 In this video, we're going to solve the oblique triangle ABC if we're given the side length A, which is 54 centimeters, the side length B, which is 62 centimeters, and the angle measure A, which is given as 40 degrees, as illustrated in our diagram. I want to point out to you that the criteria we're given is the side-side angle scenario, which is commonly referred to as the ambiguous case. We have to be cautious here because if we're not cautious, we might get the wrong solution in the end. There could be no solutions. One solution are two solutions to the ambiguous case. So we'll investigate them using the law of signs. Why do we use the law of signs? Well, because we have angle A and we have side length A, we have an angle opposite side pair in AOS. So whenever you have an AOS, we can use the law of signs to help us find the missing information. For example, we have side length B. We can then use the law of signs to find the angle B in the situation. So the law of signs here tells us that sign B over little b is equal to sign A over little a. We want to solve for b, so times both sides by little b, that is, we're solving for angle b. So sign of b is equal to b over a times sign of a, and we're going to compute this value here. Little b is 62, little a is 54, and sign of a, we're going to take sign of 40 degrees. We're going to use our calculator to help us out here. Make sure your calculator is in degree mode and feel free to follow along with me. If you plug in 62 divided by 54 and times that by sign of 40 degrees, you'll end up with 0.74 as your ratio. Now this is a critical step when you're working with the big U.S. case. You have to check, is this an acceptable sign ratio? The thing to remember here is that sign of b, like any sign function, it must be less than or equal to one, but it must be greater than equal to negative one here. 0.74 falls inside that range, so that means there will be a solution to this triangle, and so we proceed forward to find it. Of course, we're going to compute b here using, of course, sign inverse on our calculator. b will equal sign inverse of, well, 62 over 54. That fraction does reduce to be 31 over 27 if anyone cares about that right now. Sign of 40 degrees, this is the exact value of our function, but an approximate value will be very appropriate here. When we take sign inverse of 0.74, we end up with 48 degrees right here. But when you're working with this ambiguous case, like I said earlier, you could have one solution, you could have two solutions. We have to really consider both situations. So we have 48 degrees, that's one possibility, and that one will work. We'll finish that one up in just a second here. But the thing is, when it comes to the sign function, sign can't tell the difference between an angle and its supplement. That is to say, sign can't tell the difference between an acute angle or an obtuse angle, because in the first quadrant or in the second quadrant, sign gives you, in both cases, a positive ratio. So we have to consider the possibility that b is an obtuse angle. That is, we have to compute 180 degrees, take away 48 degrees that we just found right here, which of course, if we have 180 degrees and we take away 48 degrees, that would give you 132 degrees. We have to consider both of these possibilities. So let's consider the first possibility where b is equal to 48 degrees. Since we know a is 40 degrees, and we now know that b is 48, this tells us that angle c will equal 180 degrees. Take away 40, that's going to be a, and take away b, which is 48, and that will then tell us that c is equal to 92 degrees. Like so. So then proceeding from here, we can then use the law of signs to find little c, because little c over sign of capital C, this will equal little a over sign of capital A. I would always recommend you use the original AOS of a to solve this to avoid any rounding concerns. So this tells us that little c is equal to a over sign of c over sign of a. Let's compute these values here. Little c equals little a, which is 54 times sign of c, which was 92 degrees, and we divide that by sign of 40 degrees, for which then we use our calculator here, and we get that c equals, again, make sure we're in degree measure for this. We're going to compute a 45 times, excuse me, 54 times sign of 92 divided by sign of 40 degrees, and you'll get approximately 84 centimeters right here. So let's put in the information we found. Okay. So if b is an acute angle, we'll get that it is 48 degrees. Let me try that again. So it's a little bit more legible, 48 degrees. That would then imply that c is 92 degrees, and then the remaining side would be 84 centimeters like so. That would then be, that's one possible triangle here. And so I'm actually going to sketch it down below here so we can see this thing compared to what the other solution is going to look like here. So if we have our triangle, something like this, again, labeling everything, we have angle a, angle b, angle c. So all the information we found out here, I'm going to put in green the original information. So we were given 40 degrees for that 54 centimeters for this side, and then 62 centimeters for this side. And then we discovered that b was 48 degrees, c was 92 degrees, and that little c was 84 centimeters like so. That's if b is an acute angle. But what if b is an obtuse angle? We have to consider that case. If we just ignore it, there could be a second triangle we have to consider. Well, if b is 132 degrees, a is still 40 degrees. There's no variability there. What would c then be? Well, let's compute c. c equals 180 degrees. Take away a, take away b, right? In which case, when we do that calculation, you take 40 degrees away from c, excuse me, take 40 degrees away from 180. You'll get, of course, 140. Take away 132. That leaves c being 8 degrees, which is a positive angle measure. And this is the critical thing here. If your angle measure for c turns out to be negative, that would mean that this obtuse case is impossible. And the one triangle we found already is the only solution. But in this situation, c ended up with a positive angle measure that actually tells us that there is a legitimate second solution. This is an example of the ambiguous case that has two distinct solutions. Once we have angle c as 8 degrees, we can then use the law of signs, just like we did over here to, I'm actually going to separate there. We're going to use the law of signs again to find little c. So we're going to take sign here. Now of 8 degrees, this will equal 54 over sign of 40 degrees. We use the exact same values for a right here. A little c then becomes 54 over sign of 8 degrees over sign of 40 degrees. Your calculator is going to be your best friend right here. Again, make sure you're in degree mode for this calculation. 54 times sign of 8 degrees divided by sign of 40 degrees. This will end up with 12 centimeters as your final answer. And so then we get our second possibility, which I want to draw it side by side with this one right here. I'll draw a little bit. Let's put it over here so it's a little bit easier to see. So we end up with something like this, which in this situation, we have angles a, b, and c. You'll notice I drew my diagram a little bit differently here because this is the situation where b is obtuse. So the initial information, a is still 40 degrees. This side right here is still 54 centimeters. This side over here is still 62 centimeters. But what we discovered in this situation is that a or b, excuse me, turned out to be 132 degrees. C then turned out to be 8 degrees. It was teeny tiny. And then the other side down below turned out to be 12 centimeters like so. And so given the initial information, side side angle, 54, 62 centimeters and 40 degrees, it turns out there are two possible triangles that solve this ambiguous case. And this is why the case is called ambiguous. Just given us information of side side angle, we don't know if there's zero solutions, one solutions or two solutions. And in this video, we see an example where the ambiguous case ended up with two distinct oblique triangle solutions.