 Trig quiz four. I haven't done the joke for a while, so we'll do it here and then sir key. Number one says solve that using your graphing calculator. Now on your test there's going to be one like this. It's going to be multiple choice. I won't say using your graphing calculator. How will you be able to instantly recognize that you have to use your graphing calculator? What do you look for? Decimal answers. I'm going to keep repeating that and you're going to get tired of it. I hope that means you'll do it, but I told you every year one kid forgets and it drives me crazy. So I'll boot up my graphing calculator. Hello graphing calculator. And I will make certain that I am in radians, which I might not be. I am. And I will clear whatever equations I have here. Y1 is going to be the sine of 2x. Y2 is going to be negative 2 divided by x. Did they give me a domain? Now sometimes they won't. If they don't, I usually go from negative 2pi to positive 2pi, but I keep an eye out on the graph to see if it looks like there's other solutions. But here they said to go between zero and 2pi. I think 6.28 is 2pi, I think. So you know what? Amplitude 1. I'm from negative 3 to 3. That's a pretty good window, Amy. I'm not going to waste time mucking around with it. They're sine of 2x. Am I looking for four answers? I think. So if this was multiple choice, hopefully you'd cross out anything that didn't have four answers. Second function, calculate intersection. First curve, enter. Second curve, enter. Guess. Let's find the left-hand one first. Right there, ish. Enter. Don't forget to hit enter. Your guess is only an approximation. It's when you hit enter that it uses that approximation to find a better solution. Three decimal places, 2.162. There's one answer. Second function, calculate intersection. First curve, second curve. Guess. 2.731. Second function, calculate intersection. First curve, second curve. Now you can see this is a bit of a slow process. In fact, when the provincial, the provincial used to be only two hours long. They've given you an extra hour for the same exam. When it was two hours long, Dylan, I would actually teach my kids when they hit enter, go move on to the next question. Start working on it and then come back. Now the TI-84s are a bit faster, but the older calculators in particular, they would really think for a while. 4.922 and dope. Second function, calculate intersection. First curve, second curve. About there. Guess. 6.117. Is that right? How would I mark this? Well, one mark, four solutions. How about a half mark off for each one you got wrong? So if you got two wrong, sorry. If you rounded off wrong because you didn't see the three decimal places, take a half off. They will be fussy on the provincial, so I got to get you used to reading instructions. For the next questions, you must show work, solve the following algebraically. Do you see it? Period change. You have to spot it. By the way, again, I'll say this again. Every year I see some kids that go, I'm going to divide by two and that's going to make that a four. No, you can't divide out of a trig function like that. Can't be done. Not allowed. That's bad. No, no. Instead, I'm going to first of all make a big note. The period is 2 pi over b. The period is pi. I'm going to solve, well, I'm going to let a equal 2x. I'm going to solve sin, yes, a equals negative root 3 over 2. She likes to use the x because she doesn't like the a. But then I've said it confuses her because now you have x's that aren't really x's. They're just x's that are different from the other x's which aren't x's because these are x's anyways. So far, so good. These, I've solved the gazillion of cast rule. It's sin, it's negative. Sin is negative there and there. It's the 1, 2 root 3 triangle. This is so, oh, don't drop like that. Mr. Dewick dropped elongated so that it's exaggerated. 1, 2 root 3 triangle, which has a sin. You know what? This one has a sin of root 3 over 2. My reference angle is pi by 3. So my first a value is I think 4 pi by 3 and my second a value is 5 pi by 3, but that has very little to do with my actual solutions because my solutions have what letter Ari? My solutions have what letter Ari? What's your favorite letter? x, absolutely. I want to find x1. Now I would find it by looking at this little statement. It's well worth writing that because when you glance at this, hopefully all of you are good enough for your graded equation solving, Dylan, that you look at this and you say, well how would I get the x by itself? Divide by 2. Divide the a by 2. Divide the a by 2. I think one answer is 4 pi by 6 except I wouldn't write that. How will I write that if this is multiple choice? 2 pi by 3. I'll write it like that. And the second answer is going to be this value divided by 2. 5 pi by 6. Are there more? Could be. What I'm going to do is I'm going to add the period to each of these and I'll go systematically. I'll add it to this, then I'll add it to this, and I'll keep doing that until I'm past my domain of 2 pi. 2 pi by 3 plus the period. 