# Ambiguous case law of sines two triangles SSA

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OK, so right now,

ladies and gentlemen, we’re given 58 degrees. B equals 12.8 and A equals 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 degrees. 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 side 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, now is a possibility

that I have two cases. Because what we need

to be able to do is, there’s a possibility now that

I could have multiple triangles. And here’s the reason why. This isn’t really

a great triangle. I could actually

even shorten this up, but what I’m trying

to show you guys is you could have a triangle

that looks like this. Couldn’t you also just

draw the same triangle if you kind of use

this as a hinge? And it went right there? Because we don’t

know what angle C is. So my angle 11.4 could be here,

or we could say 11.4’s 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

triangles isn’t written because this side is much

longer than this side, and it’s shorter, but we’ll

just get through that. But do you guys see how I

could have two possibilities? Yes? What’s that line in there? That’s like me hinging. Like I took a hinge on a door

and I rotated it down to here. OK, so it’s kind of like

the pathway of that side. So you guys can see there’s

actually two triangles. I could have a triangle

with an obtuse C, or I could have a triangle

with it an acute C, right? So there is a possibility. So 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 here’s

this side length. Is it a possibility that

these could not even touch? 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. Let’s just go and

take a look at C. I want you guys to

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 Ms. McCoy,

what you just told us to do is just find side

side angle and so forth, or to find the 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 Sines. So we have 11.4 over the sine

of 58 degrees equals 12.8 over the sine of B. So we do our same thing. We do our cross multiplication. And we can say that sine of B

is equal to 12.8 times the sine of 58 degrees all over 11.4. I’m solving for B, so I

cross-multiplied and then I divided by 11.4. So I can do 12.8

times the sine of 58 and then divide it by 11.4. So I can say the sine

of B is equal to .9522. All right, now,

ladies and gentlemen, does that make sense

for that to be an angle? You need the inverse sine. Right, you’ve got to take

the inverse sine, correct? Right? So you take the inverse

sine of your second answer and you get 72.21 degrees. So we say B equals

sine inverse of .9522. And we can say B

equals 72.21 degrees. OK, 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 applied the inverse and the

important thing for you guys to understand about

the inverse is, if let’s say I

have a sine value. Let’s look at the

sine value of 1/2. If I say the sine

of B equals 1/2, is there one answer or

two answers to that? There’s two. Because is sine equal

to 1/2 at pi over 6? Yeah, of course it is. And it’s also equal

over here, right? So if I was going to say

sine inverse B of 1/2, we could say B equals pi over

6 and 5 pi over 6, right? Because your sine is positive in

the first and second quadrant. So therefore there’s

two actual answers. We could say here,

it’s pi over 6, which is your reference angle. Notice these are your

reference angles. But this angle right here

is 5 pi over 6, right? So does everybody

understand when I’m taking the

inverse of my sine, I’m going to have

two values that are in the first

and second quadrant? I have my original

angle, and also using it as a reference angle. So what I want you

guys to understand is if I’m going to

look at 72 degrees, where’d my market top go? So if I’m looking at 72 degrees,

and I say the inverse of .9522 is giving me 72 degrees,

which is right here, do you think I’m going to

have another angle that’s going to have that

exact same sine value? Yeah, I’m going to have

the one over here, right? So what would that angle be? We know this angle is 72.21. How can I figure out

what that angle is? A 99 7? Close. Not exactly 90, but if

we take 180 minus that, we’ll get the remaining angle. OK, so let’s go and take

180 minus our angle 72.21. And what that gives us is,

B could also equal 107.79. It’s OK, we’re not done yet. Yes? [INAUDIBLE] What do you mean? Like, the last screen on. This one? What abuot the rest? Here? Yeah. No, this from here to

here is 72.21 degrees. Oh, OK! 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. 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 sine value? OK, do you understand here, how

these both have the same sine value? 1/2 and 1/2, right? So same thing with this. If I give you one

angle, you know there’s an opposite one that

has the same sine value, right? These two angles, I don’t know

what their coordinates are, but they’re going to have

the same sine value, right? So if you find one,

you have to make sure you check

with the other one because there’s going to be

two in the first and second quadrant. OK, so you got to

check two angles. We’re going to go through now– do both of these angles work? So that’s what we do– is

we create case 1 and case 2. So right now, we have

B equals 107 degrees. So we can say B

equals 107.79 degrees, or we also said that B

could equal 72.21 degrees. So there’s a possibility now

of there being two different C values. And let’s go and

see if these work. OK, so what we do is we

write, case 1, case 2. So case 1– let’s

do this as case 1; and this will be case 2. So case 1 says this

is 72.21 degrees. That’s an acute angle, right? So we could say that’s going

to look something like this– 58 degrees. That’s going to be

something like this. So this would be 72.21. And this is– oh wait. Did I get them to be the same? Oh, I wrote it in

