 So quiz 4.5 number one will solve this by going clear clear Cosine of 2x Okay negative 2 over x is one of you need to borrow a graphing calculator. Oh Got it you need one. Oh got one. Oh, there we go Zero to two pi zero to two pi You know what I think this window will work if not. I'll just make it bigger. Well, that looks pretty good How many answers to first one second function calculate intersection first curve second curve guess first one is it four point one seven seven and The second one is second function calculate intersection first curve second curve gas 5.304 is that right folks? You know what I forgot to do You got to make sure I was in radians on a wait a minute the cosine graph looked okay, so I must be in radians solve the following algebraically and my trigger phrase exact solutions means probably special triangles and there's a two period change I Refuse to deal with a period change. Uh-huh. I'm gonna say this though. The period is two pi over two Or just plain old pi But I'm gonna let a equal to x temporarily. I'm gonna solve sign of a Equals negative one over root two and when I use my cast rule Sign is negative. Why sign is negative here and here. Oh, it's the one one root two triangle One one root two triangle where this angle here has a sign of one over root two had a cosine of one over root two And this angle here is counted pi by four So that means that this angle is pi by four and that means that this angle is pi by four Which means my whole angle is let's see my first a value is five pi by four five pi by four and My second a value is seven pi by four, but of course I'm not done They wanted me to actually find x to get x I think I would divide the a by two to get the x by itself I think as it turns out my first x value is five pi by eight and My second x value is seven pi by eight and My period is pi or Eight pi by eight so I can find extra answers Let's see x three is going to be add eight pi by eight to three at the period to three to five pi by eight Sorry, 13 pi by eight, which is less than two pi Is there an x4 probably because usually there's four answers when there's a two there seven pi by 15 pi by eight, which is also less than two pi Is there a fifth answer? No when I add the period to that I'm bigger than two pi I'm bigger than 16 pi by eight You know what to keep it simple today on the quiz now on the test if you didn't get all four I'd give you lots of part marks, but this is a quiz four answers four marks. How about one mark per answer? Uno, dos, tres, cuatro Now on your test if you didn't get this far you'd still get marks for some of this I have a different marking scheme, but it's a quiz I'll mark harder so you find the test easier easier Give the general solution. Well, it's going to be the first angle five pi by eight As well as the second angle seven pi by eight and we're going to take both of those and we're going to add multiples of the period where n is an integer and you could have written eight pi by eight there as well But I just didn't feel like it number three I Started to panic because I saw two trig functions and I went what does he want us to plug in an identity? Oh, wait a minute. Wait a minute. Wait a minute. Mr. Duke. What's the first thing that I always? Always always always always always always always always always check for There is a greatest common factor When I saw that I went boy am I sure glad that I learned that the first thing that I always always Always check for is a GCM. Otherwise, I could have been really floundering for like 10 or 15 minutes before I spotted that I might not have finished the test hint hint Yeah, anyways factor out a secant and I get root three Because cosy can't Plus two equals zero So I said, okay I guess this is going to break into two factors the first one says secant x equals zero and the second one says Cosy can't x equals negative two over root three. I don't deal with secant Secant goes with Coside now this is kind of weird This is saying cos x equals one over zero because I got to flip both sides So if I flip a zero it become a one over zero. Oh Was that from next door that wasn't the phone I thought I heard a phone for a second there I was like, I've been on a roll. I bet we had two on Friday. Great What's the answer here Tyler No solution undefined or even just crossed it out or if you crossed out this because some of you may remember the secant graph The lowest the secant graph gets is one or negative one remember it had that weird range It never touches zero you could also have just crossed out there as well I would have taken that but yeah, no solution little sneaky. Mr. Do it. What about here? I don't deal with cosy can't I'm gonna solve sign x equals negative root three over two C a s t y sign is negative there in there Why this looks like the One two root three triangle where this angle up there has a sign of root three over two Well, I as a reference angle that's pi by three pi by three So we're gonna get this x one equals four pi by three. Is that right and x two equals five pi by three. Is that right? Okay, I would probably say One mark if you got this One mark for that answer one mark for that answer and one mark for clueing in it was no solution over here Something like that. I don't know that's kind of cool. I'm no solution reciprocal. Whoo Turn the page a lovely quadratic trig now some of you are still having trouble factoring some of these and that's fine I've given you my using the force guessing method But now I'm gonna take it one a little step further You see Kyle when I look at this it could be a two and two or it could be a four and one But here's what I notice. What's four times five? What's the middle term? Those are close enough. I bet you I want the four and one sticking around How'd you figure that I you just do a gazillion of these and you start to spot patterns In fact, I'm 99% sure that it's going to be Four four x mr. Do it