 Welcome back to our lecture series Math 1060, Trigonomchi for students at Southern Intel University. As usual, I'll be your professor today, Dr. Andrew Misaline. In Chapter 8 in our lecture series, particularly in Lectures 24 and 25, we've been using the law of signs and the law of cosigns to help us solve various settings of oblique triangle. So we use the law of signs to help us out in the situation we have an angle side angle or an angle angle side criteria given. We've used the law of cosigns to help us solve side side side or side angle side situations. And there's one more case that we have to consider, and so this is the case of side side angle. With our usual triangle situation where we like to have a triangle labeled like A, B, and C, we'll just assume that we know angle A, all right. We're going to know side B and we know side A. And so this gives us our side side angle situation. We do have the situation where we have an AOS, that is there's an angle opposite side pair. So it's very likely we want to use the law of signs because we can then, since we know little B, we can compare that to angle B and go from there. And so the ambiguous case is generally solved using the law of signs. But depending on the length of these numbers A and B compared to the angle measure of A, this can affect, again, the situation. And in these six cases that we're going to talk about, we're going to consider the altitude H, which equals B sign of A in this situation. That is we're going to take the altitude associated to angle C, like so. Let's call that height of the altitude H. And so by basic trigonometry, you have opposite overhypotenuse with respect to this right triangle here. And so H equals B sign of A. So consider that in the following situations. So the thing is, the ambiguous case, we're going to see can have no solution, it can have one solution, it can have multiple solutions. And that's what really makes it ambiguous. You can solve the ambiguous case using the law of cosines or the law of signs, but it turns out the law of signs is typically simpler. And so in the subsequent videos for this lecture, I will demonstrate how to solve the ambiguous case using exclusively the law of signs, but be aware, things could be modified for the law of cosines if one wanted to. So the first situation is, let's suppose the angle you are given is less than 90 degrees. So angle A is acute. And if then the side length A, which is opposite from angle A, if that's shorter than the altitude, which again, the altitude is given as B sign of A. So one could compute this altitude here. You're going to get no solutions to the triangle. Why is that? Well, you get a picture that kind of looks something like the following. You have angle A, which is acute, you have angle B, which is given, and then the altitude is B times sign of A, and you have little A over here. Basically in this situation, the little H, excuse me, little H is too short, right? It doesn't matter how you rotate this angle right here, because angle C is not given, it potentially could be anything, right? There's no amount of rotation that'll ever make the short A side, since it's shorter than the altitude, it'll never hit over here, and so you can't form a triangle. So this case demonstrates, there are conditions for which the ambiguous case leads to no solution. It could be that, again, because the angle's too small, it's acute, and it's corresponding side length is too small. In other words, the AOS is too small, therefore no possible triangle could have those parameters, all right? Still sticking with the case where A is acute, what if A is equal to the altitude H? Well, in that situation, you would get exactly one triangle. It would necessarily have to be a right triangle, which you see right here, because the altitude H and the side length A happened to be the same length, and that's the only way to connect the triangle. That would force that angle B is in fact a right angle. Side length B is the hypotenuse of that right triangle, and then C would be the complement of A, and then by the Pythagorean relationship, our trigonometric relationships, we get the other leg as well. So there is this one situation where if A equals the altitude for an acute angle, then you get a unique right triangle, okay? Now continuing on, this is where it can get kind of interesting here. Suppose we take A to still be an acute angle, so A is less than 90 degrees, but let's suppose that the side length little A, it's larger than the altitude, which again, altitude's given by B times side of A, so we can compute that length, but it's shorter than the side length B. Then it turns out there are actually two possible triangles that could be created using these parameters. There is an obtuse triangle where angle B is obtuse, that's the one we're talking about here, something like this, but there's also the acute triangle where angle B would then be acute in that situation. In both situations, angle C would be forced to be an acute angle, but we see that there's two possibilities because after all, the angle B is unspecified, so this length that corresponds to A, it has the potential to rotate, right? You could swing it like a pendulum and anywhere it contacts the line given right here by the ray, I should say, given by this angle A that's provided, that would provide a triangle, and so there is a situation where the triangle could have two possibilities, and so side-side angle does not provide us a congruence condition because it turns out if two triangles share the same side-side angle conditions, they might be incongruent still because there could be two possibilities. Another situation for which the angle A that's given is still acute, and it could be that A is greater than or equal to B that's provided, so maybe A is the longer side in that situation. That would lead to exactly one triangle, and you get a picture that looks something like this. Again, this AOS, it's larger than B, so that would force B to be an acute angle because it has to be smaller than A, which itself is acute, and then we can go from there. Now let's consider, I'm not gonna really consider the case where A is a right angle because, again, that right triangle trigonometry, something we've already talked about, and we're really focusing on the ambiguous, or not the, well, the biggest case, obviously, but we wanna be focusing on the oblique case. So if A is an obtuse angle, so A is larger than 90 degrees, if A, if little A is less than or equal to B, then it turns out you get no possible triangle, and this is the same situation as before, right? Since A, it's just, you know, A should be the longest side because if angle A is larger than 90 degrees, it's obtuse, and a triangle could only have one obtuse angle. That means the obtuse angle is the largest angle of the triangle, and then therefore its opposite side should be the longest side of the triangle. So you can see that if, in fact, little A is too short, if it's not bigger than B, then it's not the longest length, and therefore it can't, no such triangle could exist, right? A similar statement could be said if we are greater than or equal to 90 degrees, right? And then in the other situation, if the angle A is greater than 90 degrees, but little A is bigger than B, then you get a unique possible triangle. Again, this has to be the longest side. And again, we can modify this to be A is greater than or equal to 90 degrees if we want to consider right triangles, but that's not gonna be our primary goal in the subsequent videos for this lecture right here. So this demonstrates all six possibilities. Let me zoom out for you so you can see them right here. Now, the good news is when it comes to using the law of science to consider the ambiguous case, we don't need to recognize all six of these cases. That is to say, we're not gonna be checking for like, well, if A is an acute angle, check the altitude, check the altitude, check B, right? We don't have to do any of those checks whatsoever. We don't have to worry about is A up to, so is A acute. It turns out that we don't even really need to recognize that we're in the SSA situation. We can just proceed to start using the law of science and we'll see this in the subsequent examples, because if you have this situation, let me come up to the picture we started up here, right? Because of the AOS associated to A and A here, since we know angle A and we know side A and we know side B, the very natural thing to do first is to then use the law of signs to find angle B. So we have sine B over little B and this equals sine A over little A. The data we're given suggests that we should be solving for B. And then the process of solving for B here, sine of B, we will naturally determine which case we go. It sort of naturally filters itself, like we throw the coins in the coin sorting machine. We don't have to worry about just like their shapes, they're gonna go into their different holes. And so if we follow this sorting process, we will solve this ambiguous case. And so while the ambiguous case, there is no def, I need more information about the side side angle before I can tell you what the outcome's gonna be. The good news for us practitioners of trigonometry is we just have to use law of signs and it'll naturally sort itself into these six cases.