If I gave you this right triangle on the left here and asked you to solve for the missing sides, you would be able to do it pretty quickly given everything you’ve learned in the course. Because you have a hypotenuse and an angle, you could use SOHCAHTOA to find that other side, which is equal to 3. Then you could use SOHCAHTOA again or the Pythagorean theorem to solve for the other side. Pretty straightforward. What if I gave you this triangle over here and asked you to do the same thing? The key difference here is that this angle is not 90 degrees, it's 70. So, this is not a right triangle but a non-right triangle, which means that this isn’t really a hypotenuse, and that means that you can’t use SOHCAHTOA or the Pythagorean theorem. It just doesn’t work. So, unlike right triangles, you cannot solve for missing sides in a non-right triangle by using SOHCAHTOA and the Pythagorean theorem. So, how do we actually solve for this? Well, I'm going to show you in this video how we use a different equation called the law of sines. And it sounds kind of scary at first, but I’m actually going to break it down for you, showing how it's really straightforward to use.

Let’s go ahead and get started. I’m just going to show you the law of sines. The equation here is the sine of a, big A over little a is equal to the sine of big B over little b, and that’s equal to the sine of big C over little c. A couple of notation points here: you’ll see that angles in these problems are always capital letters A, B, and C, and sides are always lowercase letters. The law of sines is really just three ratios, and it's comparing ratios of angles on the top to lengths on the bottom. It’s called the law of sines because all three terms involve sine, and there’s a pattern here: always an angle over a side, A over a, B over b, C over c, and so on.

To actually solve using the law of sines to find the missing variable, what happens here is you have three ratios. You just try to pick two out of the three in which you know or can figure out three out of four variables. What do I mean by this? In this problem, I know what big A is, big C, and little c. My job then is to choose two out of the three terms in which I know three out of four variables and I can solve that missing one. If I try to pick these two over here, for example, I only know one out of the four. That's definitely not going to work. I know nothing about big B or little b. If I pick these two over here, notice how it doesn’t actually include my target variable. Also, I only know two out of four variables, so I also can’t use this. So, basically, because I know nothing about B, I can just ignore it and choose the first and the third ratio. Over here, I have three out of four variables, and I can solve for this missing one.

I'm going to rewrite this. The sine of big A over a is equal to the sine of big C over c. Trying to solve for this little a over here, there are a couple of ways I can do this. I can cross multiply, but really what I like to do in these problems is when you're solving for a side, you could basically just flip this fraction. This becomes a over sine of A. Whatever you do to one side, you do to the other. So, this will be c over the sine of big C. And then you can move this sine of A up to the top and start plugging in numbers. So, c is equal to 6 and then sine of C, assuming angle C is 70 degrees. Then this is 6 over sine of 70 times the sine of A, which is 30 degrees. When you plug in all these numbers, you will get a is equal to 3.19.

So, a is 3.19, the side of the triangle. Notice how the answer makes perfect sense because in a right triangle with a hypotenuse of 6, when you just go straight down, that's only a length of 3. But in this triangle, you have to go a bit farther because of the diagonal. So, you get something that's a little bit longer than 3, it's 3.19. That's how you use the law of sines. One last point I want to make is that instead of seeing capital letters A, B, and C, you may see some Greek symbols like alpha, beta, and gamma, but it works the exact same way.

That's how you use the law of sines. Let me know if you have any questions, and let's get some practice.