how about four coast x mr. Do it and one coast x and I think I want to have a positive five there and a minus one there and Don't forget always drop down the equals zero that way you have a full equation Trust me. You'll like me. You'll make less sloppy mistakes. I get coasts x equals negative five and I get coasts x Equals one quarter. What can you tell me about this side? Now again on your test one of your quadratics will have solutions on both sides I won't do everyone like this, but I will tell you on your test at least one of them won't have solutions on one of the sides Well, you give us one. There's no solutions on both sides Probably not on the test. Although Dylan. It's nearly cool. All I'd be testing that is can you factor it? I'm really testing can you solve it not just factor it? Hey, do I have a triangle with a one on a four in it? Oh, that's why they said give your answer to three decimal play. Okay C a s t it is positive one quarter, which means cosine is positive there and there I'm gonna find the reference angle by going inverse cosine of one quarter inverse cosine of one quarter and I get a reference angle of one point three one eight Making sure I wrote that down, right? I did so I guess that means my first x value is 1.318 and my second x value is 2 pi minus 1.318 which is 2 pi Minus the previous answer I had on my calculator 4.965 If you got that you get four out of four Otherwise, I would probably do something like this One mark for that one mark for that one mark for crossing the puppy out And I'd probably give you one mark if I saw like the cast rule reference angle thing number five Solve tan. Okay, er. Oh never mind tan tan x tan x Numbers that multiply to five and add to six oh plus five plus one so tan plus one tan plus five so I get tan x equals negative one and Tan x equals negative five Now if this was cosine or sine I would do this But it's tangent ugly cousin as it turns out tangent does have a range of all reels This will have a solution now this one here if you did this is a decimal That's fine, but believe it or not This is actually an exact value because it's negative one over one and I do have a triangle with two ones in it The one one root two triangle, so I'm gonna do says an exact value You can then make it a decimal if you did it as a decimal because I didn't say you had to Tangent is negative there and there This is the one boy. Mr. Do it make a triangle This is the one one root two triangle where that angle there has a tangent of one over one So my reference angle is pi by four Pi by four and I get x1 equals three pi by four and x2 equals seven pi by four over here See a s t I also get tangent is negative Which is also there in there, but here I'm going to find the reference angle by going inverse tan of Five not negative five of five Inverse tan of five And I get one point three seven three four One point three seven three four, which means that angle and that angle are both one point three seven three four Means I think the first angle x3. I'll call it is pi minus Let's try that again pi minus One point three seven three four I Get one point seven six eight. Is that right and My second angle is going to be two pi minus two pi minus One point three seven three four. I guess I could have just gone second function anyways Four point nine one zero if I round off properly four point nine one zero. Yes, I think Yeah, that'll become a yep Do we have to have a zero there? No, but I did because it's three decimal places. How many answers? Four how many marks Four that works Give the general solution. Well, there's that guy and There's this guy. Now if you wrote also x2 and x4, that's fine But as it turns out by the way tangent, what's the period of tangent ugly cousin? If you add pi n where n is an integer To these two this one will generate that one and this one will generate that one But if you wrote them in that's fine as well Could you give yourself a lovely score out of 19? Please and then pass them inwards turn to page 359 it's less than I think eight. I've skipped lesson seven temporarily I'm also doing something a little bit weird normally. I've started out each day by saying hey any identities questions You want me to go over? I'm gonna be actually taking questions from homework or from big review after Good news is it's a short lesson today I'm gonna be showing you a couple of new rules And then you're gonna have the rest of the class to just whittle away at this stuff And you want to have your calculators handy We're not gonna be using of a whole bunch just at the very beginning here And you want to make sure you're in radians and here's what we're gonna ask This statement right here Don't highlight it because it's false. I'm just highlighting it so you can see what we're talking about We're asking Jesse is this true what we're really asking is can you pull a two from inside the sign? outside to the front is The sign of two theta the same as just going sign theta and double your answer Says determine whether or not the statement can be verified using theta equals pi by six So here's what we're gonna try and do let's get our calculator out and making sure we're in radians We're gonna go The sign of two times Pi by six that's the two theta. That's the left-hand side Enter and that's what I get as a decimal and I'm gonna compare that with going two times the sign of Pi by six And the question I have for you is are they the same? Oh Are they the same? No Turns out we cannot we cannot pull something from inside the trig function to the front Well, then here's the next question What is the sign of two theta the same as now? There is a fairly easy proof But I can only do the fairly easy proof After next day's lesson because I'm doing lesson eight ahead of lesson seven because I think it works better The only way it doesn't work better is I use lesson seven to prove lesson eight So we'll just have to hold off on the