 In this video, we're going to solve the triangle ABC given the data that little a equals 314, which is approximately 100 pi, of course, little b, which equals 205 feet, and an angle a is given to us and its measurement is 35.4 degrees in this situation. So you'll notice that because we know side length a and we know angle a, we have an angle opposite side pair and AOS. Whenever we have an AOS, it's very suggestive that we should use the law of signs to help us solve this oblique triangle. Notice that we have the AOS for a, and then if we look at the AOS for b, we have the side length b, but we don't have the angle b. So it's very natural that we solve for angle b first. And to do that, we, of course, will use the law of signs. So we're going to take sine of b over little b is equal to sine of a over little a. Once we want to solve for sine or for angle b, we would clear the denominators times both sides by little b. This tells us that sine of b is equal to b over a sine of a. And then plug in the data that we have little b is 205, little a is 314, and then sine of a is equal to 35.4 degrees here. And so we're going to want to compute this number using, of course, a calculator here. Help us out here to approximate this makes a follow along with me here with your own calculator. We're going to again, just throw this in our calculator. Make sure you are in degree mode at this moment, 205 divided by 314 times that by sine of 35.4 degrees, you'll get approximately 0.3782. Now this is a very important part to mention here. Notice that the original information you're given was side, side angle. This is the side, side angle situation, the so-called ambiguous case. It could be that there's no solution, it could be there's one solution, it could be there's two solutions to solving the ambiguous case. You know there's no solutions if this ratio for sine turned out to be something bigger than one or less than negative one. Because sine of b needs to be less than equal to one, but greater than equal to negative one. If you're outside that range, then b could not, there's no angle that could satisfy b here. And therefore there's no solution. Now this is an important place to check. Notice our ratio does lie between negative one and one. So we can proceed forward. There is going to be a solution to this triangle. There could still be multiple solutions. We'll see maybe that in a second, but we're going to proceed forward here. b is going to be approximately sine inverse of 0.3782, like so. For which again, consulting your calculator, if we want to be a little bit more precise of course, then we really would be like, oh, the precise value for sine of b, excuse me, the precise value for b is going to be sine inverse of 205 over 314 times sine of 35.4 degrees. That's the exact value. But again, we're going to use an approximation here. On the calculator, if we round it and there's degree, we're going to end up with 22.2 degrees, like so. In this situation, that would then give us that the measurement of angle c, right, it's going to equal 180 degrees, subtract angle a, subtract angle b. And so we end up with 180 degrees minus the 22.2 we got for b minus the 35.4 degrees we were given for a. And this would give us that c is equal to 122.4 degrees. And so with that information in mind, we can then use the law of sines to finish off finding c here, little c. So little c is going to equal sine of big c equals, we can compare this to b, we can compare this to a as angle b here is an approximated value. Even the sine of b is approximate, we're just going to use angle a again. So we're going to get little a over sine of a solving for little c, we get c equals our little a times sine of capital C over sine of capital A, little a remember was 314 sine of c, we'll plug that in there, we get sine of 122.4 degrees, and then we divide that by sine of 35.4 degrees. That's supposed to be a degree symbol right there, put all of this into our calculator as well, and approximately we get 458 feet for little c. And so if we come up here and label the information we just found out here, we would see that it's like okay, c would have an angle measure of 122.4 degrees, the length of c turned out to be like we said 458 feet, and then angle b turned out to be what was it again 22.2 degrees like so. Now this is a very important step. The next step here is a very important step when it comes to the law of sines. It's important to remember that if you take sine of b, this is actually equal to sine of 180 degrees minus b. That is to say the sine function cannot tell the difference between an angle and its supplement. You get the exact same sine ratio because in the first and second quadrant, sine is both positive. In particular, sine cannot tell the difference between an acute angle and an obtuse angle. So when your calculator tells you that b is equal to 22.2 degrees, it's given you the angle in the first quadrant. That is the acute angle. What if it was an obtuse angle? Well the obtuse angle would be 180 degrees minus 22.2 degrees. That's also a possibility that we have to consider. So we consider the situation where b is acute, that force c to be obtuse, but what if b is the obtuse angle here? We take 180 degrees minus 22.2 degrees. We end up with 157.8 degrees, but I want you to compare 157.8 versus a right here. When you started to compute c here, the idea is you take 180 degrees, you subtract from it b, which in this case would be 157.8 degrees, subtract from it a 35.4 degrees here. In that situation, you would end up with a negative value, right? Notice that you get 180 degrees. Subtract from that the combined value of b and a in this situation would be 193.22. That's going to give you something less than zero. C can't have a negative degree measure. And so this tells you that b cannot be obtuse in this situation. Therefore, there's only going to be one triangle that solves this situation. And this is what happens when you try to solve the ambiguous case using law of signs. You have to make sure you check, is the ratio acceptable? If it is, that means there's one or two solutions. If the sign ratio is too big or too small, there's no solutions. Then you continue forward. You calculate b. You have to also calculate the supplement of b. Once you get past this marker right here, the acute angle for b will always work, but you have to check the obtuse angle. In this situation, the obtuse angle didn't work because it was so big that its combined value with a was over 180 degrees. There wasn't an obtuse version for angle b, but there are situations where that is the case. We'll see that in a little bit. And therefore, this then demonstrates how we can get a unique solution when we solve the ambiguous case using the law of signs.