The Best of Ambiguous Case 2 Triangles

if there's one thing that I ran out of time inside the classroom it was teaching the ambiguous case when we had two solutions it's a lot of work and that's a lot of step-by-step examples of working through the problems to find all the missing measures of not just one triangle but two triangles well lucky for you I found my top three examples of me going through this concept inside the classroom and I compiled them all together into this one video just for you I hope you enjoy so right now ladies and gentlemen we're given 58° B = 12 .8 and AAL 11.4 all right so let's go and draw the triangle like we would any other time so I could say here's a which is at 58° all right here's C and we'll call this one b c we don't know any information for B we know this is 12.8 and a we know is 11.4 right so automatically ladies and gentlemen you can see that I have Sid side angle all right when you have side side angle rather than just following your daily task that you do every single day I kind of want your ears to perk up and say all right now is a possibility that I have two cases all right because what we need to be able to do is there's a possibility now that I could have multiple triangles all right and here's what here's the case the reason why this isn't really a great um triangle I could actually even shorten this up but what I could uh what I'm trying to show you guys is you could have a triangle that looks like this right couldn't you also just draw the same triangle if you kind of use this as a hinge and it went right there right because we don't know what angle C is right so my angle 11.4 could be here or probably a little bit or we could say 11.4 is here you guys see how that's a possibility because we don't know what c is right now all we know is what these two side lengths are and yes I know my triangle isn't written because this side is much longer than this side and it's shorter but we'll let get through that but do you guys see how I could have two possibilities yes that's like me hinging like that's pretending me like I took this like a hinge on a door and I like I rotated it down to here okay so it's kind of like the pathway of that side so you guys can see there's actually two triangles I could have an OB I could have a triangle with an obtuse C or I could have a triangle with an acute C right so there is a possibility so it's not as it's not as basic as just hey give me money take it back you know the stuff that we were talking about before so there's a possibility of two different triangles there's also a possibility what if 11.4 looked like that and let's say you know here's this side length is it a possibility that these could not even touch it is let's say if 11.4 is that long and then we end up finding C because we don't know what the length of C is what if C is short all right we'll get through that case right now um let's just go and take a look at C I want you guys understand there's a possibility of either two triangles one triangle or no triangles and this is always going to happen when we look at our side side angle so how does that work because Mr M what you just told us to do is just find side side angle and so forth or Define your missing angles well let's take a look at it so we have a ratio right we have a over a and we have a side length B so let's create the law of signs so we have 11.4 over the S of 58 degrees equals 12.8 over the S of B so we do our same thing we do our cross multiplication and we could say that s of B is equal to 12.8 times the S of 58° all over 11.4 I'm solving for B so I mul I cross multiplied and then I divide it by 11.4 so I can do 12.8 times the S of 58 and then divide by 11.4 so I could say the sign of B is equal to 95 2 two all right now ladies and gentlemen does that make sense for that to be an angle right you got to take the inverse sign correct right so you take take the inverse sign of your second answer and you get 72.2 1° so we say b equals sin inverse of 9.95 22 and we could say b equals 72. 21° okay so here's where it's going to get a little dicey so let's go back to the unit circle all right what I did is I just found the inverse right I apply the inverse and the important thing for you guys to understand about the inverse is if I'm going to look at let's say I have a sign value let's look at the sign value of2 if I say the S of b equals 12 is there one answer or two answers to that there's two because what is what is is s equal to 12 at pi over 6 yeah of course it is and it's also equal over here right so if I was going to say sin inverse B of2 we could say b equal pi/ 6 and 5 pi/ 6 right because your sign is positive in the first and second quadrant so therefore there's two actual answers we could say here it's pi/ 6 which is your reference angle notice these are your reference angles but this angle right here is 5 pi/ 6 right so does everyone understand when I'm taking the inverse of my sign I'm my domain I'm going to have two values that are in the first and second quadrant I have that I have my original angle and also using it as a reference angle so what I want you understand is if I'm going to look at 72° where my mark toop go so