 So, from page 362, any questions from the homework? Two. Yep. Okay? You're ready to be wowed, because if you do this right, it's actually about four, I think maybe five lines. I'm trying to do this in my head, but I think so. I did this, here's my little t-table. Now, cos 2 theta, I got three options. Either going to be cos squared minus sine squared, or the one with a 2 sine squared, or the one with a 2 cos squared, and I can never remember which one. I'll come back to those. What I'm going to do is, I'm going to watch on this side and if I see a 2 sine squared appear here, probably make this the 2 sine squared one. If I see a 2 cos squared appear here, probably make this the 2 cos squared one. If I have a cos squared and a sine squared, probably make this the cos squared minus sine squared one, okay? And this denominator is twigging me. What is one plus tan squared? See that's something on my sheet, is it not? The sheet that you have in front of me, you? The top row, huh? Okay, did you catch that? Oh, you did catch that, okay. Uh-oh, hang on, I got someone coming in late here. I need to interrupt my little video lesson. So one plus tan squared is secant squared. I think Victoria, the first thing I would have done is this. Why wouldn't you just made everything in terms of sine of cos, Mr. Duk? Well, first of all, cause technically there's only two trig functions. But also, anytime I can get rid of a binomial denominator, that's usually a good thing. Now I'm gonna rewrite everything in terms of sine and cos. And I can already tell I'm gonna run out of a room, so I'm gonna draw my little line over here. This is gonna be one over one plus sine squared over cos squared all divided by one over cos squared. No big, sorry, plus should be a minus here, right? Thank you, Tyler. Those are the sloppy mistakes that can come back to haunt you cause then my rest of my identity would have been wrong. No big surprise there, so far so good. Now this is a complex fraction in that it's one fraction over, it's a four level fraction. But because I only have one fraction on the bottom, I can say how do I divide by one fraction multiplied by the reciprocal? In other words, this is why I figured I was gonna run out of room. I can rewrite this as one over one minus sine squared over cos squared times how do I divide by one fraction multiplied by the reciprocal? You could have also used my canceling fraction trick. You would have looked at each mini common denominator and you would have said, oh, my mini common denominator is cos squared and you would have multiplied top and bottom by cos squared, okay? That would also get you there. But we're using the how do I divide by fraction for the multiply, which is legal. When I multiply, nothing's gonna cancel here. It's gonna be just plain old cos squared. When I multiply, what's gonna, oh, the minus sign would drop down, yes. What's gonna cancel here? I gotta be fussy about the cos, the cos squareds. What will be left behind? And lo and behold, I'm pretty sure that the cos of two theta is cos squared minus sine squared. One, two, three, yeah, four lines I figured it was, well, five if I count that one, okay? I'll do more, yo, three. These are both cos, now I can't factor out a cos sign or something like that. It's not a GCF, because it's a cos two x and the cos x, they're completely different. Now this is a period change, but I'm not gonna solve it using a period change because there's also a non-period change here. I'm not gonna go, oh, a equal two x. You know what I'm gonna do? I'm gonna replace cos two x with something that has only cos signs in it. I'm gonna rewrite this as two cos squared x minus one or plus one, minus one, and then the cos x will drop down equals zero. That's gonna be two cos squared x. I'm gonna write the plus cos x right here, Pat, because that's where we're used to seeing it, minus one equals zero. And now we're a lovely quadratic. You want me to keep going? I can, you good? Okay, it factors. Two cos squared, I bet you I had a two cos and a cos. Minus one with a positive one in the middle. I wanna go a plus one there and a minus one there. So I think I'm gonna get cos x equals a half and cos x equals negative one. And then I'll go from there. Cos x equals a half. I think the answer is gonna be pi by three and pi by three, but don't quote me on that. I'm doing the whole reference angle, triangle, drawing, cast rule in my head, which is always dangerous. And I think cos x equals negative one. I think you're gonna get x equals, if I visualize the graph, I think pi, okay? I'll let you take it the rest of the way. So looking at this one, for example, I would replace this with the one that only has signs in it. That would give me something with all signs and probably a quadratic. Looking at this strange one, well, this is a bit of a