PC 11.3 Sum and Difference Identities

All right today we are doing 11.3 wheel of fortune theme. Some indifference identities, all right, wheel fortune seemed just because I came across this and I don't know, this is one of those is a typical math mistake on wheel of fortune. Going too fast and making simple mistakes and this guy, I'm gonna show you one clip of one guy and he is amazing because he is the king of simple mistakes. We all make them, but he is definitely the king of them. All right? So let's talk a little bit about this. First of all, let's take a look at this. Is the sine of 45 plus 30 if I add those together is at the same thing as saying the sine of 45 plus the sine of 30. Well, there's an easy way to find out. Let's go to our calculator. Let's make sure we're in degrees over here. So the first thing I want to find out is what is the sum of, well, 45 and 30 is 75. So what is the sin of 75? All right, the sin of 75 is .9659. All right. So now I want to find out what is the sin of 45, 45 plus the sin of 30. And when you do that, you find out boom, bada Bing, they are definitely not the same thing. All right? A little close, yeah, for sure a little close, but not the same thing. So it makes you wonder, why? Why bro? Why? Why is it so well? Actually, here's mister brust explanation of this. And it's going to get into a very in depth process here. Now here's the deal. Just sit back, relax and watch this proof. He's going to give you a proof of why this is not true, all right? And he's going to lead into what is true. All right? Again, don't feel the need to copy this down in your notes. I didn't even leave a spot for it in your notes to copy it down. Just sit back, watch mister busts, try and stay awake, and it'll lead us into our next thing. So my goal is to prove what happens when I subtract or add two angles. So I'm going to start off with subtracting an angle. So let's start off with this blue angle here, and I'm going to say this angle from here to here is alpha. So I've got alpha right there. And then in red, I'm going to draw another angle. I'm going to call this one beta. So this is beta and the other angle in blue was alpha. So that's great. So what I want to do is subtract these two lines. If I subtract the distance, I can find the distance between alpha and beta. So I'm looking for that distance. So ideally, if I want to make up numbers, if you're more of a concrete person, this first angle could be like a 140°. And then I want to subtract another angle from it. Maybe I'm subtracting a 120°. Obviously what comes up 20°. So if I subtract these, then I get a new distance over here. That would be like my 20°. So what is this angle? This angle is actually alpha minus beta is that angle. And again, the distance between this and zero should be the exact same. It's this distance. We're subtracting by looking for a distance. So that's the basis of this whole proof. You're subtracting two angles. And really, that's the gist of this. If you understand what we're doing, we're subtracting two angles. What we're going to look for is finding exact values now. So here is the proof, get ready. This is fantastic. All right, let's start with the coordinates of this. This point right here, the XY coordinate. So I'm looking for these coordinates. Well, I can use my trig function. X is cosine. So that is really the cosine of alpha, the sine of alpha. So those are my coordinates. Those are my XY coordinates for that angle right there. How about four beta? Same thing. If I want to find this point right here, the X value is cosine beta, the Y value is sine beta. So it's depending on the angle, that if I type into my calculator, that'll tell me the number there. The one that looks weird, though, is this one. This angle is actually alpha minus beta. So what are the coordinates of this point right here? Well, you guessed it. It's the cosine of alpha minus beta. The sine of alpha minus beta. So it looks intense, but that's our ultimate goal. We're trying to prove that this is the identity. What happens when I subtract them? So all we're going to do is show that this green line over here is congruent to this green line over here. So if we're going to do that, how are we going to show that? Well, we are going to use the distance formula. Remember the distance formula? Distance is just Pythagorean theorem. We're going to subtract our X's and square them. Then we're going to subtract our Y's and square them. So if we do that, some people like to put little numbers here. If we'd like to do that, that's why I have these coordinates so I'm going to subtract blue and red to get this distance that I'm going to subtract black and what does this point right here? Well, this