if I'm looking at 72° and I say if I say the inverse of 95222 is giving me 72° which is right here do you think I'm going to have another angle that's going to have that exact same sign value yeah I'm going to have the one over here right so what would that angle be how how could you we know this angle is 7221 how can I figure out what that angle is close not exactly 90 but if we take 180 minus that we'll get what this Ang we'll get that we'll get the remaining angle okay so let's go and take um so let's go and take 180 minus our angle 72 21 and what that gives us is B is going to could also equal 10779 it's okay we're not done yet yes what do you mean this one here no this from here to here is 72.2 1° so we want to find what that angle is so we're taking 180 minusing this and that's going to give us that angle well I'm not done explaining I'm still going to kind of go through it do you kind of understand though how there's two different angles that have the same sign value okay do you understand here how these both have the same sign value2 and2 right so same thing with this if I give you one angle you know there's an opposite one that has the same sign value right these two angles are I don't know what their coordinates are but they're going to have the same sign value right so if you find one you got to make sure you check with the other one because there's going to be two in the first and second quadrant okay so you got to check two angles we're going to go through now do do both of these triangle do both of these angles work so that's what we do is we say we create case one and case two so right now we have b equal 107° so we could say b equal 10779 de or we also said that b could equal 72.2 1° so there's a possibility now there being two different C values and let's go and see if these work okay so what we do is we write case one case two so case one says that um let's do this as case one and this will be case two so case one says this is 72. 21° that's an acute angle right so we could say that's going to look something like this 58° um that's going to be something like this so this would be 72. 21 and there you go and this is oh wait did I get them to be the same what oh I wrote it in there didn't I okay yeah we said that could be 72.2 one or we could look at case two which says it's GNA I wrote in I wrote in case number one we don't know that that's one example right we don't we don't actually know what this angle is right now I wrote In what B was and I wrote In what a okay we don't know what c is though right now m let's go and take a look at this one so if I say now um B is equal to 102° so I still have 58° what you guys need to understand is do do you understand when I was taking well first of all we have side side angle do you know in the other ones I wasn't using the inverse sign right when you when you complete the inverse sign that's what's giving you your two options right because you have when you complete the inverse sign you know that you have to be able to find both values that's why that's why I drw the unit circle when you apply the inverse sign you have to understand that there's going to be two possibilities in that first and second quadrant right that's why we come up with this case so what did I find a a b so we said B could equal 17° all right so now what I want to do is I want to see do do either of these both work I know it it's not work so let's go and take a look at case one does case one does this work if I'm given A and B can I figure out what C is so 180 minus 58° right we are we're given 58° from the beginning and then minus our new angle 72.2 is that going to give us our new value equal to C so what will now angle c equal right because on my case one I figured out what B was and I now I can figure what C was so we do 180 minus 58 minus 72. 21 and that gives me yeah hey C is going to be 49 79 degrees okay2 oh it's just 7.21 okay let me go and change it then 180 minus 58 oh I did it right I don't know why I wrote that in there all right so now that's for case one what about if Case two what if we said now hey this is going to be 170° this is 58° is it still possible to create this second triangle so what I do for this C is I do 180 - 58° minus 10779 de equals c so we do 180 180 - 58 minus 10779 and guess what I get 14.21 so I could say 14 .21 de equals c and I know my triangle is kind of looking a little crazy guys but it' probably just be something like that all right so we could say this angle is 14.21 so do you guys see how I can create kind of triangle here's where it's obtuse here's where it's acute but there's two possibilities and I can still remain I can still create the same so we know a is 11.4 B is 12.8 so the last um remaining value is we need to figure out what our c is so we're going to use a law of signs for each value to find our value C so for this one I don't know I'll use uh a so I'll do 11.4 over the S of 58 equals um C over the S of 49.7 9 that's for case one for case two I'll do the same thing 11.4 over the S of 58° equals c over the S of 14.21 all right and then I'll cross multiply and divide I'm just going to kind of do this to speed this along so my last one uh for case one I'll