curve ball, but I see two cos squared something minus one. I know I didn't assign those by the way, but I was looking at them. Two cos squared something minus one. Like I think this kind of looks like two cos squared theta minus one, where theta is a half x. What is two cos squared theta minus one according to your sheet? Yeah, this is actually gonna be cos of two theta equals zero, where theta is, what's two times a half x? Yeah, in fact, you get this when I'll send them. Okay, so these are the double angles and where you can use them. Anymore, I'm thinking, yeah, unless everybody impressed the heck out of me and actually got number, what, I assigned five and or seven, seven? Okay, so look at seven. Here's what I see on the top. Except instead of a theta, what do I have sitting in my example? Three over two x. What is cos squared minus sine squared? From my formula sheet, what is cos squared minus sine squared? Don't say whoever said what, no, sine squared plus cos squared is one. What's cos squared minus sine squared? That's a double angle. Double angle, brain's going a little tired. Ian, what is it, cos? Okay, so the top is cos of two. Now, instead of theta though, what do I have sitting where the theta is? Okay, it's the cosine of two times three over two x. That's what the top works out to. Now, what about the bottom? In the bottom, I see sine theta, cos theta, which is sort of on my formula sheet, except on my formula sheet, I have two sine theta, cos theta. What does two sine theta, cos theta equal? Well, if I want to make this into just a plain old sine theta, cos theta, Tyler, I need to move the two over. How would I move the two over and get rid of it? I think I would divide. I think I would have a one half. I think the bottom is the same as one half sine two theta, but instead of a theta, what do I have sitting where the theta's are? Three over two x. By the way, what is two times three divided by two? Yeah, I think this is actually just plain old cos three x all over a half sine three x. By the way, I'm crossing out this one. Dividing by a half, what's that the same as multiplying by? See, I think that's, I'm looking at these two answers here. I think there's gonna be a two. And what is cos over sine? And I haven't asked this question in a while, but it's time to ask it. Did I really do anything new there? No, stubborn and clever phrase I've been using all year. What you kind of get good at, and this is the art part of the identities, is seeing tweaked identities in bigger identities. So I don't really even really see that three over two x and the three over two x. I don't let it scare me. I truly see that when I glance at that, and I know I can make it fit one of my identities, except instead of a theta, I'll just drop a three x into its place. I don't really see sine three over two. I just see sine cos, and I know that sine cos, well there's a two sine cos, but I can tweak it and make it a sine cos. But I've done the gazillion of these justine, so yeah, I'm not trying to compare you guys to me, but it'll come from any of you. Any more? Five C, good one. So five A and five B went okay? Awesome. Those ones I was hoping would five C. Yeah, wait, wait, wait, wait, wait, wait, wait, wait. Gonna temporarily ignore that, Jesse. Here's what I see. One minus two sine squared theta, except instead of a theta, what's sitting in mine? Three x, see it? Have I got one minus two sine squared on my sheet somewhere? I hope. What is one minus two sine squared? Oh, this is the same as the cosine of two theta. Oh, but instead of a theta, Jesse, what did we say we have sitting in mine? What's two times three x? There you go. And we have written it as a single trig function like they asked, okay? You do a bunch of these and suddenly you pretty much start to see almost every type of little curveball they can throw at you. Oh, did I give you a huge phone book sized review on Monday? You think maybe part of it was, I'm sorry, but one of the best ways to get good at this, unfortunately, one of the easier ways to get good of this is to do a gazillion of them. It's a lot of work, but it seems to occur naturally. Speaking of Monday, so we did lesson eight, double angle identities. I said that I had skipped lesson seven. We're gonna do lesson seven. So if you would be so kind as to turn back a couple of pages, lesson seven starts on page 351, page 351. And what we've been trying to do in the last couple of days dealing is establish rules for dealing with stuff inside the trig function, inside the brackets. So last day with double angles, we said, if you got a two in there, you can't just pull it to the front. It's stuck inside there. Today we're gonna ask ourselves, hey, if you're adding two things, if you're adding two things, is that the same as doing the first one plus doing the second one? And again, there's an easy way for us to check. We're gonna get our calculators out, except we are not gonna go to degrees like they suggest. I'm mortified. I'm appalled. I am shocked and defended. What is 60 degrees in radians, please? Pi by three. What is 30 degrees in radians, please? Pi by six. By the way, what is 60 plus 30 in degrees? 