point right here is over one up zero. So that one should be easier. Let's start with blue and red. Let's go for it. Going for the glory here. So let's do distance formula of let's see. It's going to be the square root and it's going to take some room here. I'm going to take my X value. So cosine alpha minus cosine beta. So cosine alpha minus cosine beta square it plus I'm going to do my Y values that we're going to be sine. So sine alpha minus sine beta. Oh, this is so fun. I love it. So that's the distance between those two. Can we simplify this? Sure. We can foil this out a little bit. I'll change colors here. But we can't foil this out. It's going to be what cosine squared alpha, and I may have to slide that over. I think it's going to take some room. Cosine squared minus cosine alpha beta, then I'm going to have another minus the inside term. I actually get two of a minus cosine alpha beta. So it's a difference of squares here so I get two in the middle. And then my last one cosine times cosine was going to give me the plus cosine squared beta, holy cow. That's the first part of this. Let's slide it over. I'm going to need more room, I think. All right, so keep this rolling. What's the second part? Well, I'm going to add same thing here. I'm going to get sine squared when I'm foil it out. Sine times sine alpha times sine beta, and I'm going to have two of them. So I'm going to have two sine alpha betas or two alpha two sine beta, then do the last term, it'll be plus sine squared beta. So that is just foil property right there. What's cool about this, though, is anything good happen from this. I hope so, 'cause that's a mess. And I don't want to do much more. Check this out. I have a cosine squared here. And a plus sine squared here, what happens when I combine these two together? It's a property. Yeah, cosine squared plus sine squared is what? It's one. Awesome. And it's alpha. They're both alpha, but check this out. Cosine squared beta, sine squared beta, what happens with these guys? They are also equal to one. So really, if I add those together, I'm going to get what. One plus one, two, so that is actually two. Unfortunately, that's as far as I can simplify it. This cosine alpha cosine beta. It just comes on down and then minus that two sine alpha sine beta. Fantastic. And really technically this is a distance and I'm square rooting it all. So that's as far as I can simplify the first part. That is this line up here. So I just found, let's say, D one. That is the first distance. Let's find D two. I love it. This one is going to be based off of this point right here. So D two is going to start with. And I'm going to give myself lots of room for D two. Here's my X value. Here's my Y value. So it's going to be cosine of alpha minus beta minus the X value of that. That's just going to be one. And then I'm going to square it. Plus I'm going to do what sine of alpha minus beta minus the Y value of this point I'm finding this D two right here, here's D two. So I'm fighting that, and that's going to be minus zero. And I'm going to square that. Fantastic. So again, I got to do a little more foiling for D two. Let's foil this up. I'm going to end up with cosine squared, alpha minus beta, distributed there. I'm going to get minus one cosine alpha minus beta. If I do this one, then I'm going to get I'm going to get minus one. So this is another perfect square. I'm just going to get two of these. And then I'm going to go plus one at the end. Negative times a negative. So if you want to expand that out and multiply it, feel free, but that's I'm just foiling it. This one feels great because that's just zero. So I'm just going to end up with sine squared. Alpha minus beta, holy cow, look at this screen. I love it. So we can call your parents and be like, hey, look what we're doing in math today. But this will look better. It'll get better here in a second. We're going to need some more room though. Okay, so get myself a little bit more room there. So again, this is still square root of all this in a distance. Does anything cancel in there? Did anything good happen at all? Actually, something does happen, doesn't it? Check this out. We've got cosine squared plus sine squared. They're the same angle. Just like it's 20. It's the alpha minus the beta. So these are Pythagorean identity that was equal one, and I already have a one in there. So check this out. I've got two minus two cosine alpha minus beta fantastic. And that's distance two. And it's the square root of that. So we're doing great here. Very good. So we've got D one, D two. What I'm going to do is I'm going to set distance one equal to distance two. That's our whole goal to show they're the same. And