do 11.4 times the s of 49.79 and then I'll divide that by the S of 58 and I get c equal 10.26 for this case I'm going to do 11.4 * the S of 14.21 and then divide that by the S of 58° and here I get C is going to equal 3.29 okay so ladies and gentlemen the main important thing you guys need to take from this all right I know it's a lot of extra work you're looking at you just need to take from this when you're given side side angle and you have to you have to apply the inverse sign there's two opportunities you could have an obtuse or you can have an acute triangle you need to make sure you look into both of them next what we're going to do is look into what if there's no triangle at all all right and that'll be pretty simple that you guys will be able to see okay so in this problem what I have is a = 60° a = 9 and C 10 so what I'll do is I'll just create a triangle here kind of see what it looks like 60° AAL 9 we'll say here's B here's c um so C is going to be 10 all right now so what we first want to do uh with this problem is obviously go ahead and take a look at um what what our uh use our law signs to be able to figure out what the angle of C is going to be so I have 9 over the S of 60 equals 10 over the or 10 over the S of C so everybody can get to this point right Elizabeth yes okay it's pretty basic and then we can solve for the S of C so we could say the S of C equals uh 10 * the S of 60 by 9 what form how we'll go through on this part of thing I know right how do they figure out that like MC so now you're just going to take the S of 60 times it by times it by 10 divid by 9 and what you have and then you take the inverse sign of that value sorry so you have S of C equals let me do this again for you 60 sin * 10 / 9 you're going to get C = 96 2250 okay now to find what C is you C equals sin inverse of9 62250 does everybody understand that to find the angles you have to apply them yeah that's what you supposed to do even on your homework last night now I do the inverse of that and I say C equals 74.2 uh 1 okay so now I have Cal 74.2 1 degrees but ladies and gentlemen this is getting a little bit more difficult than just doing this one problem so what you're going to look at is all right it could be 74.2 but what I want you guys to understand is let's go back to our unit circle let's look at two angles let's look at pi/ 6 and let's look at 5 pi over 6 all right now notice the reference angle these are both pi over 6 off of the xais right right now let's take a look at our coordinate points this is 3 over2 comma 12 this is 3 over2 comma 1/2 what do you guys notice do you notice that for each of these angles pi/ 6 and 5 pi over 6 they the sign is exactly the same correct right sign is exactly the same so when you're finding the inverse of of an if you're finding the inverse of a value you could possibly for sign you have two positive answers so there's might be two Positive Solutions so let's go and take a look then of what would be if this is 74 what could be a possible other solution so to find that I could say C = 180- 74 21 again what I'm doing is I'm taking 180 subtracting my set my angle to be able to find the other possible angle we're going to get to that in just a second so I add 180 so one we take 180 and that's going to equal 10579 so what I'm trying to tell you guys if you look at take this take this for example if you guys take the sign of 7421 okay so 7421 sign it's 9622 then attention teachers take the sign all remaining Juniors who have not been screened for VIs vision screening may go to the media center this time again all remaining Juniors who have not been screened for vision screening may go to the media center this time thank you okay so what I'm trying to say ladies and gentlemen is when you take the inverse sign there's two possible answers you could have 74.2 1° or you could have 105.7 n° if you want to check that take the sign of 105.9 or 79 you're going to get n6225 take the sign of 74.2 you're going to get n6225 why do they same work because exactly what I said here what's the sign of pi/ 6 12 what's the sign of 5 pi over 62 so they both have the same value right now we're just dealing with some decimals and some angles that are not the same but now we have 74° so we're saying the sign of 74° is exactly the same as the sign of 105° does that kind of make sense so we need to make sure we both Solutions and now we got to see can both answers work can both of those answers work yes so it's because your calculator is just going to give you that positive solution it's not going to work with the reference angle of the other one okay because it's not going to give you both Solutions it's just going to give you that one pos smallest solution well what it's doing it's just giving you the really the reference angle solution for there so so now we have two possible solutions yes um so the P 6 and P were just an example they didn't they don't have anything to do with the problem no I was just kind of showing you why when signs equal to each other I was like where did you get that okay so now let's just make sure that these two problems work so let's do case number one so if I said case number one I'm saying a equals 60 and then I could say C equals 74.2 so therefore what does B have to equal well it equals what well let's just do 180 minus 74. 