i'm going to be speaking to you about the sine rule again but i'm going to show you something called the ambiguous case that's why i like this picture here so i'm going to just try to remind you about the sine rule so if we know some sort of angle theta here and let's say we know this side a let's say we know that and we want angle c but we also know this side right here so we have this we have this and we have the angle here the idea is that you could use sine rule in order to figure out angle c and do you remember how sine rule goes there's two different versions of it one says you know sine you know a over a that's the side right so sine of the angle a divided by the side a equals sine b over b equals sine c over c or because there's another version it could be just where it's flipped right so it's side a over sine angle a and so on like this because this is the sine rule just to remind you so these are the two cases here right so i got this one or this one all right so uh so what's the big deal why is it an ambiguous case well it turns out if you're using sine rule to find an angle so if this is your goal is to do just what i outlined up above here okay if you're using sine rule to find an angle you're given two sides a short one and a long one and you know the angle that's opposite to the shorter side here it's possible to have two different solutions for angle c here's an example here the way i've shown it like this right here this right here could be length let's say that here will be length a here this will be length c this will be the same length c the same length a and the idea here is if you look at this or here there's this angle right here that's possible for c this one right here but there's also this one this obtuse one so see there's a acute angle that's less than 90 and there's an obtuse one that's greater than 90. notice they both have the same angle theta they both have the same length c they both have the same length a in fact the trick is if you draw it like here the way it works let me just do it in a different color maybe in red if you notice this side right here goes like this it's also possible to place it like that so that's the way i like to think of it it's like this one or that one do you notice that's what happened here so i could show you sort of that was the other side there so it was like this or like that so those are two different solutions possible as long as it still makes a triangle you still have to check that the sides all add up to 180. but just so you know at least these are the two different possibilities all right so let's see if we can deal with a real question like this so we've got a triangle abc we've got we don't have a drawing of it so that's why we're supposed to try to figure it out and then we have angle bac is 40 degrees side c is 7 side a is 5 what's angle bca like this in here again i had another meme like this was kind of similar the teacher said you know life has its up and downs like a sine function like nope not this one it just goes down what's that so let's try to draw ourselves some sort of triangle that could satisfy this so uh let me start with well i'll just draw a triangle let's see how that goes maybe i'll do like a long end maybe i'll do like a shorter end here like this yeah something like that and i'll draw like this all right so let me label everything so let me call this a i'll call this b i'll call this c i know that this angle b a c is 40 degrees so that's this one okay that's this angle right all right i also know side c is seven oh i know this is seven and i know that side a is five so ah there so the answer is the question sorry is find b c a in other words b c a i want to find this angle right here that's what i'm looking for so i'm just going to go along and try to use the sine rule so i know that the sine rule is going to be at the sine i mean i could call it bca i guess that's this angle right here over its side opposite to it which is that one it's going to be the same thing as a sine of 40 degrees over a oh by the way i also know a i shouldn't have just called it a i should have labeled it were you yelling at the screen when i said it because you should have uh there we go this value is actually five because we're told that's right we're told the a is five all right so if i keep going i got sine 40 over 5. well then let's find what sine bca so sine of b ca this angle over here is going to be equal to let's see i multiply by 7. so 7 times the sine of 40 degrees all that over 5. right that's what i'm doing here and that means then the angle bca i'll see how do i undo a sine i do inverse sine that's the key here right so inverse which is like this of this answer here so 7 sine of 40 degrees all that over 5. let's see what that gives me i need my trusty calculator out to do this because i don't know this by heart so let's just do it so i'm going to do the um well i'll just do the first part here i'll do the multiplication so 7 times the sine of 40 degrees all that divided by 5. double check that i'm in degree mode oops i am in degree mode here but i didn't do the 5 here that was my answer and i have to do the inverse sine of that inverse sine of the answer so do you notice i get 64.1 degrees let's just say so i'll say that i'll say that's a bca i'll maybe state my conclusion there so angle bca then b c a is approximately equal to and it was 64. let's just double check what the answer is here 64.1 all right so 64.1 degrees there we go that's my answer and i'm done right no no no that's the whole point of the ambiguous case there's something else possible because i'm going to try to draw like this and say or maybe by doing big letters or something like this and nope like this i'll just write it like this i'll say or so another solution possible and that would have been if i drew it just like this or here so i'll try to draw it like that so i could have had it like this except i would take this piece right here remember instead of going this way it could also fit that way that that piece that's five it could also fit that way maybe we'll have to double check if it works sometimes it doesn't we'll just double check that it does fit so that right there should also be five uh let me just make this one here a little bit longer just so it sort of matches the drawing omega doesn't hear match a little bit the drawing like that something like this and then of course it goes across let's see if this triangle fits so it has to still satisfy all the same criteria so a here b here this here would be c um this angle still has to be 40. this right here has to be seven this here has to be five now the good news is we already know a lot of this we know this little piece right here um well this piece right here which is this 64.1 turns out that's the same as this one right here because remember i sort of drew it going like this right here to make like a isosceles triangle here so that means this angle here would be the same as that one so let me just show you this that means this angle right here then we will do a different color here this angle right here i already know this angle this angle right here is 64. degrees well if that's 64.1 then i can find the complement of this because remember that adding up the two angles this angle plus this angle has to equal 180. so aha i know that angle bca this new bca at least will be equal to 180 minus 64.1 i mean keep in mind it's approximate right it's not exactly uh so that means let's see b c a then will be equal to let's just double check whoops trying to make it look clear here to angle bc let's see 180 minus 64. that would be 1 16 1 15.9 is that right yeah all right so there's my second answer now i have to double check that it really makes a triangle because sometimes you get this answer but it's not a real thing let's just double check well 115.9 plus 40 there will be 155.9 and then that means that yes does that mean that there's something that could fit here that's you know adding up to 180 yes so i haven't sort of broken any rules so there's another answer possible so to see in this case right here there was an ambiguous case it's a little bit obvious because i have a video about ambiguous case but here we go it was ambiguous because there were two different answers still satisfied all the criteria that's what made it ambiguous