2 pi by 3 plus pi. Really in my head, I'm not going to add pi. What am I really going to add in my head? 3 pi by 3. Let's, you know, hopefully we're good enough at the basic common denominators. I think I get 5 pi by 3, which is less than 2 pi barely, but it is. Is there an x4? Let's see. 5 pi by 6 plus 6 pi by 6 is 11 pi by 6. That is also still less than 2 pi, but 12 pi by 6 is 2 pi. I don't think I can add anymore. I think if I do, I really quickly end up going beyond my domain. Now, how would I mark this? If you got all four of these and you don't have to have them in lowest terms, but I did just to remind you that we will, so if you had instead 4 pi by 6 and 10 pi by 6, I'll give you four marks. If you got all four of them, you get full marks. Otherwise, I would probably do something like a half mark for each of these. That's 2. I would probably give you a half mark if I saw the period change, a half mark if I saw that you picked the right quadrants, a half mark if I saw the 1, 2, root 3 triangle, and a half mark for these guys probably. But if you got all four answers, you're fine. By the way, what if there was a 3 there? How many answers am I probably looking for? 6? What about a 4? 8? The tricky ones are where it's a fraction. What if it's a 5 over 2? That's a 2.5. That may be four answers. It may be three answers. It really depends on where your last couple of, your last root is. It might end up being just past or just inside the domain. For those legal, yeah, we did one with a fractional domain as our final example. Sure, wouldn't freak out. Give the general solution. Take my first 2 and add the period plus multiples of pi where n is an integer. My physics kids have probably learned this, but for those that aren't in physics 12 with me, you never leave stuff blank if you can help it. One year, they gave this equation on the provincial. It was tangent, but it was this idea. It was a period change, and then they asked for the general solution. And what bothered me was a bunch of kids left both pages blank where we had decided as markers, if kids couldn't get the first part, if they couldn't get these roots, to get the full two marks down here, and it was worth two marks for the general solution that year, all you had to do was say, assume the solutions are pi by 3 and 5 pi by 3, and all you had to do was show that you could add the period to it, and you got 2 out of 2 on the second section. In other words, if you can't get this, but you know the period and you can say I would be adding that. Don't leave it blank, right? Although I think most of you are doing good. Quadratic trig. Oh, exact solutions suggest to me, I'll be using the 1, 2, root 3, the triangles, or the unit circle. It's a quadratic, I got to make it equal to 0. So the first thing that I'm going to do is I'm going to solve 2 sine squared x plus 7 sine x plus 3 equals 0. Now, what this really is the same as is this 2a squared plus 7a plus 3 equals 0. Mathematically, those are equivalent. Now, you don't have to write that if you don't want to. I have some kids that can keep track of the sine and all the factoring. Fine. Good. I'll write this though. This is going to factor into, well, definitely a 2a, a 2 sine x, and a sine x. I want to end up with a 7, and I need numbers that multiply to 3. I think I have to go plus 3 there and plus 1 there, because that's going to give me a 1 plus 6 is 7. Middle plus outside gives me the middle of my original term. And I get this. Sine x equals negative one-half, sine x equals negative 3. Do I have a triangle with a 1 and a 2 in it? Yep. Sine is negative there and there. This is going to be the 1, 2, root 3 triangle, where I think that angle there is the sine of one-half opposite over hypotenuse. What is that angle there? Pi by 6. So if I hear you correctly, this is pi by 6. This is pi by 6. This is pi by 6. And so I think my first x value is going to be e in 7 pi by 6. Is that right? And my second one, e in 11 pi by 6. Is that right? Good. And I'm solving these two independently, so I usually draw a line down the middle. By the way, usually I'll give you more vertical space to work on this, but I was trying hard to fit this quiz onto one page. This one is interesting. Now a lot of kids when they glance at this, especially if they're using a bad substitution, but that's okay. A lot of kids, if they glance at this, they go, oh, let's see. I do have a triangle with a root 3 on it. Maybe there's some way that I can turn that into a 3. And there is no triangle that has a 3 and a 1. As a matter of fact, remember the sine graph? What's the lowest the sine graph gets? Negative 1. And this is saying, when is sine equal to negative 3? The answer area is what? The answer to this one is what, my child? And you have to write that. If you just stopped here and you didn't write no solution, if you didn't have the integrity or the guts to say, I know those two graphs don't cross, I won't give you full marks. How would I mark this? Oh, probably something like this. I would give you one mark for that, one mark for that. I would probably give you one mark for that. And then I would give you one mark if I saw the factors