21 minus 60 and equals 45 he said what well no b equals 4.79 that's case number one if we did case number two what we're saying for case number two a still equals 60° but now I'm saying C rather cuz remember C could be 74 or it could be 105 it could be either one so let's say what if C equals 10579 de then what would b equal well let's see if this will work so I could do 180 minus 105.7 n° minus 60 therefore in this case C is going to equal 14.21 these are all degrees okay so first of all the thing you guys need to understand is we can have multiple answers for C right because our signs is exactly the same then what I do is I just subtracted these from 180 so if sign if C is 74.2 that means B is 45.7 n if C is 105.7 n that means C is 14.21 De all right so let's go back and let's take a look C for this triangle does that kind of make sense well if this is 60° and then I could say C is going to be 7421 and then B is 45° all right so I didn't draw my triangle perfect but you guys kind of have an idea it kind of looks similar similar right let's go and take a look at so that's case number one let's go and take a look at case number two what would a triangle look like if what would my triangle look like if I still had a was 60° um but now my C is 14 but now um or my B but now B is 14.5 and C is going to be 105° so let's pretend it's going to be something bigger like 105° so now it's 105 degrees and we'll call C kind of something like that why are you changing here I mean all you need to understand is this could be a here let's say that's 60 right then it could look something like that okay so this is 105° and then this is 14.21 so you have C A and B do you guys see how these are exactly the same triangles but now rather than having C be 74.2 now it's a obtuse angle so it's going to be greater than 90 so it's going to be like 105° so therefore your B is obviously has to be smaller but you guys see how there's two I know I like my triangles aren't like just shape or form but you guys can see how there's two possible solutions there's two cases so this is case number case number two in case number two we're saying a = 60° C = 10579 and b equal 14.21 so when we have two cases there's two possible solutions what we need to do is now find the rest of the information for both of them so when we look at this one um we're given we already know what a is but we don't know what b or what C are so we need to use the law of signs again so I can just use um a over s of 60° equals uh we're going to have sorry that's 9 sorry 9 over s of 60 is going to equal B over yes we TR find the remaining angles we don't know what B is and we but we know what 10 is or we know C equals 10 right why can't we do the yeah why can't we can't use the Pythagorean theorem unless you have a right triangle right so that's why you have to go back using the law of signs yes so are we trying to find the last of no we well we we found when we found C we found there's two possible answers for C it could be 7421 or it could be 105 there's two possible answers for it all right so then if there's two possible answers that means your third angle is also going to be different okay yeah because look it you can see I can re I can draw a triangle with a and 60 B are a being nine B being or C being 10 but them being but them with having the same angles but within the same sides but just one being an obtuse angle and then one being an acute Okay so it's possible for me to draw a triangle with an acute and an obtuse but with the same information up given that's what I said when you guys have a side side angle there's three options I showed you when there's no triangle that can be formed we did last class Peri we worked on ones where there's only one triangle forms and today we're working we're seeing when there's two triangles can be formed all right so now the only last remaining information I don't know is what B is so I just go and solve for b so b equals 9 * s of 45.7 / the S of 60° and then here I need to figure out what B is so I could use either the same Ratio or I could use a different ratio um let's just use a ratio for our C so I could do 10 over the S of 105° equals B over the S of 14.21 notice how sign of B is different in for each one of these so therefore in this case b equals 10 * s of 14.21 / the S of 105° so let's go and figure out what each value of B is because for each triangle B is going to be different so I have 4 5.7 s * 9 / 60 sign and I get in this case b is equal to 7.4 4 okay so for in this triangle in case number one I say 7.44 and let's let's look at it does that kind of make sense well at 60° my side length is N9 right at 74° my side length is 10 so if my angle B is smaller than all three all those other two angles then length B should be smaller than right does that kind of make a little sense okay so now if I give an angle 14.21 do you think my new B is going to be larger or smaller than 7.44 should be smaller so let's go and take a look see if we did the math correctly so I do 14.21 um s of that * 10 divid by 105 sign and I get b equals 