 In this video, we'll demonstrate how we can solve the side angle side condition using the law of cosines. So side angle side means that we know two sides of the triangle. So let's say we know side B and side C. Those turn out to be 25 inches and 32 inches. And then we know the angle that's interior to the two sides. So in this case, that would be angle A and let's say we know that to be 60 degrees. Now I should first mention that we really like to use the law of signs. It's much simpler to use the law of cosines. But for the law of signs to work, we need to know an AOS, an angle opposite side. So the only angle we know is angle A, but we don't know side length A. So we can't solve using that AOS. If you try to look at B, we know the side B, we don't know the angle B. So we can't use that AOS and same thing with C. We don't know angle C, but we do know the side length of C there. So we're quite stuck there. If we want to know an angle, well, if we know two of the angles, we can find out the third one, but we only have one angle. So the law of signs cannot be utilized yet in this problem. But we can start off by using the law of cosines. Now to use the law of cosines here, let me point out to you that we know angle A. And so we're going to use the law, the version of law of cosines associated to angle A. And that one tells us that A squared plus, excuse me, A squared equals B squared plus C squared minus two times BC cosine of A. So it looks like the Pythagorean equation, A squared equals B squared plus C squared, but you have this correcting factor because it's not an actual right triangle, probably. In which case you have to subtract two times BC, the two numbers which are adding together their squares, and then the cosine of the angle of the square that's all by itself. So let's plug in the information we know here. So we don't know little A, so it's just going to be A squared for the moment. We do know little B, that's 25. We do know little C, that was 32, and minus two times. Well, B was 25, C was 32. Now the good news is if you forgot which one was which, B or C, it doesn't matter. The symmetry of the problem, you'll be just fine. And then we get cosine of 60 degrees like so. And so let's try to compute this thing the best we can. 25 squared is going to be 625. Feel free to use a calculator with the arithmetic and the trigonometry here if you need to. 32 squared is going to be 1024. Next we're, of course, going to get 2 times 25 times 32. 2 times 25 is, of course, 50. 32 is just 2 times 16. So if you borrow one of the twos, you're going to get, of course, 2 times 50, which is 100. That makes arithmetic a lot easier if you do multiplication by 100 there. And then cosine of 60 degrees. Cosine of 60 degrees is actually equal to one half. So that saves us a lot of effort right there. So you get 625 plus 1024 that adds up together to give you 1649. And then a half of 16 is 8. 8 times 100 is 800. So we end up with 1649 minus 800. Their difference turns out to be 849. Like so. That gives us a squared. To find a, we need to take the square root. So you take the square root of 849. That would be the exact value, but let's approximate that. And if we do that to, to one decimal place, the approximation would be 29.1 inches. And so this is the length a, like so. So with that measurement in mind, we can then solve for the remaining information of this triangle. So the other thing we need to know is angle B and angle C. Now we do have now an opposite side angle pair and AOS here. We could use the law of coast, I love signs, excuse me to solve for the remaining angles. That would be perfectly acceptable. And I think we can demonstrate how both of those will work in this exercise here. But the problem is, if we're going to use this AOS, angle side A is an approximation. We have to worry about compounding variables. So if you're going to do that approach, you need to use the exact value and approximate it at the very end. And so let's see how that would work. If we want to find, for example, angle B, what we would get there is we're going to get sine of B over 25 is equal to sine of A over, of course, A, the little A, which is the square root of 849, like so. B we don't know. A we do. So we can actually put in a 60 degrees in for that, of course, sine of 60 degrees. So to solve for sine of B, we can just times the size by 25. So we get 25 times sine of 60. Sine of 60 degrees is going to equal root 3 over 2. And then we have this square root of 849 in the denominator. This would then give us that angle B is equal to arc sine of 25 over 2 times the square root of 3 over 849, which you can reduce that fraction down if you want to. But we're just going to throw this all in the calculator anyways. Not a big deal. You get approximately, if you round to the nearest tenth of a degree, you'd get approximately 48.0 degrees for B, like so. And then once you know that A was given as 60 degrees and since we just found out B was 48 degrees, you can then find the measure of angle C, of course, to equal 180 degrees minus 60 degrees minus 48 degrees like so. And so we get that C would be 72.0 degrees as well. You can find out the missing information that way. And so the law of sines can be very useful to find out the remaining bit of the triangle. Once we use the law of cosines first, the law of cosines was unavoidable. But like I said, there's two cautions you need to use with the law of sines, one of which already was mentioned. That is, you're better off using the real value of A, not the approximation till the very end when you throw it all into your calculator, right? One issue you have to be very cautious about is that when you do sine inverse on your calculator, sine can't tell the