# PC 12.2 Law of Cosines

## PreCalculus

They were talking unit 12 section two. This is mister Kelly from bomb holder, talking about the law of cosines. Let's set it up for us. We got the brush disorders. That is Nacho average dinosaur. Mister Bean is an avid dinosaur hunter. In fact, his entire bedroom is decked out in dino sheets and posters. This is actually a true story. He decides to pursue his life passion, the brust asaurus. All right, so being to learn more about the breast disorders by examining its step angle, the closer the angle is to 180°, the more efficiently the dinosaur walked. That's something we know. The following breast prints were found by an ancient nachos stand. Nice little picture there. We can use the law of cosines to find the step angle. Okay, so the law of cosines, little bit more complicated than the law of signs, but not so complicated as, say, like the quadratic formula with square roots and all that kind of stuff. All right, so here's how basically works. And I want you to focus on a couple of things. We have basically, if you name a triangle, ABC, we have three different formulas that exist. And there are all the same except for the letters kind of switch around. So if you're looking for a side length, then it's this first left hand column here that you'll be using. If you're looking for an angle measure, then you'll use the right hand column. And if you notice, on the left, that you're going to take the cosine of whatever angle it is that you're solving for on the left. So if you want a squared, then you're taking the cosine of a the other two side lengths are squared and then added together, as well as you subtract two times whatever those side lengths are. Okay, that happens no matter what, so it would be noticed we go down to B squared, use a and C and C squared use B and a, it takes cosine of C so on and so forth. The formulas on the right are for finding the angle measures. All right, so if you're looking for the cosine of a, which basically means you're looking for the measure of angle a, then that is the that's the side length that you subtract. Okay? If you're looking for the cosine of B, then you're subtracting B squared, cosine of C, you're subtracting C squared. When are we going to use the law of cosines? Well, we're going to use a lot cosines whenever we have side angle side information. Or side side side information. Those two situations will warrant the use of the law of cosines. Otherwise, we'll be using the law of sines which you just aced on the last mastery check. All right, so how do we use the law of cosines? Well, here we have a triangle. It's basically labeled force here where we have our side lengths B this would be a C, this would be a, remember it's opposite the angle. And so we want to find angle B so we're going to use the law of cosine. So the cosine of B is going to equal a squared plus C squared minus B squared. Remember that part that you subtract is always whatever the angle is. All over two times a times C so we just need to plug in all the numbers from this wonderful triangle here. So let's do that. All right, I think we figured that out loud here nicely. And we get to cosine of B is equal to one 55 squared plus one 97 squared. Minus B squared, so B is three 16. All over two times one 55 times one 97. Ha ha. What does all of that equal? Well, that all equals if you figure it all out, negative zero point what do we get 6 zero 6 7? I lied to you 6 zero 6 two. Pay attention here. So how do we find that? Let's bring up the calculator. All right, so we punch all the numbers in. You hit enter, you get negative points 6 zero 6 two. Like we said, how do you find, all right, what do we want to find? That's the cosine of B, how do we find B, we do inverse cosine. So you got to look for the button on your calculator that's cosine of the negative one of X and so that should be above the cosine button. So if you notice here, here's cosine in blue, it's inverse cosine, and I want to use that last answer that we just got there. Hit the button and we're going to get a 127.3 two, we'll put it at one 27.32. And that is in degrees, and it's the measure of angle a oh, I'm sorry, measure of angle B and we're all done. That is the law of cosine. You just plug it in and figure it out. All right, so first example, solve for X this is a side length, so we're going to use the left hand column. One of these formulas here on the left hand side. So what I need to do, whoops, I'm going back and forth. What I need to do is I need to label what do we have a, B, and C, so if we label our side lengths, we have a is here, and B is the 11 and C would be 12, so I want to solve for a, so H squared is going to equal. It's going to be the other two side lengths, B squared plus C squared, minus twice the product of those two, so two times B times C times the cosine of whatever that angle is that's opposite the side length that you're looking for. So now it's just a plugin, the numbers type of deal. So B squared equals 11 squared plus 12 squared minus two times 11 times 12 times the cosine of 22, all right, we'll close that circle off there. Can we figure, what do we get at one 21 plus one 44? When we get 22 times 12 is two 64. Times the cosine of 22. And if I simplify this line, it's going to reduce to 20.2234. And that's what a squared is equal to. So what's the inverse of squaring a variable? You need to take square root. So square root of each side, we're going to get a equals 4.497, which we're going to round that to the nearest tenth. So we're going to call it 4.5. All right, there you go. That is the law of cosines solving for a side length. Let's look at an angle now. So an angle measure says solve the whole thing. We're looking for more than one angle measure. We're going to have to look for two angle measures. So let's start with, what do you want to start with? Let's start with Z because we love Z so the cosine of Z, we're going to use the right hand column here. The cosine of Z is going to equal. X squared plus Y squared, minus Z squared, all over two X, Y all right, so Z is 17. We have X equal to 15, and we have Y equal to 23. Plug in the numbers. Here we go. So 15 squared. Plus 23 squared -17 squared, all over two times 15 times 23. This is fun stuff. So if I figure that all out, we get cosine Z is equal to .6739. We're going to use our inverse cosine function here. So inverse cosine, and I have in my calculator, I'm going to take the last answer. And we're going to get 47.6°. That's going to be the measure of angle Z 47.6°. Now, let's put that in here. 47.6°. Why do I have to do to find one of the other? Okay, so let's focus our attention on X now. So the cosine of X, what does that equal? That's going to equal Y squared plus Z squared minus X squared all over two yz, so when we plug in all those numbers. We get cosine of X equals .7583, do the inverse cosine. And you'll get X equals 40.6843. All right, so I'm going to write that down up here 40.68. What are we going nearest tenth? Let's just go 40 .7. We'll call it that. What would that make our third angle? Well, we need to add up, subtract from one 80. Why then equals 91.7 slightly obtuse there? All right, so now we know X, Y, Z, we know all three sides. So let's just write out our answers. When it says solve a triangle, it's a good idea. Take all your answers, write them all in one place. So the person knows the greater that's what we want. The greater they want to know what's going on. So we have three angles. We want to find the measure of each one of those. Plus we want to find side lengths, X, Y, and Z so we're going to write all those down. So the measure of angle X is going to be equal to 40.7 degrees measure of angle Y, we found to be 91.7° measure of angle Z equals 47.6° with X, Y, and Z, they gave us. So if you screw that up, then you should probably transfer to breasts class, not saying anything about breast. I mean, I'm just saying, you know, all right, done with that one. Okay, this is crazy because whenever they tell you to solve a triangle, you need every side length and every angle measure. So start off by using the law of cosines. I'm going to find the opposite side there. We work it through, I get 20.6. Okay, once you find that, then you can use the law of sines, which is a little bit easier, less complex, so I go through I find what angle C is equal to. So the sine of C over 21 equals the sine of 66 over 20.6. I use my last answer. I solve it, sine of C equals point 9 three two zero, and then what do we get for the measure of angle C I can't move down any? That's crazy. Oh, there it is. 68.8. All right, so then once I find that, you know what you can do, subtract from one 80 because all of the angle measures equal a 180°. So I add the two that I have, subtract from one 80, I get 45.2. So long story made long. Most people go for short there, but measure of angle a 66 degrees. Measure of angle B is 45.2° angle C is 68.8° a is, of course, 20.6 and B is 16 and C is 21. And that is it. This is the easiest video you will watch all year. It is so easy, even bruss could do it. This mister Kelly bauman, remember, it's nice to be important, more important to be nice.