somewhere along the way. Before you turn the page, if you're shaky at factoring, you can cheat. I've given most of you the quadratic solver on your graphing calculator. You could write that, and as long as your quadratic solver can give you negative 0.5 and negative 3, and you can interpret that as fractions and go straight to this line, you'll get full marks. Even though I gave you one mark here, technically I don't have to make you factor. All you have to be able to do is find the roots of that quadratic. Oh, and then recognize that they're trig functions and then find the roots of the trig functions from the roots of the quadratic. I don't care how you do it. Turn the page. Now your calculators are allowed on this test. That means my trig questions aren't as limited. I have to admit it gets kind of boring only using reference angles of pi by 3, pi by 6. You noticed on your test and in your quiz's last unit, I was run out of angles. They were getting kind of repetitive. Here it says, give your answer to three decimal places. That probably means no special triangles. I added the hint. I won't put that hint on a test. I'll expect your clue and if I say give your answer to two decimal places or three decimal places, you'll go, oh, probably that means not an exact value. I got to factor this puppy. Okay. Now remember I'm really factoring 9a squared minus 6a minus 8 equals 0. Let's see. This one's a bit tougher because I got that 9. It could be a 9 and a 1 or a 3 and a 3. I'm going to start out guessing 3 and 3 and if I guess wrong, I'll just erase it. But I think maybe it's 3 sin x and 3 sin x. This is a negative 8. That's also tougher because that means I could have a 4 and a 2, a 2 and a 4, a 1 and an 8, an 8 and a 1 and they could be negative and positive. But remember, I want to be able to go that times that plus that times that gives me the negative 6. It's amazing how my brain works but somehow just by writing that, my brain saw, hey, put a minus 4 there and a plus 2 there and you get a 6. Take away 12 is negative 6. And if you think I'm lying there, I'm not by the way. As I said that, I'm looking for a, as I drew those two lines, the answer popped into my head. And I don't know how my brain does that. I think it's because I've done so many of them but I can't do that until I actually ask the question. My brain I guess needs a little jump start. Oh, you could have, I guess the minus 2 there and the minus 4 there because they're both 3x's. Anyways, here's what I'm getting is my roots. I'm getting sine x equals negative 2 thirds and I'm getting sine x equals positive cos. Why don't I do sine suddenly? Good point. See what happens when you're careless with your substitutions? Oh, sorry. Cos, thank you. Cos, thank you. Cos, thank you. Cos, thank you. Equals 4 over 3. By the way, did any of you do this right away? Is that bigger than 1? What's the highest cosine guess? Now, this one has no solution and I need to pause. There was one year they did, I lifted this question from a provincial actually except instead of sine instead of cos. You know what they had? Tangent and a whole bunch of kids crossed this out. The problem is how high does tangent get? Infinity. Just don't do that. I marked this question on the provincial. The year it was tangent and I would bet you one in five kids proudly crossed this and actually it was even meaner. Instead of having tan x equals 4 over 3, it was tan x equals 8. They were really taunting kids, tempting kids to cross that out because of course, oh, sine cosine don't get that high. Cross it out. Remember tangency as the cousin. Most of our rules don't work. This one, C A S T, cosine is negative there and there. I don't have a triangle with a 2 and a 3. I got a triangle with a 2 and a root 3, but not with a 2 and a 3. I'm going to find the reference angle by going inverse cosine of two thirds, not negative. I use the negative to figure out where we are. Clear, clear, clear. Make sure I'm in radians. I am inverse cosine of two thirds. I get, oh, three decimal places, 0.841. Perfect. 0.841. So x1 is going to be pi minus 0.841, and I get 2.301, I guess technically, yes, if I round off properly, and I'm going to get x2, which is going to be pi plus 0.801. Conveniently, I can just go second function, enter, and hit the plus sign. 3.983. By the way, what's pi as a decimal? So this is a bit less than pi and a bit bigger than pi. Those numbers look right to me. I said to you, the disadvantage with radians is it's harder to guesstimate where you are on the circle, but if you know that pi is 3.14 and 2 pi is 6.28, you can kind of rough guess where you're supposed to be. It's doable. If you got this and this and crossed that out and said no solution, you get full marks 4 out of 4. Otherwise, how did I give out part marks? I can't remember. That's physics 12. Quiz 4, version 2 answers. Come on. It's not that hard computer. You could do it. There you go. I guess one mark for the negative two-thirds, one mark for crossing out the negative four-thirds, and then one mark for the 2.301 and one mark for the 3.983. The last one. Again, for this