoria, can I cancel? Have I? Those have to go together. Don't ever, when you're doing fractions, don't ever multiply by something that's not a 1. That's why I yelled at you so much two days ago for those of you when you're finding a common denominator who just have the habit of multiplying the top only. If you start that habit, you won't catch when you're doing it illegally. Trust me, always top and bottom, always. That was so much fun. Let's do another one. One more. All right, hit me with your best shot. What do you want to try? 29. Actually, 29 is from the old curriculum. So I'll talk about 29. I won't do it. Look at the top on the left. How many terms on the top on the left? Two. Minus sine? No. However, that does factor. Plus sine? Say yes. Perfect cube? Perfect cube? Say yes. We used to teach a factoring rule. It was called sum of cubes in the old curriculum. And you would factor that. And guess what one of your factors would be? Sine squared plus two, sine, cos plus cos squared. The bottom there would end up being one of the factors and a bunch of stuff would cancel. Okay. Kind of nerdily cool. So I'd love to do 29, but it's a bit beyond our curriculum. So which one? Sorry? 24? Oh, sure. That looks doable. Probably this is over the top in terms of what you'll see on the provincial. I don't know what you're going to see on your test or what you'll see on the provincial. The big review trig package that I gave you, which is an assignment and which you can do everything except for anything with a two theta in it. That's a great idea of the level of difficulty. So 24. Ooh, I haven't done this one before. I don't think that a while since I've done a brand new trig identity, I am getting my nerdy adrenaline rush. Well, the good news is I'm pretty sure the right side's done. Right? Oh, and this is where some of you may be so tempted. Why can't I plus one over? Because that would be assuming they're equal in order to prove they're equal. And that's really what makes identities tough is we can't use our equation solving rules, which are glorious normally. I mean, normally if I was solving as I'm plus one over and all sorts of stuff. Anyways, I'm pretty sure we're leaving the right as is. Suggestions. Ian is correct in that we are going to write everything in terms of sign and coast, but I'm going to give you some advice. Do you think we're going to eventually also have to get rid of brackets? I'm going to suggest we do it now without fractions because you guys are become your mediocre and you're becoming tolerable. But I know you make more dumb mistakes when there are fractions than when there are not. So I'm actually going to get rid of brackets first. Then I'll rewrite everything in terms of sign and coast and bring in the fraction because I'm a good test writer. Always trying to think, how can I minimize my sloppy mistakes? We've clued in this year, one of our biggest enemies on tests has been, we'll call them dumb mistakes, right? When we go through the tests, all of you are like, holy smokes. Can't believe my math A is coming back to you to haunt me. But it is. So I'm going to go boom, boom. C can't squared. Whatever the heck that thing is looks sort of like a V. I should know what that is. I can't remember now though. Plus C can't squared. Coast C can't. And then it's going to be minus tan C can't and minus tan squared. All over. Coast C can't. You guys asked for this, didn't you? Plus coast C can't sign. By the way, you just got a half mark. If I did give you one this nasty, nice thing about trig identities is, as long as you can do the basic substitutions, basically impossible to get zero. Anything cancel? I was kind of hoping I'd end up like it with a plus C can't squared and a minus C can't squared. But I don't, I didn't. Sorry, what do I forget to what? Oh, and the minus one. Sheesh. Except I'm going to do that because it's going to be a fraction. I'm sure it'll somewhere along the way. Well, actually, I guess what this is saying is that this whole thing should work out to what? One, because you're supposed to get one minus one is zero. Okay. So I may not need to combine fractions. I may just be able to make this work out to one. I notice I have a C can't squared and a tan squared. Are those related? Can someone look along the top row? Are they in the same identity or not? They are. Okay. What's C can't squared the same as? Sorry. Ooh. I'm going to do that for starters. I'm going to go one change colors, Mr. Dukes, so it stands out a little easier so they can read it better. One plus tan squared plus C can't squared. C can't minus tan C can't minus tan squared minus one over cosecant plus cosecant sine. And something good happens. I got a plus tan squared and a minus tan squared. So now I have one plus C can't squared cosecant minus tan C can't. What was that? Are you being witty and funny? If it was nerdy, I'll like it. Okay. Now what? I think now we'll try writing everything in terms of sign and code. Do you see why I said there's kind of an art to this, even though I called that hint too? I think we saved ourselves some yuckiness by saving that off. Well, I can factor it out of that and that, but there's a one there. I was looking at that actually. I was going, this is where I start to do some algebra in my head. If I pull a C can't out here, I'd have a C can't cosecant minus tan, what, this and this? With the other bracket. Well, I'm going to keep it around. Okay. Here's Ty's suggestion. Don't write this down because I don't think it's going to get us somewhere. This is what I'd be either doing in my head or a scrap piece of paper or whatever. I'm okay because I thought that too. I went, well, if I factor out a C can't, I'll have a C can't, and I'm writing X's instead of that stupid Greek letter because I can't write that Greek letter, cosecant minus tan. That's what I would have on the top. Or I could