 In this video, we'll demonstrate how we can use the law of cosines to solve a triangle with the side-side-side criteria here. That is to say, we know the three sides of the triangle, and we'll then find the three angles of the triangle using the law of cosines. So let's say that the side opposite of angle A is 34 kilometers, the side opposite angle B is of 20 kilometers, and the side opposite angle C is 18 kilometers. We need to find the three angles here. And so it doesn't matter really which angle we choose whatsoever. We're going to use the law of cosines to find out the missing angle. I'm personally going to choose angle A because angle A is the largest angle. How do I know that? If I look at its opposite side, that's 34 kilometers. The bigger the side, the bigger the angle. The bigger the angle, the bigger the side. These things are directly proportional to each other. So the biggest side will have the biggest angle. And when you have an oblique triangle, because the angle sum adds up to 180 degrees, you can have only at most one of two-sangle or one right angle. Two of the angles have to at least be acute, and the two smaller ones will be acute angles. And so later on in the problem, we could solve all of these using the law of cosines, but it turns out the law of signs is typically easier. We use it when we can. And so the law of signs can't tell the difference between an acute angle and a two-sangle. So therefore, if I can take care of angle A using the law of cosines, then the other two angles I know are going to be acute. Angle B has to be acute, angle C has to be acute. So it avoids some technicalities that arise using the law of signs. So how are we going to use the law of cosines here? Which version? Well, since we're focusing on angle A, this will look like A squared is equal to B squared plus C squared minus 2 BC cosine of angle A. And plug in the information we know. A is 34. B is 20. C is 18. Of course, if you swap 18, if you swap the B and C around, it makes really no difference whatsoever. Minus 2 times 20 times 18 times cosine of A. So let's try to solve this thing going forward here. 34 squared for you to calculate to help you with arithmetic if you need to. 34 squared is 1156. We're going to then get 20 squared, which is 400. And then 18 squared is 324. Like so. And then we're going to take 2 times 20 times 18. That gives us 720. And then cosine of A, we need to solve for the angle A like so. And so what I'm going to do is I'm actually going to, since we have a negative in front of the cosine of A, I'm actually going to move to the right hand side of the equation. We'll move the 1155 to the other side as well by subtraction. So we get 720 cosine of A. This is equal to 400 plus 324. That's 724. And then we're subtracting from this 1156, which then gives you negative 432. Don't really worry that this is negative at all. Not a big deal. Cosine doesn't care about that. It just means that the angle is going to be obtuse. Because the cosine ratio, if it's negative, means we're in the second quadrant. Not a big deal. Divide both sides by 720. This gives us that cosine of A is equal to negative 432 over 720. We can simplify this fraction. 432 and 720 are both even numbers. Turns out they actually simplify dramatically. In the end, it simplifies just to be negative 3-fifths. And then to solve for A, we will take cosine inverse of 3-fifths. If you want your answers, it doesn't actually say the angle is here. Pay attention to your calculator. Are you in degrees or radians? I'm going to do this one in degrees. If your calculator is in degree mode, you can take cosine inverse of negative 3-fifths. Don't forget the negative sign. You would end up with approximately, we're on to the nearest degree, 127 degrees like so. The negative sign is important. If you take cosine inverse of a negative value, you will get something in the second quadrant. It'll be something between 180 degrees and 90 degrees right there. So it did turn out that angle A was obtuse. That's great information here. And so when we add that to our triangle, we have 127 degrees right here. Now we have an angle in opposite side pair. So we have an AOS. To find the other two angles, B and C, we could use the law of cosines again. And we would do exactly what we saw on the screen just a moment ago. That's a perfectly great strategy. I'm perfectly happy with that. No big deal whatsoever, but be aware that we're going to use the law of sines to try to do this thing a lot simpler for us. So to do the law of sines, let's find angle B next. So we get sine of B over little b is equal to sine of A over little a. For what information do we know right here? So we don't know sine of B because we don't know B. So we need to come, we'll come back to that. We do know the side length B, which of course is 20. We know the side length for A was 34. That's all given to us. Do we know sine of A? Well, we know A sort of right 127 degrees, but we actually do know sine of A. Because we know cosine inverse is negative three fifths are better yet cosine of A is equal to negative three fifths. We could use the approximation of 127, but when you use an approximation, then the errors can kind of compound over time. To make life a little bit cleaner for us, we can avoid this approximation entirely by just doing some type of right triangle diagram, which in this diagram, we're thinking this is angle A. We're going to do a right triangle with angle A here. The cosine ratio is negative three over five, like so, which then by the Pythagorean relationship since we're in the second quadrant, then the other side would be four. So this is just a three, four, five triangle. And so if we did sine of A in that situation, sine of A would actually be four, excuse me, sine of A would be four over five, like so. And so we can use that in this situation. It turns out we could do sine of A to avoid some unnecessary rounding problems. It's just going to be four fifths, which four fifths, of course, would be the same thing in this situation as four over five times 34, which would come back to that in a second. Let's cross multiply. We end up with five times 34 times sine of B. This is equal to four times 20. Like so I'm delaying the multiplication till I actually need to do it. Sort of a lazy computation, but it's an efficient thing to do. Divide both sides by five times 34. We end up with sine of B is equal to four times 20 over five times 34. And this is sort of the advantage of delaying multiplication here because I can actually cancel things across the fraction bar. For example, five goes into four times four times four over 34. 34 is two times 17. So you end up with eight over 17. We actually never needed to multiply those things out until the very end. We just needed two times four there. And so now we can do sine inverse of 817. That's going to give us angle B right here. That's the exact value of sine inverse of 817s, which will be approximately 28 degrees like so. We found A which approximately 127. We find B which is 28. And so the thing is, B doesn't have any worse rounding than A does, right? So A we rounded to the nearest degree, but there is some error there. It's not exactly 127 degrees. It's exactly arc cosine of negative three-fifths. As such, B is not, it's the same thing there. It's not perfectly 28 degrees. We rounded it. The exact value for B is sine inverse of 817s. To avoid the compounding errors here, we didn't want the rounding error of A to infect rounding B to make it even worse, right? So we did this exact value here. Now to find C, because A has a little bit of error, B has that same amount of little error, not a lot. To find C, we can just take C to equal 180 degrees minus A, which is 127, minus B, 28. The subtraction here isn't going to have a huge error. It's not going to be worse than what we had before. So we end up with 25 degrees for C like so. And so we're able to find all these values. What's it necessary to go through all this triangle business to get the exact value? Well, we like exact values because we don't like error. But you can check with your calculator that if you had done sine of 127 degrees right here, how would that affect things? Again, I'll let you try that on your own. But again, the rounding causes error. So using exact values helps us out here. And so being comfortable with exact values definitely helps us work through these problems and avoid the errors that a small miscalculation can cause.