proof I'm just gonna give it to you as it turns out whenever you see the sign of two theta You couldn't replace that with two sine theta cos theta which is oh hey get out your pink identity sheet and your hints for proving identities Unless you got them all memorized if you got them all memorized then you don't need to get them out And I think about one third of the way down on the right hand side justine you do see sine two theta equals two sine theta cos theta Okay, so if I'm doing an identity one of those t-table thingies and they give me a sine two theta You know what I replace it with? Two sine theta cos theta. Oh, and you know what the cosine of two theta is the cosine of two theta is Coast squared minus sine squared it is But do you notice that next to coast two theta? It doesn't just have one answer Ian on your formula sheet How many answers does it have? Go find coast two theta. How many possibilities are there for coast two theta actually three? Okay, this is the original but because Coast squared equals one minus sine squared If you plug that in right there, you'll get I think you have one identity with a two sine squared in it That is one minus two sine squared. Oh and because sine squared Equals one minus coast squared if you plug that one into there with brackets because there's a negative in front You distribute the brackets you get these two so there are three choices There are three choices to replace cosine of two theta with Which one's the right answer all of them, but one of them will always be better Than the other two and so this is where I always try and do a little bit of the math in my head To see oh does some nice stuff cancel here now try the other double-angle identity for cosine Does some nice stuff trans cancel here now? I'll try the other one. Oh nice stuff cancels specifically If in my identity I have a positive one I'd probably plug in this so that the positive one minus the one would cancel or If in my identity I have a minus one I'd probably go with this one so that the minus one plus the one would cancel ideally but Even if you pick one of them and it's not the easiest one, they'll still be fine So those identities are on your formula sheet that the double-angle identities. I'm not gonna prove them right now They are actually really easy to prove. It's two lines, but you have to have the sum of difference identities to do it Let's use them Turn the page page 360 number one says use an additional identity to prove the double-angle identity That the sign of 2a equals to sine a cos a oh heck you got your formula sheet See the sum of difference identities See them Find the one that looks like this the sign of alpha plus beta See it there Victoria can you read it out to me, please sign of alpha plus beta equals and read to me the right-hand sides Believe it or not. There is remember how when you add two logs together, it's multiplying well when you add two signs together It's that sign of a plus a what is a plus a in lowest terms when I gather like terms To a this side You'll get this Sign a cos a plus cos a sine a And when you gather like terms you'll get to sine a cos a that's the proof for what it's worth It comes out of the addition identities, which I haven't taught you yet By the way, have you just kind of sort of figured out how to use the addition identities you kind of Find an alpha plus beta or an alpha minus beta and depending whether it's sine or cos you plug in different stuff That's next lesson really we want to use this So even if you didn't get that if you don't write that down no big whoop I'm not gonna ask you to prove it. I'm gonna ask you to use an example to this is what I'm more interested in Says consider this identity. Yeah Says describe how to use a graphing calculator know what well I Would graph the left side Jesse? Carefully I would graph the right-hand side carefully and if they're the same how many graphs should I get on the screen one? Prove it. Okay What is sine 2x the same as? to sine x Cos X and we're done on that side Mr. Duke aren't we supposed to start with the uglier side? Remember I said there's a science and an art to this I noticed that on the right-hand side. I only had to do one thing and quit So I've done the one thing and quit now on the left side. I have Tangent sine and cosine so I may Rewrite everything in terms of sine and coast but that denominator that one plus tan squared is twigging me Is that something from the top of your formula sheet? Oh? And that gives me instead of a binomial yucky fraction that gives me a single term on the bottom You know what I'm gonna replace one plus tan squared with secant squared. So this is gonna become to tan x over Secant squared x that's much nicer Now I'm going to rewrite everything in terms of sine and coasts and you've noticed I prefer to work downwards when I can But I haven't got much room. So I'm gonna do something. I hate doing so I'm gonna go sideways. I don't know Because tangent is What is tangent? Sign of a coast and there's a two on the top because there's a two in the front to sine x over coast x all over What's secant squared in terms of sine and coasts? One over don't forget the one over one over coast squared Just got my nerdy adrenaline rush because I did a little bit of math in my head How do I divide by a fraction? I'll get two sine x over coast x times Coast squared x over one I Would say to myself self how many coasts on top to how many coasts on the bottom one? How many left and where why I think I'll have one single solitary coasts on top and low and behold Those are ready a to the sign would drop down the coast would drop down I would smile and I would spike the football after the touchdown So remember that great big identities package that I gave you that has I think 21 Identities in it or 24 or something like that and I told you that you couldn't do any of the ones that had a 2x in them Now you can oh Those are