one, I'm looking at a squared plus 2a minus 3 equals 0. I am also screaming in my brain, ugly cousin, ugly cousin, ugly cousin, like be aware. Hey, this is tangent. It misbehaves a little bit. It misbehaves. It's yucky. Specifically, for part b, when they want the general solution, what's the period of tangent? Yeah, I'll be adding multiples of pi. Anyways, let's factor this. It's going to end up being, oh, this is an easier one to factor. Nothing in front of the square, thank God. Numbers that multiply to negative 3 and add to positive 2, subtract to 3. Oh, a positive 3 and negative 1. Tan a plus 3, tan a minus 1. What are my roots? Tan a equals negative 3. Oh, I did do one like this, equals 1, except it's not a, it's x, Mr. Dewick. Why are you using the a's? Because I don't know. Somehow, someone else's x's have just rubbed off on me this morning, not mentioning any names. I just wouldn't want to put this on the internet. Hey, did any of you cross this out? This one, I can't. If it was sine or cosine, yeah. But tangent, ugly cousin, has a range of all reels. This is going to have a solution. Draw the line down the middle. This is going to be, let's, come on, come on. There we go. C, a, s, t. Tangent is negative here and here. The reference angle, now this is negative 3 over 1. I have a triangle with a root 3 into 1, but not a 3, so I'm going to have to use my calculator. The reference angle is going to be the inverse tangent of 3, inverse tan of 3. 1 point, what did it say? How many decimal places? 3. Okay, fair enough. 1.249, 1.249. So, let's see. I'm going to have my first x value, oh, by the way, that means that this angle is 1.249 and this angle is 1.249. So, my first x value, how will I calculate that? Pi minus, minus 1.249. 1.893, and my second x value is going to be 2 pi minus, 2 pi minus, 5.034. Okay, there's that factor. Now, let's do this factor here. Now, I don't know if you caught it. If you did this one with your calculator, that's fine, but this is technically, I can do this as an exact value because I do have a triangle with a 1 and another 1 in it. It is 1 over 1. So, I'm going to do this as an exact value and at the end, I'll just make the answers decimals and you can see if you got them right. But I caught that tangent, ugly cousin. I know once it actually handles just fine. As a matter of fact, it's telling me that tangent is positive, which of course is here and here. And a 1 and a 1, why this is the 1, 1 root 2 triangle where this angle here is pi by 4, where this angle and this angle are both pi by 4, where you end up with x1, hang on, x1, x2, I'm going to call this x3 being pi by 4 and you end up with x4 being 5 pi by 4 or as a decimal, pi by 4 is 0.785 and 5 pi by 4 would be 5 of those, 3.927. How did I give out part marks for that one? Oh, it looks like for 4 marks, I gave out 1 mark for each root. And the last question, the general solution. What's the period? Pi, x1 equals 1.893, x3 equals pi by 4 plus pi n, where n is here. Mr. Dewick, why didn't you write x2 or x4? Now you could have put x2 right there and you could have put x4 right there and you'll get full marks. But as it turns out, if you add 3.14159 to this, guess what you get? That one. And if you add pi to this, guess what you get? That one. So you can write them but they're redundant. That will always happen with tangent because when you draw this line like that and like that, which you always will, it's always all positive or all negative, you're going to find you're cutting the circle in half, you're cutting it into sections of pi for what it's worth. So you can see, I wrote the first two, I started to write this and I said, no, I don't need those ones, they're redundant, a half mark for each. Give yourself a score, please, out of whatever this quiz is, out of 19. Holy smokes, 19, please. And once you have done so, pass them inwards. So we're going to be doing some of the heats, the back of the hints for proving identity sheets. We're going to be practicing some of these. In terms of the hints, we only got to hint number four, I think, factoring. And I want to show you some where we use the other hints. And I also today want to try and get rid of the most common problem. The most common problem, the most common problem, if I can beat this out of you, you're in good shape, is kids cancel when they're not allowed to cancel. And as soon as you do that, you've changed the identity, it can't be true and you can't make it true, sadly. So the first one we're going to try doing from the identity sheet is number five. Okay, you can copy this out somewhere, I'm just going to copy and paste it. This one here, make it a bit larger. You have it in front of you. Instead of using theta, they're using, hey, does anybody know what Greek letter that is? What lowercase letter sort of looks like that? Yeah, alpha, that's the Greek letter alpha. Hey, you know what that is? Beta, alpha beta is literally where we get the alphabet, the word alphabet from it actually does mean ABC is from Greek. Hey, you learned something new today. Anyways, talking about these, when kids do a bunch