difference between an acute angle and an obtuse angle, right? That is, sine can't tell the difference between an angle in the first quadrant and the second quadrant. And so when you consult your calculator and calculate sine inverse of this ratio here, it'll give you the reference angle in the first quadrant. But what if it was the angle in the second quadrant instead? What if it's an obtuse angle? This is the problem because we don't actually know. We're not in a right triangle right now. These oblique triangles could have obtuse angles. There is a reason why I started with angle B using the law of sines right here instead of using angle C instead. Look at their opposite sides. A we now know is 29.1. B was given as 25 and C was given as 32. C has a longer opposite side and because of the law of sines, the longer the opposite side is, the bigger the angle is. The bigger angle coincides with the bigger length and the bigger length coincides with the bigger angle. So the biggest angle of this triangle would have to be angle C because the biggest side is opposite of that. And now if a oblique triangle has an obtuse angle, it has only one obtuse angle. So I guess what I'm trying to say is because the side length of B is smaller, I know absolutely, I know that angle B is going to be an acute angle. It has to be. Angle C could be obtuse or it could be acute like it was in this case. We're not sure. So when we have a choice of angles to solve for, like we could choose angle B or angle C, and we're using the law of sines to do so, it's best to solve for the smaller angle. And this is because the triangle can have a most one obtuse angle, which if it's present, must be opposite the longer side. It cannot be opposite of the smaller side. And this is because sine is positive in both the first quadrant and the second quadrant, which given a triangle, the angles have to be acute, right or obtuse. So they have to lie in the first or second quadrant. Inverse sine can't tell the difference between an acute angle and obtuse angle. Now, I should mention as an alternative approach, we could have actually used the law of cosines to solve for angle B because angle B can tell the difference. Excuse me. Cosine inverse can tell the difference between an obtuse angle and an acute angle. So one of the advantages of using the law of cosines again is that you don't have to worry about this. Do I pick the smaller angle or not? In that situation, you'd get B squared equals A squared plus C squared minus 2AC cosine of B. Like so, little b, we know it's 25. Little a, we know it was the square, but I'm going to use the exact value here. We have the square root of 849. Notice when you square the square root, you'll just get back 849. And then we have 32 squared, like so. Then we're going to get minus 2 times the square root of 849 times C, which was 32 times cosine of B, like so. Squaring these things, we squared 25 before we got 625. When you square the square root of 849, you'll just get 849, like I mentioned. 32 squared is 1024. We already know that. 1024. Then we subtract from that 2 times 32 is 64. We'll just leave the square root there for a moment because what else can we do with it? And then we have cosine of B. So now it's our goal to solve for cosine of B right here. Of course, since we have this negative sign, I'm actually going to move the cosine of B to the other side. And we're also going to subtract 625 from both sides as well. So we get 64 times the square root of 849 cosine of B on the left-hand side now. So we get 849 plus 1048. That's 1873 and we have to subtract from that 625, which that simplifies to be 1248, like so. Next, we're going to divide both sides of the equation by 64 times the square root of 849. Again, we're keeping the square root as long as possible so we don't get any rounding errors here. That should be 649 right there. Sorry. 849 as well. And so then we get cosine of B is equal to 1248 over 64 times the square root of 849. We can simplify that for action if possible. It turns out that 32 does go into 1248 39 times. So it simplifies to be 39 over 2 times the square root of 849. And so then that tells us the measure of angle B is going to equal cosine inverse of this value. 39 over 2 times the square root of 849, which like we saw before, this would give us 48.0 degrees. Then you can find out C with 72 by subtraction as well. So we'll get the same angle measurement for B either way. And so there are some advantages and disadvantages of these two different approaches. Consider them for a moment, seeing them both on the screen simultaneously. I want to definitely argue that using the law of signs is a simpler algebraic approach to finding angle B than the law of cosines. But the thing is, in both cases, you're going to want to use the exact value for angle or for length A to avoid rounding mistakes. But the thing is when you're using the law of signs, remember, you have to choose the smaller angle first so you don't make a mistake. And you can see that by looking at the smaller of the two sides, B with shorter than C. So that's the only real thing you have to watch out for. So you can use the law of cosines if you don't want to worry about this concern whatsoever. But if you use the law of signs, make sure you're focusing on the smaller angle. This is the whole idea behind the so-called ambiguous case, because there are some possibilities, because sign inverse can't tell the difference between acute and obtuse angles. So you have to watch out for that. In this situation, of course, where we're solving the side angle, side situation, just choose the smaller side if you want to use the law of signs or just use law of cosines and you don't have to worry about it.