I'm going to go ahead and square them both. They're both square roots. So I'm going to square both sides. So if I can do that in one lump sum, I think that'll help me out here. So I'm going to bring down, let's bring D two and left. I've got two minus two cosine alpha minus beta. And this is my whole goal. I'm looking for a difference identity. I want to get cosine alpha my beta by itself. On the right side, I'm going to have two minus two cosine alpha cosine beta minus two sine alpha sine beta, still barely fit. So I'm just bringing this down as my other distance. What's good about this? I see a lot of twos in there. So that seems like it's pretty cool. So hopefully I can do something with my twos. In fact, I can. If I subtract them from both sides, right off the bat, what's going to happen minus two minus two, these guys are just gone. Then what else am I going to do? Well, this is negative two left over. I'm going to go ahead and do we need more room here. I'm going to divide that by negative two. Whatever I do to one side, I got to do it to the other, divide that by negative two. So these will cancel. Here's my final identity. You could have fast forwarded to this point. Cosine alpha minus beta, what do you get when I divide this by negative two and divide that by negative two? I'm going to get cosine alpha cosine beta minus sine, not minus plus negative divided by negative plus, sine alpha, sine beta. That is as far as we can go. That is the identity that is amazing. Holy cow. So that was great. Thanks a lot, mister brust. He led it to a lot of these things all right. He didn't give us all these. He gave us one of them, but it's true for all we could go through the same process and come up with all of these. It would be really long and boring and I'm sure you don't care. But if you do again, go over to Ramsay in high school, find Mister Bean, and you should be able to find him as his skin in sky over there. He's got a big smile on his face, and he's just as nice as can be asked him, hey, can you teach me on the board? And he'll go over to the board and he'll show you a long proof. If you need a good nap, I suggest going over there after school, giving him 30 minutes and then going home, you'll sleep like a baby. All right, anyway, these are our new identities. If we add two angles together and we're doing the sign, it's sine times cosine, and I keep whatever I was. If I'm adding, I add sign and cosine I switched the angles. If I subtract, I keep the subtraction. Over here, you notice when we do the cosine one, let's do a cosine one right now. When we do a cosine one, what's gonna end up happening is that we are gonna end up switching the signs. When it's cosine of alpha plus or minus beta, and then it becomes minus or plus beta, that means you switch whatever you originally had, all right? So let's do the cosine of 15. Well, the biggest trick to this well is finding what adds up to 15 or subtracts to 15. I know that that is 60, so we're gonna do the cosine of 60 minus the cosine of 45. So that's R two. So we're gonna do cosine 60 -45, all right? Now, a lot of times there's a lot of different ways to do these and get to that number 15. This is just one way. So let's do our cosine formula, so I have cosine of the first angle, so cosine of 60 times cosine of the second angle, all right, now it says if I am subtracting, I'm on the bottom. I actually am going to add, see if I'm on the bottom, I'm on the bottom here, so I'm going to add these two together. So I'm going to do the sine of 60 and the sine of 45. All right, that's plugging it into the formula. Now we have to look it up. We have to know that cosine of 60 is one over two, one half. The cosine of 45 is radical two over two. The sine of 60 is radical three over two. And the sine of 45 is radical to over two. And we just have some math. One half times radical two over two is radical two over four. Radical three times radical two is radical 6 over four, and we have a common denominator. So I'm going to put radical two plus radical 6 over four. All right? And there you have it, that is the exact value of cosine of 15 using the cosine difference identity. All right? Strange looking answer for sure. All right, they're going to be a lot of these because you think about it, you have half radical two over two radical three over two. A lot of them are very similar so they're going to look a lot alike. Let's go over here to the sign. Sine of one O 5. Well, there's a lot of different ways I could do this. I could do one 50. Oh, that's not one 50. One 50 -45. That would definitely give me that, right? But I'm going to do a different one. I'm going to do sine of 60 plus 45. All right? 