Hi, everyone, welcome back. We're going to learn our last bit of identities here, and this will be our foundational identities that we can go use to do anything else we want. So the next identities is that we're going to do, they're not as easy as to derive as the other one of the Pythagorean identities where we just look at the triangle and say, aha, Pythagorean theorem in disguise. These identities will involve sine and cosine of the sum and difference of two angles, and these should be kept handy. You don't have to memorize them, but they should really be kept handy because these are going to be frequently needed as we go through and work with our things. So let me just give these to you and you can copy them down. So this is called the sum and difference formula, sum and difference identities, and here we go. We are given that sine of any two values, x1 and we'll do plus or minus, doesn't matter, x1 plus or minus x2 is equal to sine of x1, cosine of x2, plus or minus cosine of x1, sine of x2. Let me give you the other one, and then we'll talk about it. So if I have cosine, again remember, just sine and cosine, and the rest, you can always manipulate from these. x1, we'll do plus or minus, x2, we get cosine of x1, cosine of x2, minus plus, I'll talk about that in a second, sine of x1, sine of x2. Okay, here we are. In these sum and difference formulas, so they're called sum and difference because they're plus or minus, when the top or bottom sine is chosen, then you choose the top or bottom sine on the formula on the right. So what does that mean? It means if it's a plus coming in, then on the side over here, it's a plus, and if it's a minus going in, then it's a minus going out. So whatever the sineis, you just match the top or bottom. For the cosine sum, it turns into a minus, so it switches, and we'll do examples of this to drive this home. But if it's a plus coming in, it's a minus over here on the right side, if it's a negative coming on the left, then it's a negative going in, positive on the right. So sines, same, this is how I remember the sine, s for sine, s for same, same sine, sine same. Cosine, gotta switch, okay? So these values, to prove this, we'll put a link somewhere, but it's not as obvious why this is true, so if you were looking at that and saying why, it's a very nice clever geometric argument that could take up a whole page that we're not going to do. I'll put a link to it, a video where they explain it quite nicely, but for now, just use these, keep these handy as we go through our calculations. But this, along with the, I'm going to put it all on the same side, so you have it, the Pythagorean identity, this is your core identity, so the Pythagorean. These three make up everything. I'll write it one more time just so we have it all on the same slide. So sine squared of x plus cosine squared of x is 1. If you know these three identities, everything else follows, right? We already saw that you can get the other two identities for free, and your even and odd identities, maybe we'll throw these on there as well just so you have it all. So stare at this slide, let this sink in. This should be what your review sheet kind of looks like, and and try to, every single time you have to use something else, come back to this. So here they are, sum and difference, Pythagorean, even odd. Know these, the rest you get for free. How are they used? Okay, so let's do an example. We're going to keep those handy, so pause video back and keep those handy. Let's find an exact value here. Let's do the sine of pi over 12, so pi over 12, this is not on your sheet, right, and I want to know the exact value. I don't want it in decimal, I want it in closed form. And what's nice about pi over 12 is that you can write pi over 12 either way, and check this here, but you can write pi over 12 as a sum or difference of our known values. So pi over 3 minus pi over 4, I said for 4 but wrote 12, that's a good trick. So pi over 3 minus pi over 4 is in fact pi over 12. Pause the video, convince yourself that that is true. Once you write an angle as a sum or difference of two values, well, then you can use the formula for the sum or difference of values. So sine of pi over 12, I want to think of pi over 12 as now sine of the difference of pi over 3 minus pi over 4. We have a formula for this. Remember, sine keeps the same sine, so it says take the sine of the first, sine of pi over 3, cosine of the second, so cosine of pi over 4. Keep the same sine, so minus, and then you switch it, cosine of the first. Cosine pi over 3, sin to the second, I always say sign of the first and second whenever I have a formula, I don't want to think of formulas in terms of the variables that they're written with. The reason for that is because you can also see this in terms of theta or maybe alpha or beta or some other crazy variable. And so just a little vice to you whenever you have a formula, say the first or second. So the sign gets the same sign here, minus as a minus, so that doesn't change, sine to the first cosine to the second. Keep the same sine, cosine to the second sine to the first. Again, if you just trying to just as you say it and then different parts of the brain are activated and maybe one day you'll have to look it up as much. But for now, keep it handy, check my work, make sure you agree, and then I claim all these values are known. So I'll