Understanding the Ambiguous Case in Triangles Using Law of Sines

so there are situations where the information we're provided with for a triangle make it possible for there to

actually be two different triangles created even after satisfying all the pieces of the given information we call

a situation like this the ambiguous case so in this lesson we'll explain more about this ambiguous case and see how we

can identify these situations using the sign law [Music]

before we begin this lesson we highly encourage you to watch our previous video on determining if a triangle

exists if you haven't already done so before moving on with this video in order to get some more context on what

we're about to teach here otherwise let's get right to it so if we have a triangle where we're given the acute

angle a the length of the side next to the angle and the length of the side OPP opposite to the angle it is important to

compare the two values of these side lengths if the opposite side length is longer than the side length of the side

next to the angle then we would definitely only have one possible triangle created AKA one solution so we

would know how to solve this angle here with the help of the sign law which is just sin a over aals sin b/ B however

what happens if a is shorter than b well first of all we've already learned that the a which is generally the side

opposite to the angle must be longer than the height of the triangle if it's not then we wouldn't even have a

complete triangle alt together but assuming that it is longer than the height but shorter than the length of

the side right beside the angle we'd have ourselves what we'd call an ambiguous case which is when it is

possible to either have this length drawn like this to make a triangle or drawn like this to become a completely

different triangle notice how both triangles still maintain the values of every side and

angle that we were required to respect so let's learn how to solve a problem when we're given a situation like

this let's say that we're given the following triangle first of all even though the

triangle was drawn for us and even though it's probably okay to assume that this triangle does exist let's go ahead

and check for it just in case so here we identify our height and use SOA to solve for

H using the so in this situation we know we can do sin of x equals to opposite over hypotenuse which gives us s of 36

is equal to H over 8 multiplying both sides by 8 gives us this so let's just rearrange and Computing for H gives us a

final value of 4.7 for the height of this triangle so so we know that this is indeed a complete triangle since value

of this side is greater than the height good so now that we have a situation where H is less than a Which is less

than b we can expect to see two different triangles that satisfy the 36° the B of 8 and the a of 5 so let's

try to find the two different angles that produce these triangles through the use of the sign law so to find the first

one is easy all we need to do is plug in our values into the sign law to get the following then we multiply both sides by

8 and rearrange to get this finally we take the S inverse of all of this to get a final value of roughly 70.1 de

therefore we have 70.1 de as one of our angle B's so the way to algebraically find out if we have an ambiguous case is

this we first assume the side length of five on this side now this makes for an angle over here that is congruent with

this angle over here since these two side lengths are the same this would automatically mean that this angle over

here would be 180- 70.1 the answer to this angle becomes 10

19.9 now here's the key when we add these two angles if we get a value less than 180 then we know that there is room

for this angle over here and that we have successfully found our second case in the ambiguous case so since 10 19.9 +

36 is equal to 145. n Which is less than 180 we know that this angle over here will be 180 minus

145. giving us 34.1 de for this angle so there we have it our two different triangles based on the same information

provided so let's try one more example of an ambiguous case together here is our triangle with the

following information so to begin with let's check to see if the side opposite to the angle

namely side a is greater than the height of the triangle what we do again is identify the height and use S of xal

opposite over hypotenuse to get the following after simplifying and Computing for H we get the height as

roughly 6.43 which is less than seven so would we be able to expect an ambiguous case

here well the answer is yes because we can see that we have the length of side a being greater than the height as well

as the length of side B being greater than side a along with our angle being aute giving us the perfect situation

where an ambiguous case can occur now that we've confirm this let's solve for the two different angles that produce

the two triangles so let's let's use the sign law to get the following simplifying gives us this and Computing

this gives us roughly 66.67 de so we know that this angle here is equal to 66.67 De assuming the side

length of seven on this side as well we know this angle is also 66.67 now what is the process we would

use next to find this angle over here well it would be to subtract 180 by 66.67 De to get

11333 de here again if these two angles added together are less than 180° then we can confirm for a fact that

we have ourselves another triangle and we can tell right away that these two do not add up to

180° so it seems to us that we have ourselves that second case with angle b as either 113 18. 33° or 66.67

de awesome so now we know how to identify whether the ambiguous case occurs and we also know how to find the

two different angles that make up the two triangles within it so that's it for this lesson make sure to practice some

more questions and we will see you guys in the next one