 In a previous video, we talked about the classic area formula for a triangle. That is area equals one half base times height. And in that formula, we're supposed to understand that the base is just to be any of the three sides of the triangle. And then the height is just the length of the corresponding altitude to said triangle side. And so that works great if we have a right triangle because you take the leg to be one of your sides and then its altitude would be the other leg. It's pretty easy to do. For oblique triangles, it's a little bit more challenging because the length of the altitude is not necessarily known at the start of the problem. So using a little bit of trigonometry, we mentioned, assuming we know side B, side C, and angle A. That is we have this sort of side angle side situation. We showed that the area of the triangle is going to equal one half BC sine of A like so. So again, that works great when you know a side angle side. But what about it's another situation? For example, what if we have this angle angle side situation? Let's suppose we know two angles. We know angle A, we know angle C, and then we know a side of the triangle. This time, let's call it little C. And it's AAS, meaning that the side is not between the angles in question right here. OK, so if we did like A, BC, that would be that would be an angle side angle. I'll actually get to that one shortly. So how do you deal with something like that? Well, again, to compute this formula, we need to know angle A, which is great, we know that. We need to know a side like C, but what about B, right? B was necessary to compute the length of this altitude right here, but we don't know B. Well, basically, because we have an angle angle side situation, we can use the law of science to start solving for the triangle. But our goal is to get the area. We don't necessarily need to find all of the dimensions of the triangle. We just need to find enough so that we can compute the area right here. So we're going to pursue here B, the quest for B. By the law of signs, we see that B is equal to sine of B, that is B over sine of B is equal to. Well, let's pick another angle here. We already used angle A. We haven't used angle C yet. And we also know the side length for C, right? Notice we have the side length for C and the angle C. This is an opposite angle side pair in AOS. So AOSs are great for law of signs. So you get B over sine B is equal to C over sine of C like so. If you solve for B, you'll end up with B. It'll equal little C sine of B over sine of C like so, for which then we could insert this in for little B and we get our modified formula area equals one half. The B will be replaced with C sine B over sine C. Since we already have a C there, we actually get two C's and so we end up with a C squared. I'm actually going to write this as one big fraction. So one time C squared, we have a two in the bottom. Then you're going to get a sine B. There's already a sine A on top. I'll put an alphabetical order sine A, sine B like so. We already accounted for this C. It showed up in there twice. There's one and two right there. And the denominator is going to have a two from the one half earlier. And then there's a sine C in the denominator as well. The sine C comes in here. And so then we get this formula right here. Area equals sine C squared sine A sine B over two sine C like so, producing this formula that you see now on the screen. These other formulas are derived if we started with other initial information, right? Because there's other types of A, A, S as you could do. So focusing on different angles and such, you get these other equivalent formulas. Area equals A squared sine B sine C over two sine A. You have area equals B squared sine A sine C over two sine B. And then the one we derived just a moment ago. Notice what's similar about each of these formulas, right? Each of them has a single side length. You have a squared B squared and C squared. They should, it's squared, right? There's a single side length. And this will come from the fact that you know only one side of the triangle, right? Then all three angles come into play. There's a sine B, sine C, sine A. Sine A, sine C, sine B. Sine A, sine B, sine C, right? All of the angles are in play here. We have to take the sine of them. Who goes in the denominator? Notice you have sine A. Oh, A was the angle you knew. Sorry, A was the side you knew. It's sine goes in the denominator. In this one, B is the side you know. It's angle goes in the denominator. C is the one you know here. The angle goes in the bottom. So whatever side length you know, its opposite angle is going to be in the denominator as a sine. And so that can help you keep track of these things. And of course, you'll have the number two right here because it's one half base times side. And so if you have an angle angle side situation, you know two angles in one side, we can use this formula to compute the area of the triangle. But wait a second. We knew angle A, we knew angle C. What about angle B, right? This formula requires B. In fact, all