now go to the table and find with these are a sign of pi over 3, nice root 3 over 2, cosine pi over 4, this is root 2 over 2, we've done all these cosine of pi over 3 is a half, and then sign of pi over 3 is root 2 over 2. You can clean this up a little bit, what do they have here? If you clean this up you get root 3 I'm going to keep the root separated, and just combine over a common denominator of 4, and the reason for doing that is that again, all these answers are fine if we're just trying to clean it up as much as possible. There's a root 2 and a 4 in both expressions, so will do root 2 over 4 we'll factor that out and we get root 3- 1. So this would be are closed form. This is our answer in terms of square root, again a calculator will never be able to tell you that this is the expression. Sure, it'll give you a 5 decimal, 6 decimal, 7 decimals, but it will be able to tell you in I doubt that you would be able to look at that decimal and tell me that its root 2 over 4, times square root 3- 1. So this is just an example here of using the sign formula, let's do one other example, we'll switch it up and we'll do the cosine formula just to get practice using these things. So let's do another one, how about cosine of 7 pi over 12, okay? Again, this is the tricky part, sometimes if you get something off the table you're looking for, how do I write this in terms of something that I know? So 12 is usually a common denominator for these sort of practice problems, because, they usually somehow written as plus or minus of pi over 3 and pi over 4. 7 pi over 12 you come off on the side and you can check this 7 pi over 12 is pi over 3 + pi over 4. So again, the pause video, convince yourself that that is true, so when I want to compute cosine 7 pi over 12, I really want to think of this as the sum of pi over 3 + pi over 4. In that regard, I have a formula for the cosine of the sum of two values, so here we go, it's cosine of the first, say with me cosine of the first, cosine of the second, say first and second don't say x1 and x2. Now cosine we have to change the plus to a minus, that's why it's minus plus in the equation, sine same, sine keep the same SSS cosine you've gotta switch. Cosine first times cosine second minus sine of the first, so sign of pi over 3 times sign of the second, pi over 4. All these values are known, all these values are known, so you can look him up if you need to, but you get one-half times root 2 over 2,- root 3 over 2 times, root 2 over 2. Clean that up again, combine things and you get a very similar root 2 over 4, (1- square root of 3), okay? So this is just working with these two formulas, all the other identities that we're going to need are obtained from these core values. For example, if you wanted to know what the formula for the tangent of two things were, so maybe we can work that out, again I would never want you to like know this or memorize this, but just to show you because you'll see it on the big scary sheet. If I gave you tangent of two things, well, I'm not going to look that up, I'm not going to do that work, I'll just use my formulas for sine, and I'll use inside over cosine, so I'll just do this. Yeah, it's going to be a little longer, and sure if I do a lot of algebra I could work it out and clean it up, and if you want to go look like the cleanup formula, I'll leave it as an exercise if you want. But you can check, this will turn out to be tangent and this is what you see, sometimes tangent of x1 plus or minus tangent of x2 over, 1 minus plus from the cosine, tangent of x1, tangent of x2. So this is where I'm going to say it 1000 times like yes this exists, but I don't think about it this way, I don't work with this, I don't have this memorized, I wouldn't even care about this, I would just do this. The cost of doing this, of not memorizing these shortcuts is that I have. Two sort of things to calculate, sine of a numerator and cosine of a dominator and I've got to use a formula choice. But I'm happy to do that and don't mind doing that 'cause as long as I keep coming back to sign in cosine, I can answer every single question they throw at me. In particular if you want to just go through this we can look at a tangent of pi over 12, remember pi over 12, we've seen this before. Pi over 12 you can write as pi over 3- pi over 4, and you say, well, do I need to know tangent? Do I need to know this angle? No just write it as sine over cosine just write over cosine we have the values for sine cosine, so let's write sine of pi over 12 divided by cosine of pi over 12. Now we worked outside apt 12 in the last slide, remember that was root 2 over 4 parenthesis 3- 1, so the work is going to come in and finding the denominator, so maybe I'll skip ahead here and do that. So root 2 over 4, root 3- 1, we did that one already, so the work is going to come into finding cosine of pi over 12. So let's go off on the side, we'll do a little scrap work here, let's go find cosine of pi over 12, cosine pi over 12 we're going to write this as cosine of pi over 3- pi over, this is our difference formula here. So here we go, remember it's cosine of the first, times cosine a second, now cosine have to change the sign says plus, sine to the first, sine to the second. All these values are known, so cosine of pi over 3 is a half, power 4 or root 2 over 2 plus sine of pi over 3 that is, root 3 over 2, and then cosine