90, what's 90 degrees in radians? Pi by two. Here's what we're asking. Is the sine of pi by two, 30 plus 60, is that the same as the sine of 60 plus the sine of 30? Is that true? Find out for me, please. Try doing the left side on your calculator, sine of pi by two. And then do the right side on your calculator. If you get the same answer, then it's true. I don't think it is. Is it? No, sadly no. So it says, what can we say about this statement? Is the sine of alpha plus beta? If you're adding two angles inside the brackets, can you do the trig first and then add the answers? And the answer is it's a false statement. Nope, you can't. Which leads, Matt, to the obvious question. How do you add angles? This all comes from the pre-calculator days. Remember I told you that your folks in the pre-calculator days either had trig tables, but if they wanted exact values, I've only given you two triangles. I've given you a one, two, root three and a one, one, root two. And just for a moment, not on our calculators, but in our heads, we're gonna go back to the wonderful world of degrees. I've given you 30 degrees, 60 degrees and 90 degrees and 45 degrees. What if I wanted 15 degrees, half of 30? Is there a way that I can get that mathematically? And it's not go sine 30 and then divide your answer by two. It's not. Or what if I wanted 60 plus, let's go 60 plus 45. What if I wanted 105 degrees? Could I get that? And this is how you can start to build your exact values. So it says use exact values to verify the following statements. We're gonna do degrees one time and I guess we're gonna need the one, two, root three triangle where this angle down here was how many degrees? 30 and this angle was how many degrees, Jesse? 60 and let's see if this works here. What is 60? Because brackets, we're gonna do the brackets first. What is 60 plus 30? 90, what's the sine of 90? Well, behold, the human unit circle. 90 degrees is right here and my arm is one long and sine was your Y coordinate. So the sine of 90 would be how high my hand is right now at 90 if my hand is one long. How high is it? Okay, sine of 90 is one. You could also have got that by sketching the graph. Let's see if we get a one over on this side because it is suggesting that these are the same. What's the sine of 60? Root three over two. What's the cosine of 60 plus? What's the, sorry? Oh, coast, sorry. Coast 30, ha, Mr. Deweyck. What's the cosine of 30? Also, root three over two. Read these carefully. By the way, do read these carefully because it's easy to make a dumb mistake like that. What's the cosine of 60? Cosine of 60 is, a half. What's the sine of 60? Sorry, sine of 30, read it properly, Mr. Deweyck. A half. Let's see what this right side works out to. Multiplying fractions is the easiest operation. Top times top, bottom times bottom. What's root three times root three? What's two times two? Four plus, what's one times one? One over, what's two times two? Four, so here's the question. Is that the same as that? Yeah, there's a bunch more we could play with but I'm just gonna give you the sum and difference identities next page. Here they are. There are four of them and they all look really similar. So like I did not do, you should do read really carefully when you do these. If you're adding two angles together, the sine of alpha plus beta is sine alpha cos beta cos plus cos alpha sine beta. The sine of alpha minus beta is the sine of alpha cos, well, it's those things. And this is gonna be very, very careful plug and chug and these are on your formula sheet. You don't have to memorize them. Although I had a friend when he was taking his equivalent math 12 as a grownup, their teacher made them memorize all these and I have to be honest, I lost a lot of respect for that adult ed teacher. Does it all look alike? Where do you use this is really the question. Example one says, use a difference identity to find the exact value. Now exact value is my trigger phrase of the sine of what? Okay. Have I got a triangle with a 15 in it? No. But it did say exact value. What's going on here? Well, I'm gonna list the angles and we're gonna go back to degrees a tiny bit here. The angles that I know. So think about the two triangles that you have. What are the angles that appear in there? Aside from the obvious 90 degrees, the two triangles that we have, what are the angles that appeared in degrees? What are they? 30, 60, and 45, right? Here's my question. Using two of those numbers and either adding or