13. Non-Right Triangles
Law of Cosines
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So in earlier videos, we learned the law of sines, and then we learned how to use the law of sines to solve 3 different types of triangles. Well, similarly, now that we've learned the law of cosines, I'm going to show you in this video how we can use the law of cos to solve the last two types of triangles you'll run into, the SAS and SSS triangles. I've come up with a list of steps to help you solve these types of problems, and, basically, what it is is we're just gonna use the law of cosines to solve for missing sides and angles. Let's just go ahead and jump right into this problem, and I'll show you how it works. So we have this triangle here in which we're given some information. Little a is 4, little b is 3, and big c is equal to 60 degrees. Now if you haven't already drawn this triangle or if it's not already shown to you, the first thing you'll do is sketch the triangle. But I've already done that for you. And what we can see here is that we have 2 sides, a and b, but not their corresponding angles, and then we have one angle, but not its corresponding side. So in other words, we never have a pair of a side and its corresponding angle. And if you look at this, what happens is we have a side and we have two sides, and we have an angle that's between them. So this is an s a s type triangle. In fact, we've actually seen something like this in a previous video. So remember what happens in these problems is we actually can't use the law of sines because it just doesn't work. We're gonna have to use the law of cosines. So that actually brings us to our second step here, which is we're gonna use the law of cosines to solve for some different variables. If you're using an SAS triangle, if you have an SAS triangle, then you'll use this to find that third side. If you have an SSS triangle because you have all the both sides, then you would just find any angle. Alright? So we're gonna go ahead and work out 2 a over here because we're dealing with an SAS triangle, and I'm just gonna go ahead and find the 3rd side, which is really just that little c variable. So if I'm solving for c, then basically what I would do is go go to my law of cosines over here, and I'm gonna have to use this equation here that's for c squared. And I'm just gonna write this out. Remember, this is just a squared plus b squared minus 2ab times the cosine of big c. We have every variable on the right side of this equation. So this is pretty simple, pretty straightforward. It's just plug and chug. So this is just going to be c squared equals. I've got a is equal to 4. So this is 4 squared plus 3 squared plus 2 times 4 times 3 times the cosine of angle c, which is 60 degrees. Alright? I'm gonna go ahead and clean this up a little bit. If you work out these two numbers, this is 16+9, which ends up being 25, and this is 2 times 4 times 3, which ends up being 24. And then what happens is this is the cosine of 60, which we can again reduce because this is basically a half. So this works out to be 25 minus 12, and that just equals 13. So that's what c squared is equal to. It's equal to 13. And what that means here is that c itself is equal to the square root of 13. Now you can go ahead and round this or sort of approximate this if you want. Plug this into your calculator. You'll get something like 3.6. But I always just caution against doing this. It's nice to leave things in terms of square roots, and you'll see why in just a second. Basically, you want to minimize the rounding errors. Okay? So that's the first variable we're gonna solve for, that c is equal to square root of 13. So we're basically done with step 2 a. And now what happens is we're just gonna use law of cosines again because no matter what type of triangle that you have, by this point, whether it's an SAS or SSS, you have all three sides that are solved for and at least one angle. So what you're gonna do in this third step is you're gonna use the law of cosines to find a second angle. So that's what we're gonna go ahead and do over here. It actually doesn't matter which angle that you solve for. It'll work out the same whether you're solving for a or b. I'm just gonna go ahead and pick solve and and solve for a. So if I wanna find a in this third step, then what I'm gonna do is I'm gonna have to go over here to my law of cosines and figure out which equation deals with the angle A. It's going to be the first one. The first variation of this formula is going to be that A squared. So we're gonna have that a squared actually, let's do this over here is equal to a b squared plus c squared. And this is minus 2bc times the cosine of big a. Now remember, unlike step 2a, where we actually solve for a side, here we're actually solving for an angle. And And so what happens is everything to the left of this sort of variable, all of these variables over here, we actually know. So I can just go ahead and plug and chug this. My a squared is gonna be 4 squared equals this is gonna be 3 squared plus c squared. Alright? So what is c squared? You may be tempted if you've plugged this in as a as if you've written this as a decimal to try and square this, but I'd like to warn you against that because, again, you rounded this number. So if you square it, then you're gonna introduce a bunch of rounding errors. We actually already know what what c squared is equal to because we solved for it. It's just 13. So I'm just going to plug this over here. That's why, again, it's nice to leave things in terms of square roots. So this is 3 squared plus 13 minuteus 2 times this is 3. And then c by itself, not c squared, is just equal to the square root of 13 times the cosine of a. Alright? So I'm just gonna go ahead and clean this up. I'm gonna subtract rescuared from each side and then 13. And then basically, what we're gonna see here is this is 16 minuteus 9 minuteus 13. And then on the right side, I'm going to get negative 2 times 3, so it's ends up being negative 6 times the square root of 13 times the cosine of a. Alright? Now what I'm gonna do is 16 minuteus 9 minuteus 13 ends up being negative 22. I'm sorry. This ends up being negative 6, and then we're gonna basically divide by negative 6 radical 13. So that cancels out on the right side. So I'm gonna have to do this on the left, and this is going to equal the cosine of a. Alright. So now basically what I can do here is I can say that a is equal to the inverse cosine of and what happens is this negative six will cancel out with the negative six on the bottom, and all you're left with is 1 over the square root of 13. Alright? So when you plug this into your calculator, 1 over the square root of 13 and take the inverse cosine, what you're gonna get is 73.9 degrees. Alright? So that is your angle or angle a. That's gonna be this one over here, 73.9. Notice how that actually kind of looks pretty consistent with the drawing here. Now one thing you might be wondering is, okay. Well, just like the sine inverse, don't we have to then figure out the second angle? Because, the inverse sine will give us 73.9 for this number, but there's also another angle that might get us this number. And, actually, no, because we're actually taking the inverse cosine here. So what I wanna sort of what I have a note here is that in step 3, or actually it's 2b as well, it's always better to use the law of cosines to solve for angles instead of the law of sines. And this is basically because the cosine inverse only yields one angle between 0 a 180 degrees. You don't have to worry about there being a second angle that you have to go and test to make sure that it works. So, actually, this is the only angle that works for this specific number. Okay? So we don't have to do any extra work. Alright? So now that we're done with step 3, basically, I'm just gonna move on to step number 4. And, really, this is from the the most straightforward step because by this point, you have 2 of the angles, so you can always solve for the third one just by using the anglesum formula. So this is going to be pretty straightforward. We've done this a bunch of times. A plus b plus c is equal to 180 degrees. All the angles have to add up to 180. We're now solving for the last angle, the one that you didn't solve for in step 3. And in this case, that's gonna be b. So in other words, we're gonna subtract both of these things from 180. 180 minus 60 degrees minus 73.9 degrees. And what you should get over here is 43.1 degrees. Alright? So that is 40 I'm sorry. This is not 43.46.1 degrees. This is 46.1 degrees, and that is your final angle that's missing. So 46.1. So with that, we've actually solved for all of the three remaining variables, square root of 13, 73.9 degrees, and 46.1. That's really all there is to it. So let me know if you have any questions, and let's get some practice.
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