pi over 4 is root 2 over 2. Three all together, and factor you get root 2 over 4, 1 + root 3, so I'll take that scrap work, and I'll go back to my fraction and I get root 2 over 4, 1 + square root of 3. What's nice about this is that the square root of 2 over 4 square root of 2 over 4 they will cancel, so say number numerator and the nominator cancel, and you're left with a cleaner version of root 3- 1 over, root 3 + 1. So notice I never needed this fancy formula. Yes, they are nice to have, but you don't need, you absolutely don't need them final answer root 3- 1 over root 3 + 1. If you want checking on a calculator, make sure it's in radians and you can compare that the decimals are absolutely the same. There's another set of formulas identities that come with these sum and difference formulas, they're called double angle formulas, and they so let me write down the first one here. Let's write this down as x + x, it happened when the two values are the same. So like what if I wanted to add these two things together? So in that case, from our identity remember its sine of the first, cosine of the second, now sine keeps the same sign so + and then sine of the second cosine like they are the same. They are exactly the same, so when you put it all together you get 2sinx cosx. This guy comes up a lot, I'm going to write an x plus x you atend to write that a sine of 2x. But notice how like if I know the first one, I get the second one for free, so this one comes up enough that I'm going to put it down, but it's an immediate consequences of the sum formula for sine, its immediate, which is kind of nice. So if we do the same thing for cosine, so like what happens if I take the same thing for cosine and I want to add x + x, is also cosine of 2x you get cosine of the first cosine of the second, gotta put a minus sign there now, and then sin of the first sin of the second. Cosine times, cosine is cosine x squared,- sin square root of x, and so this is another way to see the identity cosine of 2x. What's interesting about this one is that you can write in a bunch of different ways. It's very versatile, so short cosine of 2x is cosine squared, x- sin squared x. Because it's a minus sign, be careful order matters. You have to have cosine squared first, but if you combine this because now you're starting to see sin of square root of cosine squared, remember you have your pythagorean identity that says cosine squared x + sin squared x is 1, so you can rearrange some things. This means you can write this as cosine squared, x is 1 minus sin squared x, and you can certainly rearrange this as sine squared x = 1- cosine squared, so these are all these are all equivalent. And you can plug these in and substitute, and sometimes you see it that way as well. So let's substitute the first one in for cosine, so you can write this as 1 minus sin squared x, and then minus sin squared x, and when you put that together, you have one minus sin squared minus sin squared, this becomes 1 minus, There's 2 sine squared x. And that's another equivalent way to see cosine of 2x. So I'll summarize that in a second, but I want you to see these both ways, so you can substitute for cosine squared. Or you can substitute for sine squared. And if you do that you get cosine squared of x minus, and watch out for parentheses here. 1- cosine squared of x. Clean that up and one of the minus sign is going to distribute to both pieces so you get cosine squared plus cosine squared. So put that upfront, that's 2cosine squared of x- 1. So there's three ways to see this, just depending on what flavor you want it. Maybe I'll summarize it down here, so cosine of 2x is, using the sum formula for x + x cosine squared x minus sign squared x. Perfectly good way to do it. It's also absolutely equivalent to 1- 2 sine squared x. And also equivalent to 2cosine squared x- 1. So there's a couple ways, a couple flavors that this formula can come in. Pick your poison, whatever one you need, whatever looks like at this could, sine of 2x cosine 2 cosine x. This one comes up a bunch. We're going to use this one later, so keep these in mind as you go through. Let's do a couple of examples with these in the more rapid ease up. So let's solve for x. And let's put x, just keep it one lap around the unit circle. So like 0 to 2 pi and let's do as our first example cosine of x is sine of 2x. All right, so now our strategy for here is going to be to write things similar, but it's already in terms of sine and cosine. It's already in terms of sine and cosine. So what's the next strategy to look at? Well, sine is with a 2x, so I have a double and cosine is with a single. Once everything is turned to sine and cosine, you want to get things out of double. You want to make the insides match so that you can cancel stuff or do whatever you want to do. Sine a 2x is our double angle formula, we just saw this. So we really want our sine it as we rewrite this as cosine of x is 2 sine x cosine of x. Very good. Okay, so be careful here. Some folks want to divide by cosine of x. It's a little dangerous and the reason is because if you divide by a fraction, it could be zero. So I would caution against that. You're going to lose some solutions here. Instead, why don't we write this as cosine x- 2sine x cosine x? So don't cancel functions. Move it over to zero. Factor out cosine, 1- 2 sine of x is 0. And then you have two