60 plus 45 is one O 5. Let's go to our assigned formula. Now, notice if I'm adding, I stay adding. All right, so I'm going to do the sine of the first angle, which is 60 times cosine of the first angle, which is 45, plus now I'm going to do sine of the second angle 45 and cosine of the first angle. Sine cosine Z cosine, we just switch the angles. So sine of 60 is radical three over two times cosine of 45 is radical to over two. Plus sine of 45 radical two over two times cosine of 60 one over two. So simplify this, we get radical 6 over four, plus radical two over four, which is radical 6 plus radical two over four. If you notice, same answer. Now that's just a quick and indie coincidence. All right, it's not going to happen every time, but like I said, a lot of these come out the same, all right? There's only so many ways you can add 45, 30, 60, there's only so many ways you can add those up and get something, all right? So let's go to our first clip of this gem of a champion on a wheel of fortune. Watch this guy. Now the first thing you have to understand is the very first thing he picks up is a $1 million thing. So if he gets the puzzle right, all he has to do is get the puzzle right he gets a $1 million, all right? Watch what happens. Pick that up, turn it over. Lay it down right over that London trip down there. Switch switches. Why? There's a Y. 500. G one G. 700. C two CK is well. Wow. Okay. And I saw. Yeah, yeah. Mythological hero, HUS. You can't accept that. Okay. Shovel. I'll solve. Yeah. Mythological hero Achilles. Yeah, that's it. I mean, that's pretty funny. Like, if you go to college, you think you know who Achilles was, you know? You think you would have run across it, maybe not. I guess not, and that's just unfortunate. All right, so now we have a tangent one. We want to get to 11 pi over 12. All right, now these can be tricky because they're fractions. No one likes fractions. We need to remember though that we need two things that add up to 12. And it's got to be 11. You just don't want to randomly have numbers over here because one of the things you want to think about is our angles are in terms of over 6 over four or over three, right? So I mean, I could obviously put one in ten, but I don't have a special 90 it's not in the unit circle for pi over 12. In other words. So I need one that's an odd and one that's an even. Well, I'm going to go with 9 pi over 12. And I'm going to go with two pi over 12. All right, now the reason I do that is because 9 plus two is 11. Obvious reason, right? The second reason I do that is because now I can reduce these down and this is in fact going to be the tangent of three pi over 12. They're reduced that plus pi over 6. All right, and those are ones we know. So let's use our formula. So I'm going to start with plus, so I'm on the top here. Now, on the top, I do what it says, but now on the bottom, it switches. So we have the tangent of three pi over four, three pi over four, plus tangent of pi over 6, all right, I'm going to divide that by one, now I switch it to minus minus tangent of three pi over four. Times tangent of pi over 6. All right? At this point, I'm going to plug in my value, so I know that the tangent of three over three pi over four is negative one. The tangent of pi over 6 is radical three over three. All right, on the bottom I have one minus. So again, tangent of three pi over four is negative one. Times tangent of power 6 is radical three over three. And I forgot to put my three here. Way to go, Solomon. All right, so let's do a little simplifying. So we know this is negative one plus radical three over three, divided by that's one, one that's plus radical three over three. All right. Here, this is a trick breast taught me for the most part by three over three. So negative one times three, I'm gonna come down here, negative one times three gives me negative three. Radical three over three times three, these cancel, so I get radical three, so plus radical three. Divide that on the bottom three times one is three. Plus radical three, these three cancels. That's radical three. So very similar now. It depends on your teacher, so many teachers are gonna be very okay with this, all right? I think you're better than that. I'm gonna multiply by the conjugate because we do not like radicals in the denominator. So I'm gonna multiply by the conjugate three minus radical three on top and bottom. All right, so let's see what we get. Negative three times three is negative 9. Plus three radical three. Radical three plus three radical three again. And then radical three times negative radical three is just minus three. All of that over. Now I know the middle's gonna cancel, so I'm just gonna do the first times last