of these Aries, sometimes they do get sloppy and they stop writing the symbol. If you do that once or twice, they won't take marks off. But if you never write the theta, you just write sine or cos all the time, you will lose marks. And the reason I say that is I've marked these twice on the provincial exams at SFU in the summer time, somewhere there's a bunch of teachers that are telling their kids not to bother writing the fetus to save time and they haven't read the exam rules because we have to, we'll get like 40 papers in a row, all wonderfully done and they're all getting four and a half out of five because they haven't bothered writing the fetus. Specifically, you have to make sure you have it on the first line and last one. Ready? Here's our wonderful tea table. If you really want to, you can do that as well. I usually don't bother, but here it is. Which side's uglier? Well, right now, I'm kind of looking at the left side as being uglier because it's a fraction. But are there any squares on the left side? Nope. Is everything already in terms of sine and cos on the left side? Are there any tangents or cotangents or secants or cos secants? We can't do anything with the left side, even though it's uglier. That's why these are hints and not rules for proving identities. This is kind of my checklist. This will always get me there. But how many trig functions do I have in this equation, grand total? Four. I'll rewrite everything in terms of sine and cos. Hopefully, you have this one memorized by now. What is tangent? Minus. What's secant? This one we do have memorized. Secant goes with... And as soon as I finish a line, I always flip my eyes back and forth, back and forth. Look at the left. Look on the right. Here's what I noticed. Maybe some of you see it already. How many fractions do I have on the left side, grand total? One big one. How many on the right side, Justine? Two. I think probably I want to try combining them as one. By the way, what's on the bottom on the left-hand side? Cos. What's my common denominator on the right-hand side? That also tells me I'm sure that's my next step. That can't be a coincidence. And this one actually is pretty good because I got a common denominator already. I can simply write this as all over cosine of alpha. What do I multiply cos by to get cos? Nothing. It's happy. I'm good. And the minus one also drops down. And we're done. And no, you don't have to write QED. I just do because I'm a nerd. One, two, three, four, five, six. Next one's going to take us anywhere from seven to ten lines, depending on how we write it. Okay? Ready? We're in the deep end now. Don't worry. Piece of cake. We're going to do number 18. This bad boy here. Cosy can't beta minus one. One plus cosy can't beta. Cosy can't beta, cos beta, all over secant beta sine beta. Is that okay? All right. Step one, hint one says, start with the uglier side first. I bet you we're working on both here. I'll be stunned if we're leaving either of those as is. Now, hint number two. What was hint number two, Kyle? Which I do. Now, I am going to, but I'm going to use a little bit of intelligence first. You see, I'm pretty sure somewhere along the way we're going to be foiling this out. If I replace this with sine and cosine right now, Justine, that's going to be a fraction. That's going to be a fraction. Do you like fractions? So you know what? I'm going to suggest we foil it, then fraction it, like then replace everything in terms of sine and cos, okay? That's not a rule. That's me being a bit clever and saying, I know what I'm good at and where I'm going to make dumb mistakes. Let's foil first. Yeah. This would be one over. No, this is not one. This is one over sine minus one. Hold your thought and I'll come back to it, okay? You're making a look up. Don't write this down. I don't see any zeros anywhere. And then we multiply. Remember, it's top times top, bottom times bottom. So it's going to be boom, boom, boom. You're going to get a middle term cancelling, but you're still going to have all sorts of algebraic fractions. I'm going to suggest we do the multiplication first, then the fraction. Is that okay, Dylan? Even if it's not, trust me. I'm going to go we're going to have cos times one, which is just cos beta. We're going to have cos times cosy can't, which is cosy can't squared beta. And we're going to have negative one times positive one, which is negative one. And we're going to have negative one times cosy can't, which is minus cosy can't beta. And Dylan, maybe this is what you were seeing and I misunderstood you. Can you see I have cosy can't blah, blah, blah. Take away cosy can't. These are definitely going to cancel. In fact, what I'm left with is cosy can't squared beta minus one. And I have to ask, because I have a squared and a one, is that on the sheet that's screaming to me to check that top row? Kyle is nodding. I think you have to do some algebra. What is cosy can't squared beta minus one the same as? Cotangent squared. Now, I'm going to argue that looks much, much, much nicer than that, except since we're going to rewrite everything