of the formulas require all three angles. If you know angle A and you know angle C here, then it's not too hard to find angle B because B is just its measure as an angle will just equal 180 degrees or pi radians if you're using radians there. Subtract the measure of angle A, subtract the measure of angle C. Just subtract the two knowns and you find the third angle there. And so basically if you know two angles, you know the third one. So it's really not that big of a deal whatsoever. So let's suppose we're in that setting right here. Let's say we know the measure of angle A. We know the measure of angle B and we have the side length A right here. Notice this is an angle angle side situation where we have this AOS. We have angle A and side A in the situation. So we should figure out what is the remaining angle right here? That's an easy calculation. C is gonna measure to be 180 degrees minus 30 degrees minus 70 degrees. 30 and 70 come together to give me 100. Therefore C is equal to 80 degrees. We take 100 degrees away from 180 right there. So we're left with 80 degrees like so. And so then we can compute the area that kind of looks like 880 degrees. There you go. So the area is gonna equal. We're gonna use the formula that uses side length A. So this will look like A squared times sine of B sine of C over two times sine of A like so. In which case little A is eight. So we get eight squared times sine of B which is 70 degrees times sine of 80 degrees. That was C all over two times sine of 30 degrees like so. And so some of these we can do by ourselves. We're ultimately gonna need to use our calculator for things like sine of 70, sine of 80 degrees. Eight squared of course is gonna be 64. Like I mentioned, sine of 70 degrees, sine of 80 degrees. These are not our special angles from the unit circle. So I'll need some help computing those. You get two times sine of 30 degrees. 30 degrees is a special angle. Sine of 30 is equal to one half. So you get two times one half. They cancel each other out. And so we get as the most precise answer right now 64 times sine of 70 degrees, sine of 80 degrees. At this moment we really do need to use our calculator. We can't go any forward without it. Make sure your calculator is in degree mode right here. But the area of the triangle then would be 64 times sine of 70 degrees times sine of 80 degrees. And you'll end up with 59 centimeters squared. We are talking about area after all. So the area unit should be length squared as we previously measured length using centimeters. You can see that right there. This area formula will produce some quantity of centimeters squared. So that shows us how we can compute the area if we have angle angle side. What if we look at the angle side angle situation? This would mean that we have two angles A, or in this case C and B. C is 82 degrees. B is 34 degrees. But we know the length of the side between the two. So this is not angle angle side. This is angle side angle. But it turns out and for area, it really doesn't make much of a difference whatsoever. Because like we observed earlier, the measurement of angle A is going to be 180 degrees minus the 34 degrees from B minus the 82 degrees from C. Not 82 squared, 82 degrees like so. And so therefore we can subtract 34. We can subtract 82 from 180 degrees and we end up with 64 degrees. That's the measurement of A there. And so the thing is if you have two angles, you can find the third one easy enough to subtract them from 180 degrees. So in terms of area, there's really no fundamental difference between angle angle side and angle side angle. When we were solving a bleak triangles earlier using a lot of signs again, there really was no distinction between angle side angle and angle angle side. Because if you know two angles, you can find the third one easy enough so you can switch between the two. Since we know the length of side A, again, we're going to want to use the area formula that relies on the side length A. So that looks like A squared sine of B, sine of C over two times sine of A. So plug in the appropriate information here. We're going to have 5.6 squared. We're going to get sine of 34 degrees. We're going to get sine of 82 degrees like so. And this sits above two times sine of 64 degrees. And really there's nothing. There's nothing that I feel compelled to do without my calculator at this moment. I mean, 5.6 squared, I could do it. But with the decimal, it kind of feels like you're squaring a two-digit number. I mean, I could do it if I had to, but why? I'm going to have to use my calculator eventually. 82, 34, 64 degrees. None of these are special angles. So at this moment, I'm just going to throw it in my calculator. I advise you to do so as well. Check it yourself with me. Make sure your calculator is in degree mode here. But you'll end up with 9.7 if we round to the nearest decimal place. 9.7 centimeters squared as the area. So if you know two of the angles and a single side, then use this area formula or AAS formula or ASA, whichever you prefer. Use this formula to find the area of the triangle.