things that multiply together that give you zero. So that means cosine of x is 0 or 1- 2. Sine of x is 0, cosine of x is 0. Where does that happen? That occurs at x equals, remember cosine of the value zero when x is zero, top of the circle? Pi/2 or bottom of the circle, 3Pi/2. We just do a one lap around, so just those two values. The other one, if you do some manipulation you get sine of x equals two a half. There you're going to get two values there as well. We've seen these before, pi/6 or 5pi/6. Okay, so corresponds to quadrants one and two. So we actually have four solutions, four solutions total to this. And you can go back and plug them in and check, they all work. How do you divide it out and cancel the cosine x with the last two of the four. Be careful, don't do that. Same thing, let's do another example. Let's do let's practice our cosines. So again, solve for x. Let's go 0 to 2pi. And I want a sine find out solutions sine of x is cosine of 2x. Now cosine of 2x, there is many ways to do. We have three versions of this double formula here. Same idea, I don't want it as a double. The one on the left side is a single. So let's convert this, I think to put in terms of sine, it possible. So why don't we pick any of the ones with sine and wouldn't it be nice if we had only sines? Again, there's probably a few ways to do this. But I'm going to think of this as sine of x and I'm going to pick the identity where I have absolutely no cosine. So it's like can I get it in terms of just sine would now be better. The one I'm going to use here, of course, is 2sine squared of x. Move things around, same idea. Let's write this as sine of x- 1 + 2sine squared of x. And will set everything equal to 0. It's a little weird to see this in out of order. So let's rearrange so 2sine squared x, will put the higher degree upfront plus sine x of second. Minus 1 is 0. And now this may not be obvious, but stare at this for a minute. This factors sine of x+1 and 2 sine of x-1. And this is all equal to 0. So with two things multiplied together to equal 0, we're going to get to that sine of x+1 is 0 or 2 sine of x-1 is 0, which means that sine of x is -1 and 2 sine of x is positive 1 or sine of x is one-half. Sine of x is -1 that of course occurs, think of the y value. This bottom of the circle that's at 3 pi over 2 and sine of x equals a half. So we just saw that's x equals pi over 6 or 5 pi over 6. So in this particular example, there are three solutions that make or solve this expression. Let's do one more just to practice our formulas. Let's solve for x same thing, so solve for x between 0 and 2 pi, so one lap around the circle, 1 equals sine of x + cosine of x. Now this equation, this one doesn't look like we're going to use some form as we know, there's nothing squared in this example. So sometimes it math when you don't have the thing you want, you'll force it there. So I would really love for stuff to have squares so I can use either Pythagorean identities or some other formulas that have sine or cosine squared. So what's the way to make squares appear when they aren't there? Well, you can square both sides. When you square both sides is perfectly legal thing to do, you get 1 squared is equal to (sine of x + cosine of x) squared. This is tricky because most students don't think about this or don't try this first, but you can certainly square both sides. When you do that again, whatever you want to do the other side you get 1, 1 squared is 1 we're going to foil this out. So it's like first sine squared of x outside inside plus 2 sine of x cosine of x plus cosine squared of x. And now you can see why this probably was a good idea, because now I have the Pythagorean identity applying to the first and last term. So sine squared, plus cosine squared is 1 and then all of a sudden I have my double angle, right? This is kind of nice so you get both things appear. 2 sine and cosine x is my double angle so plus sine(2x). So a very clever move, very cheeky move. It's kind of cool, but once you see it, you keep this little strategy in your back pocket. The 1's cancel, so subtract 1 from both sides, and basically this question that looks tough to begin with, simplifies to 0 = sine of 2x. So to have sine of 2x equal to 0, we need 2x, then has to be so like sine, it's either add 0 or pi. We'll say pi or 2 pi since we included 2 pi in our solution set. So just set up all the equations, 2x = 0, 2x = pi, 2x = 2 pi. That gives you that x = 0, x = pi over 2, and x = pi. These will turn out to be all of our solutions is never a bad idea to go back and check that they actually work, and I leave that as an exercise. So check it two ways if you want just because it's such a neat little example, graph this thing. And see the graphic at a handle of it, and then of course go back and plug in or substitute. So I leave that as an exercise, if you didn't see this strategy to sort of square things again, I wouldn't expect you to until someone shows it to you. But this little tricky, square it, keep in mind, user identities, combine identities as needed and solve for your solutions. All right, great job on this one, try some more and just keep at them. And remember if you ever stuck, think of some things outside the box. Check graph plugin, have fun, no bad ideas and brainstorm. Try stuff, try stuff, try stuff. All right, good job on this one I will see you next time.
