Law of Sines... How? When? (NancyPi)

Hi, it's Nancy. I'm going to show you how to solve a triangle like this. How do you solve for all the sides and angles if it's not a right triangle? So, if it were a right angle triangle, we could use Pythagorean theorem, or a trig function to solve, but it's not a right triangle. Well, if you're given two angles and a side, any side, that's enough to solve using something called the Law of Sines. Sounds heavy. I know, just when you thought there couldn't be any more trig identities, properties, laws. But, don't worry, I'm going to show you all about it. You got this. What is that? Don't panic, it's going to be really easy for you. Here, capital A, B, C are the angles in the triangle, and little, lowercase a, b, c are the sides, the lengths. And all the Law of Sines says is that the sine of one angle over the side opposite it, equals sine of the next angle over its opposite side, equals sine of the last angle over its opposite length. That those ratios all have the same value, that this proportion is true. So let's use it to solve. Here in this triangle, what's missing, what we're looking for, is side a, side c. We also don't know this third angle, angle A. We can get that really quickly though, just because the sum of the degrees in a triangle is 180 degrees, always. So, we can just subtract the angles that we do know, from 180. And we get 36 degrees for that missing third angle. We didn't even need a special new law for that. But, to find these missing lengths, we do need the Law of Sines. So, first let's just write out all these ratios, the sine of each angle over the length opposite. And it doesn't matter which one you start with. We can start with A, so sine of 36 degrees over an opposite side, that we don't know yet, so we call it lowercase a. So that equals sine of the next angle over its opposite side. So, this is sine of this angle, 34 degrees, over the length opposite, 14, equals finally, sine of the last angle, 110 degrees over the opposite side c. So, those are all the ratios from the Law of Sines. And, if we look at just this part, just this equation, it looks like we could solve for a. Because, if there's only one unknown, only one variable, it means we can solve. So, we separate that part out to solve and then... what did I just do here? An easy surefire way to solve a proportion like this, that you can always rely on, is to cross multiply. If you have two fractions like this, equal to each other, and it's not immediately obvious to you how to solve right away, or how to rearrange it in your head, you can get it in a simpler form by multiplying these two diagonally, 14 times sin 36 degrees, and setting it equal to the other two terms multiplied diagonally, a times sin 34 degrees. And, this is still a true statement, this is a valid equation after you do that. Now it's really clear how to solve for a, right away. By dividing out what you don't want there, sin 34 degrees, from both sides. So, that's the answer for a. This is a little bit wonky as a length, so you can plug this into a calculator and get a decimal number for that. Just punch in 14 times sin 36 degrees, in degree mode, not in radian mode in your calculator, or you'll get the wrong answer. Divided by sin 34 degrees. And, we get that a is approximately 14. 72 for that length. By the way, you could instead take the reciprocal of each side here, if cross multiplying isn't your thing. You could flip each side and do it that way. Someone out there knows what I'm talking about and prefers that. But most people find the cross multiplying to be easier. Just one more unknown to find. I skipped ahead, but you might have already guessed it. Yeah, you can use another part of this, a different pair of ratios, to solve for c. And, it might not always be so neatly... this half, then use that half... Well, that was a bad robot impression, but basically, use whatever ratio has what

you want to find in it, as the only unknown, and set it equal to a ratio where you know everything already, and you'll get the answer. So, yeah, we could have set it equal to this ratio now, because we do know a, but then you'd have to make sure to use the unrounded version of that, maybe you plug it in again, and it seems like more work, more risk, seems like a garbage idea to me. But, you can. Go rogue. Just proceed at your own risk. So, just like before we cross-multiply to solve and we get this. A little deja vu for you, but we divide both sides by what we don't want here. So, divide by sin 34 degrees. And when we put this in a calculator, we get that c is approximately 23. 53, for that length. So, now we've totally solved the triangle for all the missing sides and angles. Lovely. How do you know to use the Law of Sines? When do you use it? If you know two angles and one side, you can do exactly what we just did. whether what you're given is AAS, angle, then an angle, then a side, in that order. Or, ASA, angle, side in between, then an angle, you can do what we just did. First use the fact that there's 180 degrees in a triangle to find the missing angle, but then use the Law of Sines to find the missing sides. And, there will be one solution, one answer for your triangle. You can also use the Law of Sines for another kind. If you have SSA, two sides and an angle that's opposite one of the sides. You can use the Law of Sines, and you might get one answer, one solution for your triangle. But you also might find that you have no solution, or that you have two answers. That there might be two triangles with the measurements you were given. Yeah, that can happen. This SSA case is also called the ambiguous case, very mysterious, because of all the possible outcomes. That's a whole other video, how to handle that. This video is an intro to Law of Sines, when you're given two angles and a side, and there's one solution. I know it's a little shady to mention that to you, and then just say goodbye. I just wanted you to know the other time that Law of Sines comes up. It's a little bit of foreshadowing, so you can see the whole picture, even if you are just starting out learning Law of Sines, with these more common kinds. And, I mention it, in case you're sleeping too well at night. So, I hope that helped you understand how to use the Law of Sines in trig. I know trigonometry is everyone's favorite. You don't have to like math, but you can like my video. So, if you did, please click Like, or Subscribe.