2541 and there you go there's two possible answers uh we notice here is whenever you guys are given the information before you even try doing anything let's go ahead and sketch a picture right and we can all do this no matter what we're doing because you're going to be given information and what we have to do is of identify so let's just sketch the triangle now again we can assume that this is an acute oblique triangle meaning all acute angles but we know there could be an issue here is 68 that's going to be 18 a would be 19 okay but we know that if in when we have the case of our um our ambiguous case we could have an issue with that being um we could have an issue with this being it could be an obtuse triangle as well right and we'll and we we'll look at that uh when that comes up however based on this information we see that we have uh angle side side so it is the ambiguous case meaning we could have information that doesn't even create a triangle we could have just this one triangle that I drew up there or we could have two possible triangles and again I'll REO over that again as we get to it but the first thing is we could all just sell for a right here a we can just least solve for a okay um so again to solve for a we're going to want to use the law of signs and we're going to want to make sure a is in the numerator so I'll say s of a over a is equal to we have the ratio of C so that'd be the S of 68° all over 18 and again we've already worked on this last class period so I'm just going to write this next one to solve for a oops I'm sorry we know what a is which is 19 so therefore a equals sin inverse of um AAL sin inverse of 19 * the S of 68° all over 18 now do you guys need me to show you again how to do that on the calculator or can I just type it in and figure the answer and you guys can verify with me is that how works okay so let's go ahead and type that into our calculator you can type it all at one time um just be careful when you guys are typing stuff in make sure you're using your parenthesis so make sure it's already make sure your calculator is already in degree mode so I have um sign and if you guys didn't already get calculators you guys can grab on the back if you want to verify your stuff with me instead of walking right in front of the video camera but that's okay so we have sine of 68 divid 18 * 19 and then we do sin inverse and I am getting a equals 78.1 15 let's round this to the nearest thousandth now the next thing we'll want to go ahead and do is uh we're going to want to store this answer okay we're not going to want to use uh you know we can use this as our final like we can use this angle as our answer but we want to make sure we store a so I'm going to go and get store and then I'm going to so stow and then I'm going to hit Alpha a and again when I go over the next Pro I can show you guys this again but we'll have go ahead and store Alpha a okay now do you guys want to finish solving this and then check for the second triangle or yeah let's go and finish this let's just go and finish this the next one we got to find B so to find angle B we know that all the angles in a triangle add up to 180 so to find angle B all I'm going to do is say 180° minus my stored angle a minus 68 so that's why we like storing it because we don't have to retype the problem again so we do 180 minus Alpha a minus 68° and I get here 33 848 that means I stored it in my calcul so I'm going to store that answer as angle b as well so that's going to be 33848 okay yes no maybe so good okay now the next thing though guys is we need to solve for like little B right so if we're going to finish this we got to go ah aead and solve for little B so therefore I can use a law of signs um again and now I'm going to solve for b but I'm going to want to make sure B is in the denominator and when I'm solving on this example um what I'm solving in this case we're going to want to have B in the uh numerator because that's what I'm solving for over s of B which is stored in my calculator equal now again we could use a A or C over C but since c um was given to us C and Big C was given to us that's what I'm going to use now go ahead and solve in for b b equals the sign of stored B time 18 over the S of 68° so we go ahead and type that into our calculator s of stored B * 18 / the S of 68 and I get 6657 we'll round that all right now it's important that once we do this we want to check using kind of the hinge theorem of geometry to make sure that does the smallest angle have the smallest side and does the largest angle have the largest side does this kind of work yes yes it does work right okay so now we've kind of solved for this one but what if my question was saying hey is there one two or no triangles right is there another triangle that works we know that no we know that it makes a triangle because we're able to confirm it if we did this first sign inverse and we got an err then we know the problem there would be no triangle exists like the information provided does not create a triangle that happens when you guys have that err off the bat um but in this case we know one triangle exists what about a second triangle so again we got to understand guys what