subtracting them, can you get 15 as an answer? Sorry. This is gonna be the same as the sine of 45 degrees minus 30 degrees. Oh, you could also go 60 minus 45. That would also get you there. But we'll use this one, because it's smaller numbers. What is 45 minus 30, 15? What does this expand into? Well, I'm gonna go here and I'm going to find this line, sine of alpha minus beta. And what we're really saying is that's alpha, that's beta. And I'm going to carefully substitute in the values. It says sine 45 cos 30 minus cos 45 sine 30. Now what? I'm gonna draw a couple of triangles. How about over here? 1, 1, root 2, 1, 2, root 3. Double check to make sure I plug this in right. Beta minus alpha sine beta. I did. What is the sine of 45 as an exact value, please? What's the cosine of 30 as an exact value, please? Root 3 over 2 minus. What's the cosine of 40 as an exact value, please? Isn't it one and a half? Are we okay? We're wrong. We're right. We're good. Well, then Troy, redeem yourself. What's the cosine of 45? I agree. Troy, what's the sine of 30? It's a half. Let's tidy this up. How do I multiply fractions? Thankfully, it's the easiest operation. Top times top, bottom times bottom. 1 times root 3 on the top. I'm pretty sure is just plain old root 3 all over. 2 times root 2. It's numbers times numbers, roots times roots. 2 times root 2 is just 2 root 2. Or 2 times root 2. Minus. Matt, what's 1 times 1? Woo-hoo! See, I gave you a nice curveball. Nice one. Nice easy one. Oh, and 2 root 2. Now, I think I can go further because I can't help noticing I have a wonderful common denominator here of 2 root 2. I think I can write this as a single solitary fraction. Final answer. Root 3 minus 1 all over 2 root 2. I need to write that a little bit bigger. I'll do it over here. Equals root 3 minus 1 all over 2 root 2. Do they always work out to a common denominator? Frequently. While we're on the topic, what if instead of asking you to find the sine of 15, what if I asked you to find the cosecant of 15? How are cosecant and sine related? Cosecant would be 2 root 2 all over root 3 minus 1. Again, I don't deal with cosecant. I do it as a sine question and then at the end I flip my answer, which leads us quite nicely into example 2. And this is going to be a chunk of work and I already know that I'm going to have to write small so I'm going to enlarge my screen. So, secant, I don't have a sum difference identity for secant or for cosecant or for tangent for that matter. Why don't I have one for tangent because tangent is what over what? Dine over cosine. So if I really needed to, I'd use the sine one on top and the cosine one on the bottom, which would really be overkill, I think. There is one actually for tangent that we used to teach, but it seems to have kind of vanished from the curriculum. Secant, what do we say secant goes with? Okay, so I'm going to make a little note here. Find cose, then flip it. I'm going to find the cosine of five pi by 12. In fact, I'm going to write that. I'm going to find the cosine of, oh, I'm not going to write five pi by 12 just yet. I'm going to write cosine though to remind myself. I need to turn this into an addition question. So now we're in radiance. What three angles in the triangles do I know in radiance? In radiance. Sorry, what are they? Pi by six, pi by four, pi by three. Miguel, what's my denominator here? What should I write these all over to make my life easy? Why not? This is really the same as two pi by 12, three pi by 12, and four pi by 12. Can I write any of those as either an addition or a subtraction question that gives me five pi by 12 as an answer? What, how? Okay, add these two together, I'll get five pi by 12. So here's what I'm going to do. I'm going to go to cosine of pi by six plus pi by four because that is five pi by 12. I've got my template. There's my alpha, there's my beta. Now, normally I would draw the triangles, but do we have them on the previous question sitting there? You'll notice I didn't label the angles on purpose because I wanted to be able to use those triangles for either degrees or radiance. Anything I want to think of there? Okay, let's go to the double, or the addition identity. Cosine of alpha plus beta, what is that? Can someone read that from the top of the page? It's cos cos, okay, is it cos cos cos minus sine sine? And is it cos alpha beta minus sine alpha beta? Because I also got to get those right too. I can't just look at the trig functions. So cos cos minus sine sine. Cos pi by six, cos pi by four minus sine pi by six, sine pi by four. What's the cosine of pi by six? As in exact value, please. Root nine, root three over two. What's the cosine of pi by four? Minus, Pat, what's the sine of pi by six? Sine of pi by four, one over root two. Now, Pat, you would ask, will there always be a common denominator