- In this video, we're going to look at the law of cosines. And we're going to do two examples, one where you find a side length and one where you find an angle measure. And so here are the three forms of the law of cosines, basically the same thing, just rearranged. You might be intimidated when you look at it at first. But what you'll recognize is it's just describing the relationship between the three sides of a triangle, which we'll call a, b, and c, and then an angle, which it shows here, cosine of A. So notice a squared is here, that side length, and the angle opposite over here is cosine of A. If we have b squared over here, then there's cosine B. And then if there's c squared right here, it's cosine of C. 

The next thing I want you to notice is our general notation for sides and angles of a triangle. If I'm referring to this lowercase b as side b, then the angle opposite we'll refer to as angle B. Or here, it's showing a beta, second letter in the Greek alphabet. If you call this side a, then this will be angle A, or we could use angle alpha. And if this is side C, then this would be angle C, or we can use the Greek letter gamma. 

Now here's our first example. We're going to use the law of cosines to find a side length. And we're looking for this side right here. And while it isn't labeled, if we call this angle A, then we can call this side opposite it side a. So if we're thinking about our three different versions of the law of cosines, you might be thinking, which form of it are we going to use? Well, we're given angle A here, and we're looking for side a. So what's going to be the most helpful is this form right here. 

So let me just copy it down real quick. And now that we're here, let's fill in what we know. a is unknown. Angle A is 35 degrees. This right here we'd refer to as side c. And this right here we'd refer to as side b. So let me just substitute those into our formula. 

The great thing about this problem is there's not really anything to solve for. Really, all of this is ready to punch into your calculator. My one thing I'd warn you about is to make sure that your calculator is in degrees. When I punch that into my calculator, I got that a squared equals 13421.5281. Keep in mind, obviously, that's not reasonable to be a side length for this triangle. But we got to take the square root of both sides. 

And so whenever I take the square root of both sides, we get that our a value here equals roughly 115.85 units. So that's how to use law of cosines to find a side. Let's use law of cosines to now find an angle. In this particular example, I haven't designated yet which angle we're finding, but what you'll see sometimes is you'll see a question that asks you to find the largest angle or the smallest angle. So let's find the largest angle of this triangle. 

If we're finding the largest angle, you know the largest angle is going to be opposite the largest side. So we're actually looking for angle B here because side b is the largest side. So if we are looking for angle B, we need to use the form of our equation that uses angle B. So we're going to use this second form of our law of cosines right here. So let me copy it down real quick. 

And now if this is our side c, if this is our side a, and this is our side b, let's fill in everything we know into our equation. And remember that angle B is still our unknown. Now what we're going to do is we're going to solve for angle B. And so we've got to be careful about here is-- a common misconception is people will try to substitute all this into our calculator. 

We cannot do that because these are not like terms with this. This would be like adding 3 plus 2x. This is like our variable. It's our unknown. So since these are not like terms, I cannot combine them. So I'm going to simplify all of this together. And then I'm going to simplify those together. 30 squared plus 44 squared is going to be 2,836. 2 times 30 times 44 is 2,640. And then we still have our cosine B right here. 

So now once again, we cannot combine these. They're not like terms. I'm going to solve for our trig function. So I'm going to start by subtracting 2,836 from each side. And that's negative 235. And then I got to divide by the 2,640. 

Now over here, we have cosine of beta or cosine of B equals roughly 0.089. So our way that we actually find that is we do cosine inverse of 0.089. And we punch into our calculator, it's going to spit out angle B at us. And I get that the measure of angle B equals roughly-- we did a little bit of rounding, but 84.89 degrees.