in terms of sine and cos, is that right Alex or not? You're looking a little bit, you can follow it though now. I think he just minus one from both sides, right? I don't have them in front of me. So I have to go from whatever you guys say. And by the way, let's convert this to sine and cosine because we're going to be doing it over here anyways. Tangent was sine over cos. So what's cotangent? Cos over sine. What's cotangent squared? Cos squared over sine squared. Okay. I think I've taken that left side as far as it'll go. But that definitely looks better. And that gives me lots to keep an eye out for. As I go along here, for example, if I suddenly end up with a sine squared showing up somewhere in the bottom, I'll start to smile. Let's rewrite this in terms of sine and cosine. Let's see. This is going to be cosy cant times cosy cant cosy cant cos, goes with sine. This is going to be one over sine times. And I've told you if I have one fraction, I'd like everything to be a fraction. So I'm going to write cos over one. And that way I can clearly see the cosine ends up on the top, not on the bottom. Again, Justine, if you're silly and foolish, and if you just write the cosine like that, you're going to be tempted to put it on the bottom or somewhere because you got it sitting in the middle. Or worse yet, if you write it down here, you're going to be tempted to put it down there. So always write them as fractions. All over. C cant is one over cos times sine over one. Well, how do I multiply fractions? This isn't too bad. I'm going to get cos beta all over sine beta all over sine beta over cos beta. Ooh, I just got my nerdy adrenaline rush. I see where we're going. Do I have one fraction over one fraction? Yeah, now this is a complex fraction, but a complex fraction using the trick that I showed you last day was when you had plus something and plus when you had binomials on the top and the bottom. Here, since it's just one fraction divided by one fraction, we can use good old math eight. How do I divide by a fraction? Multiply by the reciprocal. This is going to be cos beta over sine beta times cos beta over sine beta. And lo and behold, I get cos squared over sine squared q. Okay, I think that was only one, well, one, two, three, four, five, six, okay, seven lines. Eight if you count that as a separate line. What does step five say? Hint five. Okay, don't cancel unless you've factored. I'm going to show you one where we require that in a little bit. What's hint six? Okay, the complex fraction trick I showed you last time. We'll be practicing that a few more times still. What I need to do is show you hint seven. Kyle, read to me hint seven. As a last resort, that's important because whenever I show kids this, they go kind of, it's called the conjugate. They go kind of conjugate loopy. They see everyone. No, as a last resort, keep reading, Kyle. Let's find out. Let's look at number 14 from this sheet. Number 14. Looks like this. Does anybody know what letter those are? Nope. What lowercase letter does it look an awful lot like? It's a lowercase delta. That's where our lowercase d comes from. It's this. Very similar to a lowercase d. Kyle, can you read hint seven again? As a last resort, stop. Look at this. Do you see a binomial denominator? Do you see any squares? This is how I was able to glance at all of those identities and find one where I'm probably going to need this conjugate trick. Now let's walk through our hints. Step one, start with the ugly side. They're both pretty ugly. Step two, if I have more than two trig functions, I do. We're going to rewrite everything in terms of sine and cos. The right side is already in terms of sine and cos. This is going to be one over cosine of delta plus sine delta over cos delta. Did I just finish one line of work? Then I always look at the other side, look back here. I notice on the left side I have two fractions. On the right side I have one fraction. I'm going to write the left side as one fraction. To do that, I would need a common denominator. Lo and behold, I already have a common denominator. I'll take that. This is going to end up all being over cos delta. It's going to end up being one plus sine delta. Have I got any squares? Like this here, if this was instead of one minus sine, if it was one minus sine squared, that's on my sheet. I think that's cos squared, if I recall. Right? Or if this was sine, like I could do stuff if I had squares. This is why the conjugate is your last resort. We've tried everything else. Can I factor? Is there a GCF anywhere? No. Have I written everything in terms of sine and cosine? Yes. Are any of these fractions, complex fractions? Well, no, these are all over one. I mean, I could make them fractions and putting them all over one, but my common denominator would be one. It wouldn't change anything. Do I have a binomial denominator with no squares? All right. Here's what the conjugate is. Find that binomial denominator with no squares, this guy. We're going to multiply the top and bottom by one. So draw the little fraction bar right there. We're going to multiply the top and