all fair game. Oh and The other question they'll sometimes ask you to do is to go backwards And I'll show you what I mean example three here now usually these would be multiple choice questions on a test Three a says to sine 4x coast 4x, but I'm gonna read that in my mind Just a little bit differently. I'm gonna read that as to sign something coast something On my formula sheet do I have to sign something coast something? Why yes? I do it's the right-hand side of the sine 2x except I'm gonna then write this just a little bit differently Right above this. I'm gonna say this looks like to sign a Coast a except instead of an a or when what's sitting in my original question? Instead of an a a 4x What is to sign a coast a according to my formula sheet going backwards now why that's the sign of? 2a Radiant, but what was a actually in my question here sign of 8x? to 4x's So this is the giving you the right-hand side of the identity and saying go backwards For example, I look at B And Dylan when I look at B. I see coast squared something minus sign squared something I'll deal with the something's later But I would say to myself self on my sheet somewhere. Do I have coast squared something minus sign squared something? Yeah Coasts of two something What's the something? Okay, we're gonna try this one without the little substitution trick We're gonna see we can go there one step if I heard you correctly you said this was the coasts of two Something coasts of two times a half a Okay, what is two times a half times a just plain old a C is really the tricky one and unfortunately these are the ones they love You see when I look at question c I see sign something coast something What I don't see is two Sign something coast something But justine on my sheet. I do have a sign something coast something. Do I not? Except what's in front of that sign something coast something the number Two you know what? Can you read to me that identity so sign of two Yeah, how could I make this side look a little bit more like this what oprah tell could I do with this too? Yeah It's an uglier version of the same identity. I think this is going to be A one half in front a sign open bracket A two And then justine I love the way you said it you said theta or something because yeah, I'm not going to put a theta here I'm going to put the something which in this case is the Times five over two x Is that a fraction? Then you know what I'm going to make this to a fraction because I think what I can really see alex is it's 10 over two see it What is 10 over two in lowest terms? This whole thing simplifies to one half So Five x and that's a pretty tricky one to get you can also solve equations using double angle identities Turn the page. Oh you can do it with a graphing calculator, but let's look at b What's ugly about b? to trig functions Now normally I would say oh decimal answers graphing calculator, but ah It says as exact values, which means apparently I can do this one by hand And not only is it two trig functions. Can you see the double angle? The two x Let's get rid of that What other trig function do we have? In this question sign Let's get rid of the cos two x using the identity that only has signs in it In other words, I don't want to bring in cos squared minus sine squared And I don't want to bring in the two cos squared minus one I think I want to bring in One minus two sine squared. I want to rewrite this as three sine squared Plus bracket one. Is it one minus two sine squared? Yep minus two equals zero Is there anything in front of the brackets at all here? By the way, whenever you substitute these in hint hint wink wink judge, judge though Always substitute with brackets to be very very careful because sometimes you'll have a negative right there Which when you distribute it will make it this a minus and this a plus And that can be important. Anyways, um, I think I get this Three sine squared plus negative two sine squared. What is three sine squared plus negative two sine squared? Or what's three sine squared take away two sine squared? One sine squared or just plain old sine squared x And then I have positive one take away two minus one Equals zero sine squared x equals one How would I get rid of a squared? By the way, this is a quadratic. I could have also made it equal to zero this factors But we said, you know, remember we just have a plain old quadratic with no middle term We can just get the squared by itself and then square root both sides except when I square root both sides What had I better remember? Plus or minus and what's the square root of one? Here's what I get sine x equals one sine x equals negative one alarm bell, right Ones and negative ones means this is not going to be cast rule. This is not going to be Special triangles. This is going to be either unit circle or the graph. What do you want me to use the unit circle? Or do you want me to just sketch the graph? Oh, let's sketch the graph then here's the sine graph that we know and love And this is saying when is it one high? Well, it's one high right there Oh, it's also saying when is it negative one high? It's negative one high right there Can you see what my two exact value angles are going to be? What's that there? Pi by two And what's that there three pi by two? So That's how you could use a double angle Substitution to solve a trig equation I won't put one like that on your written maybe multiple choice i'll be honest Way more like this on your multiple choice Oh c says describe how to use graphs to solve that graph that graph that find where they cross Solve that forget it. We've done one. So we're going to go like this. We're going to go like this And in fact number one wants you to prove the double angle identities. Nah two And i'll give you one three eight one trig equation seven One three a five seven and i'm almost done So i have just given you This bad boy here, right? And you may notice if you go to the