for the basic ones? Yeah, I can think of some really bizarre, strange ones where perhaps not, but I think. And in fact, I think I get this, do I not? Now we're gonna try skipping one step. Do you see I do have a common denominator? What is my common denominator? Okay, I'm gonna write it all over two root two. And it's gonna be root three minus one. Now that is the cosine of five pi by 12. It's not what we wanted. What did we want? By the way, in example one and example two, did I have square roots in the denominator? The textbook answers will rationalize the denominator again. So if you're trying to compare your answer to the textbook answer, change that to a decimal, change their answer to a decimal, and they should match. Now that's going, sorry, this will be a multiple choice question, so I don't believe so, no. That's going forwards, excuse me, that's going forwards. They can also ask you to go backwards. Example three, and this is where I like what Miguel said earlier. He started speaking of these in terms of just the trig functions. He said, oh, it's sine sine minus cos, or cos cos minus sine sine, or sine cos plus cos sine. Because I look at example three, which says simplify, and I see sine cos minus cos sine, and I go, ooh, I think that's one of my sum or difference identities. Is it? Look at the four that are on your sheet. Is there one of them that's, oh, let's make it a little easier. Alpha, beta, alpha, beta. Is there one of them that's sine alpha cos beta minus cos alpha sine beta? What is that on the left-hand side equal to? Sorry? No, no, I need the trig function, too. I need the whole thing. So you're telling me this is the same as the sine of alpha minus beta, which is the same as, what's alpha here? Oh, you know what, in our notes, let's write it. I was gonna do it in my head, but I'm gonna write 100 minus 10 in our heads. What is 100 minus 10? 90? And what is the sine of 90? Hey, let's try B. Again, this looks really scary, but I'm gonna treat this whole expression here as one big alpha, big beta, big alpha, big beta. I guess alpha is, quarter pi minus theta, beta is quarter pi plus theta. But here's what I see, cos cos minus sine sine. What is cos cos minus sine sine? Sorry? Cos alpha plus beta? This is the same as cos of alpha plus beta, which is gonna be the cos of, now alpha is this very strange one quarter pi minus theta plus, and beta is this very strange one quarter pi plus theta. I had no idea where this was going, but now I'm starting to smile a little bit cos I notice, well, what do you notice? Ooh, the theta's, theta plus, negative theta plus theta is gone, and I have one quarter pi plus one quarter pi. What's a quarter plus a quarter? A half. Or you know what, Justine? I think we've actually said it as pi by two. That's what that whole mess simplifies. Oh, oh, and that's one of my pi by two, up here. What's the cosine right here? You mean this whole thing just works out to nothing? Yeah. What's that, Ari? You would like to turn the page? Yeah, that's a great idea. Let's turn the page. I like number four, I like number four, I like number four, number four is a nice question, I like number four, and this is probably gonna answer Pat. Says, given cos of alpha equals three over five, and cosine of beta equals five over 13, and alpha is in the first quadrant, and beta is in the fourth quadrant. Find the exact value. Now there is my trigger phrase exact value, but this is the one time when exact value does not mean special triangles or unit circle, because I don't have a triangle with a three and a five in it, or a five and a 13. This is where we're going to go back to x, y, and r. Did I already say I like this question? I like this question? Okay, I like this question. First of all, they want me to find cos alpha plus beta. What is cos alpha plus beta from my sheet? Cos of alpha plus beta equals what? Cos, cos minus sine sine. Well, some of this I can already fill in. According to this question, what is the cosine of alpha? Three over five, and what is the cosine of beta? Five over 13 minus, what's the sine of alpha? Wait a minute, do I know the cosine of alpha? Here's alpha. Cosine is what over what in terms of the graph? I think when they told me this, they told me that x equals three, and that r equals five. And they told me that alpha is in this quadrant here. In that quadrant there is y negative or positive. Positive. How can I figure out how big y is if I know x and y and r? Sorry, x and r. It's gonna be the square root of r squared minus x squared. And yes, some of you recognize our old friend, the three, four, five triangle. I think you get four. So I think I can do sine of alpha because sine is what over what in terms of the graph? This is going to be four over five. Over here, I'm gonna do beta. Now beta, they told me I was in this quadrant