three times three is 9 and radical three times negative radical three is minus three. I know the middle cancels because I'm multiplying by the conjugate. I'm doing it to get rid of the radicals, all right? So negative 9 minus three is negative 12. Three radical three plus three is 6 radical three. All right, all that over 9 minus three is 6. All right? We can reduce. We can take a factor of 6 out so that's going to be negative two plus radical three. All right, quite a bit algebra there, but the pre calculus part is this initial part that we have to worry about, all right? Let's see what our guy up at two on a wheel of fortune now. Wow. Ow. There's an L picking up. You got both hands. John? Four 50. D yeah, there's a D. 300. C. Can I solve? Yeah. The world's fastest man. Yeah, that's it. All right, this is a real quick one easy. Right the expression has a sign cosine or tangent of an angle. All right, so the first thing I'm looking at is I have opposite things here. Actually, the first thing I'm looking at is I don't have a fraction. The tangent one is a fraction. So we know it's not tangent. All right? We know it's not cosine because remember cosine is cosine cosine sine. So it's got to be sine of something. All right, the next thing I'm looking for is remembering the formula for sine in the middle, stayed the same. So this is definitely a subtraction, so this is sine 42 -17. And 42 -17 is sine 25. All right, signed 25. So that is, again, not that difficult just reason yourself through it. Don't overthink like that last guy did, right? All right, next one, this is a little bit trickier. We want to find sine of X minus Y given the following conditions. All right, so a couple of things. So we're saying this is in the third quadrant. Pi over two, oh no, excuse me, second quadrant. So negative three, so I have a triangle that looks a little bit something like this. Negative three. Over 5 is cosine, right? And then we would know this would be four, three, four, 5 triangle. All right? So then we have over here tangent of 8 over 15, and this is in the fourth quadrant, fourth quadrant. So this triangle would look a little bit something like this. Tangent Y would be 8, actually I should be negative 8 over 15. Right? So this would be a 8 15 and 17 triangle. Okay? So we want to do this, so we're going to do the sign of this. So we do the sine of X, then the cosine of Y and then we're subtracting so we're going to minus, we're going to switch it to sine of Y and the cosine of X so we come down here, so there's the sine of X so I go to my X triangle, this is my Y so I go to my X sine of X is four over 5. And the cosine of Y is adjacent over hypotenuse. That's 15 over 17. All right. Minus. The sine of Y so I go to my Y go opposite over hypotenuse so that's negative 8 over 17. And the cosine of X that's negative three over 5 all right, so we have 60 over 85. Minus negative times a negative so we have 24 over 85. All right? 60 -24, what's that give us 36 over 85? All right? So again, it's very important that you set up these triangles, set them up correctly, and then just write it out, the fraction part shouldn't be that terrible. It might be a little bit trickier with the tangent function. All right, so let's see how this guy finishes up on the wheel of fortune here. C one C. Julien. Everything it just follows. It's soft. Go ahead. On the spot, dice spin. No. Shelby. On the spot decisions. Yeah, that's it. Man. I don't think anyone is ever taking a more secure this route to victory. What blows my mind there is he had three of the worst guesses ever. And still won that game. Like, how did he doubted that other girl not win? I have no idea. He still won that game. Blew a chance at a million. Blue a chance that a car and still won. All right, luckiest guy ever, I guess. All right, we want to verify this. So we're doing our tangent difference formula. So I'm going to do the tangent of my first, which is X minus the tangent of my second, which is pi. Then I do one minus oh, not minus. I was doing minus so I do the opposite on the bottom. Tangent X times tangent pi. All right? Let's see, tangent X, we don't know what that is. It's a variable, but we know that tangent pi is zero, so tangy and X minus zero. One plus, and again, we don't know tangent X we know tangent pi is zero. I love zero. Love it. You know why? Minus zero just tangent X times zero is zero. Zero plus one is one and tangent X divided by one is tangent X so we just showed that that is in fact an identity that tangent of X minus pi does in fact equal tangent of X all right, that's a great little one to end on. All right, best of luck on your mastery check. I hope you do well and I'll see you on the flip side.