very very end there are lots of questions 136 multiple choice And then a bunch of written questions as well And I did attach the answers to this yes Okay, I haven't got a hand written answer key. I'm working on it. I'm now in number 74 and counting This is a review of all of trig. This is not going to be me assigning every one of these questions I'm going to tell you which ones are for trig part two And I'll get you to circle them I don't want you to cross out the other ones though because I think the other ones would be terrific provincial exam review In other words, I would put this somewhere and later on once I I'll mark You don't give you credit for doing the trig review for the questions that I've assigned But this is going to be terrific trig review for you when we get towards the end of the year as well so Go to number one for example Number one is talking about the point m comma n point of intersection. You know what that's x y and r Which is not going to be directly on this next test So I'm not going to assign that but that's a great review question when you're studying Anyways, I'm going to give you some right now that you can try. So what's your homework today textbook You can work on the big identities review But I want to let you start whittling away at this because if I give this to you all over four days I've spent so far to get to number 74 now. I'm doing every question But to get to number 74 so far has taken me about three hours What means it's a good review Yeah, two is good Three is good Four is good Five will be good, but not yet. I haven't done that lesson. So we'll leave it blank for now We'll come back and circle it later Six is good. Hey, mr. Dewick, we did graph's last test How come you're giving us graphing questions because we're going to be doing applications of graphs this test Which means this is great review for that um 10 glance at 10 and tell me quickly how will you solve that graphing calculator right Oh, and hopefully to do number 10 you would count your solutions And if there was two you'll cross off a and b if there's four you'll cross off c and d and then hopefully you'd even be smarter You only need to find the first one Don't you Because if point four nine is a solution It's either a or c and you'd be able to figure out whether there's four or two if it's not point four nine It's either b or d and you'd get one of those answers Start thinking cleverly 13 14 radians to degrees as a decimal will be on this test because remember your last test was Completely non-calculator this test. I'll ask you a couple of calculator questions Number 17 looks kind of familiar I didn't assign it but number 17 looks kind of familiar We're not on your last. Oh Something like that. Yeah, that's where I got it from 18 What the no no look at the answers. What will you do? Okay, it's graphing calculator. Ah, but with the period change. See it Oh, but I could figure out the general solution really quickly. What's the period of this graph? Oh, come on too slow. What's the period of this graph? It's two pi over five Which means oh, I would do that and that right now because I'm not adding two pi Okay, you're gonna find on the multiple choice You rarely have to complete the whole question. In fact once I I mean Oh, by the way, I also wouldn't bother finding the first root because the first root clearly is 0.07 Which is a waste of my time. I would right away find the second one and in fact It's either 5.94 or 0.56. I can probably figure out just by looking at my graph Whether it's far to the right 5.94 or close to the left 5.5 now I don't even need to solve this one really you'll start thinking like that. I hope 19 but later Ooh double angle identity 21 is good 22 is good 25 is good 26 is good 30 is good 34 is a quadratic and quadratics will be on this test 36 39 And I'll give you a bunch more a bit later. But if you're looking for a head start, that's what you can start working on as well Okay, can you scroll all the way down to the written section? Which is uh, well find it It's after 100 number 135 Right here two Three the written you'll have to do on a set of piece of paper. Sorry, but I was trying to concert I mean this was big enough already. I had to conserve some paper 4 lovely identity 5 quadratic trig 6 lovely identity 7 quadratic trig And again, are all these due tomorrow? No, but by the time we get to the test, this is what you'll have worked on Most it's going to be almost all of the written 8 is good 9 is good eventually 10 but not yet because I haven't done applications 11 lovely identity 12 solve algebraically quadratic trig equation and a general solution Is number 13 an identity? Yeah, that's fair game Many more are there. Oh boy. Did you hear tyler? Was it tyler? miguel we used to all to be You'll be able to do this one. It's a quadratic trig this one this one This one this one this one 20 is a word problem. I haven't done word problems with you yet 21 is a word problem. I am going to do one thing. I hate number 22. It's bad physics And I got it I got it banned from the provincial exam In the last curriculum, and I've just seen it's on the new curriculum coming up So I got to get it banned again. It's You're assuming that energy is conserved and there's no friction But you haven't said that in the question and if kids take physics they'll assume it's friction And then it's not a trig function It's actually a decaying function and they'll use an exponential and they'll get zero because they did the physics, right? Sorry, is that a lot? Yeah Oh, well, no because the test isn't for two weeks and you can start whittling away at it and I'll be giving you more I know along the way, but really over the next few days You're working on today's homework, but then it's going to be the big trig Identities review. That's do the day of the test and here is your great big trig review Okay, now i'm officially shutting up. You got 20 minutes