here because it says we're between three pi by two and two pi. And beta, they gave me that x is five and that r is 13. What's y? Well, first of all, look at my quadrant. Is y going to be negative or positive in that quadrant? Negative, and I gotta put that negative in. It's not gonna come out on my calculator or anything like that. And let's see, y is gonna be the square root of 13 squared minus five squared, which I think is 140, squared 144. It works out evenly actually, doesn't it? What is the square root of 144? So y is negative 12. Sine is negative 12 over 13. I keep going here. I get 15 over 65 minus negative 48 over 65. What's a minus minus the same as? A plus. What is 15? Because I have a lovely common denominator. This is very nice. 63 over 65. This is almost, well, no, I was gonna say this is on the provincial, usually a multiple choice. I think I might make this a written. Anyhow, if it was multiple choice, you better believe one of the answers would have a positive there. So you'd have 15 minus 48 and negative answer. Last one. I'm doing a completely different question. And the fact that I'm getting rid of this question and making up my own if you're a good student should tell you something. What was that Troy? Sorry, I didn't hear that. I didn't say anything like that. You must just be paying attention. Here is a question that they love. They'll do something like this. Simplify, pick a trig function, Victoria, sine or cosine. Doesn't matter, pick quick. Coast, okay. Put your pencils down, folks. They'll do something like this. Cosine of, they'll put a minus sign or a plus sign here. They'll put an X here or an X here, or a theta or a theta. I'll do a theta just for that. And they'll go, don't write this down because I'm gonna show you all the variations and then we're gonna do one of them. They'll go theta minus pi. Or pi by two. Or three pi by two. Or which angles are they picking? The corner angles of our circle. Or what's the other one? Two pi. Or instead of putting it in that order, they'll put the theta right there. Minus or plus, doesn't matter. And they'll put a pi by two there. Or a three pi by two there. Or a pi there. Or a two pi there. You know what, we're gonna do that one. This one you can write down. Simplify, cosine of three pi by two plus theta. And kids freak out because they're like, I don't know theta, don't worry. First of all, can you see this is a sum and difference identity where that's alpha, that's beta. So cosine of alpha plus beta, what is that? Coast, coast minus sine, sine? So they'll always pick one of these four. Pi by two, pi, three pi by two, or two pi. Or they'll really go crazy, negative pi. But they'll be somewhere on the corner. Cosine of three pi by two. Now we could get this using a sketch, but let's try our unit circle. So three pi by two, there's zero pi by two. Three pi by two down here. And if my arm is one long, cosine on the unit circle was what? The x coordinate or the y coordinate? X coordinate, so if I'm down here, what's my x coordinate? What is the cosine of three pi by two? Turns out it's zero. I get zero times the cosine of theta. By the way, what is zero times anything? That whole term vanishes. The minus sign would drop down. And then they gave me the sine of three pi by two. I'm down here again. What's the sine of three pi by two if my arm is one long? Negative one. And I'm just gonna drop the sine theta down. Can you see what this actually, this big mess here, what it actually simplifies to? What's a minus minus the same as? Plus one times sine. You know what? This whole mess simplifies to that. And that terrifies kids for some reason in the multiple choice when they get something like this. Oh, by the way, so I told you that they'll either put the multiple of pi by two there or there. I'll even tell you right now what the answers will be. Negative sine, positive sine, negative cos, positive cos. That'll be the four-picture. Because for these, it'll always work out to negative sine, negative cos, positive sine, or positive cos. There's your sum and difference identities. And believe it or not, that's it for identities. So whole shebang. That's the top half of the formula sheet. Get to practice these, yes. Oh, you know what? Number one is very similar to the ones that I was talking about. Can you see they're either using multiples of pi by two or multiples of 90 if we go to degrees? Two a and two c. Six, pause there. Because I also, did I not gave you a gazillion questions to do on the review? Just curious, and I'm being partly serious and partly humorous. How many of you have started that great big review? In a couple of you? Okay, so you know what? Assuming, have you finished all the ones that I signed yet, none of you? So I won't assign any new ones today. On Friday, I'll give you a gazillion more, okay?