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 cosines 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 going to 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 SAS 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 going to have to use the law of cosines.
So that actually brings us to our second step here, which is we're going to 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 both sides, then you would just find any angle. Alright? So we're going to go ahead and work out c over here because we're dealing with an SAS triangle, and I'm just going to go ahead and find the third side, which is really just that little c variable. So if I'm solving for c, then basically what I would do is go to my law of cosines over here, and I'm going to have to use this equation here that's for c squared. And I'm just going to 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 2 = 4 + 3 2 + 2 × 4 × 3 × cos ( 60 ° ) . Alright? I'm going to 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 going to solve for, that c is equal to square root of 13. So we're basically done with step 2a. And now what happens is we're just going to use the 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 going to do in this third step is you're going to use the law of cosines to find a second angle. So that's what we're going to 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 going to go ahead and pick solve and and solve for A. So if I want to find A in this third step, then what I'm going to do is I'm going to 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 going to have that a squared actually, let's do this over her