bottom by the conjugate. What's the conjugate? Ian, can you read the bottom to me, please? Conjugate is one plus, and same on the top. What if it had been a plus sign here? Conjugate is one minus. Or if it had been sine minus one, sine plus one, it's take the binomial and just change the sine, not sine or cos, change the plus or minus sine in the middle. Oh, make sure you put the same thing on the top. This is still a one, yes? Which means I'm not changing this. I'm just going to make it look different. Always if you're multiplying, that'd be multiplying by a top and bottom one. Now watch what happens. On the bottom, I get one times one is one. I get one times sine, because this is foil, plus sine delta. I get one times negative sine minus sine delta. Then I have negative sine times positive sine. What is negative sine times positive sine? Negative sine squared delta. On the top, I get cos delta plus cos delta sine delta. The bottom is interesting. Those two middle terms are interesting. What? Someone said it, you're right. Minus sine plus sine. That will always happen. That will always happen if you multiply by a conjugate. That's why we picked the conjugate, is we wanted to make the two middle terms cancel. It's going to give us a much nicer denominator. It's going to give us one minus sine squared. Remember I said if it was one minus sine squared, I could do something with it. On the top, I have cos delta plus cos delta sine delta. What is one minus sine squared? That is something. Sorry, say that louder. What's on the bottom over here? What did we just end up with on the bottom over here? Cos square? You know what? I think we're on the right track. See how I figured that out? Put your pencils down and look up for a second. The most common mistake is kids drop this down and they go, oh, I can do that. Don't write that down. I said to you at the beginning of this class, here is the mistake I want to beat out of you. If you ever ask yourself, can I cancel? There's a second question that immediately follows it right away. Can I cancel? Have I factored? Those two questions go together. Can I cancel by factored? If you want to cancel before you start crossing stuff out top and bottom, can I cancel by factored? Can I cancel by factored? When I write this, I say to you, can I cancel? Have I factored? If the answer is no, I cannot cancel. If you remember that, if you remember that, you'll get rid of about 90% of the most common mistakes and identities. Can I cancel if I factored? Can I cancel if I factored? And the top does factor. What's the first thing that you always, always, always, always? GCF, pick your pencils up. See, we can rewrite this as factor out a cosine and I get one plus sine delta. By the way, what do I have on the top on the left-hand side? What do I just end up with all by itself in a bracket? See, this is how I also know I'm on the right track, but if you cancel this cosine, you cannot possibly make it work because the most common mistake kids do this. Look up for a second, we'll go, oh, I'll cancel out a cosine with one and, oh, I'll cancel out that, but there's still a cos left. They try making five or six math mistakes to fix one. Oh, how many coses on top? How many on the bottom? Can I give you a little time saver trick? I always show this after this question. As you get good at this, whenever you're using the conjugate, always do the bottom first and leave the top in brackets. Don't bother multiplying it out unless you can clearly see good stuff happening because if I had left this factored as that, isn't that what we ended up writing down over here anyways? A little trick of the trade. You don't multiply stuff out unless you absolutely can see good stuff is happening. Like here, I could see, I did a bit of math in my head and I said, ooh, a middle term is going to cancel. I'll take that, but if you can't see something cancelling, for one line, leave it in brackets. It's usually a good thing. We need to practice this. So here's your homework. From the textbook. From lesson six. Page 346. Page 346. I think you can do number one, but skip A. I'm not big on just plugging in the numbers. I'm assuming you all can do that. Number two, but skip A. That also gives you lots of room to write, which I think you need. Number three, all except don't bother doing it with a graphing calculator. Skip part one. Four is good. What is number five saying the instructions? Now, they didn't need to tell me that because I noticed I have a binomial denominator with no squareds and a binomial denominator with no squareds. I figured I'd probably have to use it. Not sure. I try everything else first. Anyways, number five. Number nine. And if you finish those in class, you can start whittling away at that identity sheet. The solution to that is now online, except you still can't do anything yet with a two theta in it. Next class, I'm going to start out by doing a few more. I'm also going to look at solving equations that require an identity substitution. And then that's the last depressing pause. Then I'm going to show you some new identities the class after.