is happening with sin inverse when we're doing sin inverse our calculator so we're doing the sign inverse of or here's our angle a is 78 Dees 78 degrees is right here let's just round it to 78° okay now if we were to think about that like on the unit circle is there another angle where you take the sign of that angle you're going to have the same y-coordinates wait where did you get 78 from that's the angle that we found a okay right and I'm just rounding it for easy ease of use w't you guys agree that these two angles have the same y-coordinate or at least they're supposed to if I drew them correctly right now so that means now what exactly is this angle though like that's the reference angle so what is this angle it'll be the like huh what is that angle what is that measure of that angle 102 right because 78 and 78 and this angle 102 are supplementary so all if I got to do now again let's we come back to this in a second you would you guys agree that the S of 78° is equal to the S of 102° would you say that their y-coordinate is exactly the same yes yes right yes oh well why don't you guys say something so maybe let's make sure I didn't oh I didn't store my B oops no wonder so 180 let me make sure my Alpha a is correct sorry about that I didn't store big b s of alpha B what' you get what is your B B is okay and then s of B so the S of alpha btimes 18 did you guys divide it by S of 68 okay there we go I don't know what I typed in wrong thank you oh good catch so that's 10.8 13 okay now let's go back to the second case you guys agree that the S of 70 degrees and S of 102 is the same right now here's the problem when you guys do sin inverse which we did over here when you do sin inverse your graphing calculator is only going to give you s 78° it's not going to give you 102° even though the sign of those two angles is is the same why does it only give you that one angle because again remember the inverse function is restricted it's only going to give you that angle in the first quadrant or the fourth quadrant right it only gives you that positive or the negative right here it's not going to give you the obtuse angle however is it possible that our triangle our our triangle that we create could have an angle of 78° or it could also have an angle of 102 is it just possible yeah those are both angles that could be a possibility so what we need to do is sense the acute or the obtuse angle have the same sign sin inverse only gives you the acute so we have to check the obus so what we're going to do is we're going to call this A2 so it's really A2 of our second case so I'm just going to call this my case number two so we solved case number one now we're going to work on case number two so what I'm what we mean by case number two is we have the exact same triangle this is still 68° this is still 19 this is still 18 but a we're going to say is now obtuse because could a be obtuse as well as be acute it could and let's see what it is so now rather than rather than um simple rather than estimating we don't want to do that we want to take 180° minus a that we stored and we're going to store that as or we're going to figure out that value so now I'm going to do 180 minus Alpha a and I get 101.8 448 so does TH could this be a possibility could that angle be 1. 1848 and then have 68° yes it works right now so that means this example has two triangles however let's just pretend what if what if this angle ended up being like 150° is it possible to have 150 plus 68 no so therefore only one triangle would exist so even though you could test the obtuse angle it wouldn't create another triangle but in this example two triangles can be created all right so I am going to store this as my new a so I'm just going to take this new answer and I'm going to store that as my new a so store Alpha a okay so now Alpha a is 101. 848 4628 don't want to use the rounded answer make sure you use the whole answer um so now I need to figure out a new B right so we're going to say B2 so what would B2 be well B2 is 180° minus my new stored a minus 68 degrees right you guys agree with me on that hopefully so we do 180 minus stored a minus 68 and I'm getting 10152 if I round it and I'm going to call that my new stored B so now I'm going to store that as Alpha B are you guys okay with the storing and everything do you guys want me to go back over that next problem yeah sure okay so now we have a and we found B so the only thing I need to check is small B now again since we're solving for small B we're going to put small B in in the numerator here so I'll have B over the S of stored b equals let's again use C which is 18 over the S of 68° so therefore b equals the S of my new stored B * 18 over the S of 68° so it's really the same thing I did way over here it's just with new B's though right so I do s of alpha B * 18 ID the S of 68 and I am getting b equal 3.42 and does it make sense guys for B to be kind of small in this example right because look how small that angle had to be right so a really small angle is going to produce a really small side right which is different than this one but this one made sense and so does this triangle make sense Yes No Maybe So Okay and there you go