just put what? One over C can't X. If I factor out a C can't X from one, aren't you clever? You're saying that this is C can't over C can't. I think that's more complicated than I want to get to right now. I think if we write everything in terms of sign and cosecant because I see lots of fractions, I'm hoping some stuff will cancel. Let's try that first. I'm going to go like this. One plus C can't squared is one over cosecant squared. Cosecant is one over sign. Since I just wrote a fraction, I'm going to make that first one a fraction just to see what's going on here. Minus tangent is sign over cosecant, and secant is one over cosecant. All over. I'm just going to write some ditto marks right now. We're going to leave that for a while. I would never do that on the test, but in my homework. Sure. This is going to look like when I tidy this up, one over one plus one over sine X cosecant squared X minus sine X over cosecant squared X minus one all over blah, blah. Well, I could write that as one single solitary fraction. What would my common denominator be? Sine X cosecant squared. Now let's pause and let's try and think this through. To turn this into a sine X cosecant squared, what would I multiply the bottom by here? What would I get on top then? A sine squared is kind of nice. It's on the top of my sheet somewhere. This would be a one. In fact, I'd have a one minus sine squared, which is a cose square. I think we're going to try that. I saw some nice stuff potentially happening. Let's try writing the top as one single solitary fraction. We're going to write the top as a fraction all over sine X cosecant squared X. What would I multiply a one by to change it into a sine X cosecant squared? I'm going to go times sine X cosecant squared X sine X cosecant squared X and I'll get sine X cosecant squared X. Don't cancel because we're still doing some math. Then there was a plus sign, so the plus sign would drop down. What would I multiply a sine X cosecant squared X by to get a sine X cosecant squared X? It's going to stay as is. The one's just going to drop down. Minus sine would drop down. What would I multiply a cosecant squared by to get a sine X cosecant squared X? Sine X sine X and ooh, I get a sine squared X. Ooh, what's one minus sine squared? Cose squared and then I'd have a GCF on the top of. Cose squared and I have a cosecant. Ooh, good stuff happening here. I better rewrite the bottom and the minus one. And I better draw the line. Isn't this fun? You know what we need. Yeah, let's put on the rocky theme in the background. That'll help us, I think. That'll set the tone. Little pump us up, get our brain going music because this is, yeah, this is in the big leagues here. That's assuming my iTunes decides to open up anytime soon. I think I have them under my test music, don't I? No, I don't. Good gosh. Oh, yes, I do. No, I don't. Team Rocky. There we go. Here we go. All right, that's better. Let's keep going. So we got sine X, cose squared X plus cose squared X. That's this guy right here. All over sine X, cose squared X. All over that thing, that thing. Minus one, a little too loud. What do I have on the top here? GCF, all right. Doing math now. No, okay, fine. Victoria, can I cancel? I can. All over that mess. So here's what I end up with on the top. Sine X plus one over sine X. And remember we said our goal, our thought was this great big thing will work out to just plain old one. So what I'm hoping is the bottom works out to sine X plus one over sine X, because then I'd have something over itself. Let's rewrite the bottom finally, which was cose can't plus cose can't sign. Okay, now it's annoying. Cose can't is what over what? One over sine? Did you get a sign on the bottom? And will that give me a sign on the bottom? I think we're on the right track. In fact, I'm willing to bet in two lines this is going to turn into sine X plus one over sine X. And we're done. Feeling a pride. Let's see, let's see, let's see. You guys asked for this one. I've never done this one before. So cose can't is one over sine X plus cose can't times sine X is going to be one over sine X times sine X over one. Oh, what is one over sine times sine over one? Okay. So much fun. Can you see where we're going now? Oh, don't forget the minus one. There's a line dropping down over here. Oh, and equals zero. Miguel, common denominator. I would write this as one fraction. What would my common denominator be? What would I multiply this by? And I'd get one plus sine over sine. On the top, I have one plus sine over sine. We are basically done. Holy smokes. Let's see, we would go time sine X, sine X. And we get this. Sine X plus one all over sine X all over one plus sine X all over sine X minus one. What's anything divided by itself? So I would go like this for the marker. And then I would go one minus one with zero question really easily done. Well, no, that really isn't a proper abbreviation. I don't know. Did I answer that already? Did I answer that already? Oh, I'm telling you, my little math nerd heart just darn right. That was excellent. That was the same rush I used to get playing frozen tag when I was like six years old with my friends. It was like, oh, I unfroze somebody? Yay! Now I'm their hero. Hey, I just got a trig identity. Now I'm their hero. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen. Only thirteen. In answer to your question, Dylan, I'll answer it again. I've seen nine and ten line ones, but not this ugly. They were more lots of algebra, routine, substitution. No, another substitution. Oh, they made a substitution inside of a substitution inside of a substitution, but they were easy to spot. But this was tough. Anyhow, the rest of the class is yours. I was hoping to be finished by 25 after. I went a bit longer, but you can work on the quiz. You can practice the identities. Hopefully, you're starting to swim some laps.