there, didn’t I? OK, yeah, we said that’s

going to be 72.21. Or, we could look at case 2. I wrote in case number 1. That’s one example. Yeah, minues the– We don’t. We don’t actually know what

this angle is right now. I wrote in what B was, and

I wrote and what A was. OK we don’t know what

C is though, right? Now Mackenzie, let’s go and

take a look at this one. So if I say now, B is

equal to 102 degrees, so I still have 58 degrees. [INAUDIBLE] What you guys need to

understand is, first of all, we have side side angle. In the other ones, I wasn’t

using the inverse sine, right? When you complete

the inverse sine, that’s what’s giving

you your two options. Because when you complete

the inverse sine, you know that you have to

be able to find both values. That’s why I drew

up the unit circle. When you apply the

inverse sine, you have to understand

that there’s going to be two possibilities in

that first and second quadrant. That’s why we come

up with this case. So we said B could

equal 107 degrees. All right so now

what I want to see is, do either or

these both work? I know, it’s not working. So let’s go and take

a look at case 1. In case 1, does this work? If I’m given A and B can

I figure out what C is? So 180 minus 58 degrees. We’re given 58 degrees

from the beginning, and then minus our

new angle, 72.21. Is that going to give us

our new value equal to C? So what will now angle C equal? Because on my case 1, I

figured out what B was. Now I can figure out 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. The value is 72.21, not 72.27. Oh, it’s just 72.21? OK let me go and change them. 180 minus 58 minus–

oh I did it right. I don’t know why I

wrote that in there. All right, so now,

that’s for case 1. What about case 2? What if we said now, hey, this

is going to be 107 degrees. This is 58 degrees. Is it still possible to

create this second triangle? So what I do for this C is,

I do 180 minus 58 degrees minus 107.79 degrees

equals C. So we do 180 minus 58 minus 107.79. And guess what? I get 14.21. So I could say

14.21 degrees equals C. And I know my

triangle is kind of looking a little crazy,

guys, but it’d 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 two different triangles? Here’s where it’s obtuse;

here’s where it’s acute. But there’s two

possibilities, and I can still create the same. So we know A is 11.4, B is 12.8. So the last remaining

value is we need to figure out what our C is. So we’re going to

use the Law of Sines for each value to find our

value C. So for this one, I don’t know, I’ll

use A. So I’ll do 11.4 over the

sine of 58 equals C over the sine of 49.79. That’s for case 1. For case 2, I’ll

do the same thing. 11.4 over the sine

of 58 degrees equals C over the sine 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 for case 1– I’ll do 11.4 times

the sine of 49.79, and then I’ll divide

that by the sine of 58, and I get C equals 10.26. For this case, I’m going to do

11.4 times the sine of 14.21 and then divide that by

the sine of 58 degrees. And here I get C is

going to equal 3.29. OK? So ladies and gentlemen,

the main important thing you guys need to

take from this– I know it’s a lot of extra

work you’re looking at. You just need to take this when

you’re given side side angle, and you have to apply

the inverse sine. 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. And that will be pretty

simple that you guys will be able to see. OK? so this is your example. Yes? You said that if this

is to be the sine, it’s there, but number

five, we did that together, and we did it normally. I’ll show you on number

five what exactly to do. I just want you

guys to get this. So I’ll explain number

five here in a second.

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Charlie WinkelmanPost authorIf Sean Evans from hot ones taught math

Pizza StevePost authorThank you very much, sir!

No_NecKZzPost authorAnd if the last angle we find in case two is a negative, it’s not a real triangle and there’s only one triangle, which is the first case?

RamenPost authorDuh hello

v chPost authorI could actually binge watch these, these videos are oddly so interesting. Our teacher can't explain anything even if his life depended on it, he over complicates things waaay too much. Thank you for the videos!

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Joan StaffordPost authoryou explain things so well. thanks for the help!

Mark WangPost authorWOW u explain it better than my actual teacher

Kevin ZhuPost authorMr.Mclogan, are you a teacher? can you email me your teaching place address, maybe later I going to learn.thank you

deplorable infidelPost authorthe remaining angle is 58deg + 72.21deg – 180deg = 49.790deg a triangle has a total of 180 deg on 3 sides

Livia ReynaPost authorI don't understand how you go the inverse

Alexander FettermanPost authorJust want to say that the way you teach and elaborate helps simplify this material for better understanding. You seem like a great teacher. Wish I could have had you when I took this class. Currently reviewing for the SAT and your videos are extremely helpful, thank you.

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joeyrogerson83Post authorThis guy is a good teacher, watch and learn.

Jayloni BarekzaiePost authorThank you! Finally understood this

Jorge GuizarPost authorThanks!, this helped as a good review to understand the little things that confused me

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Dalton MaxwellPost authorYou're honestly the best math teacher I've ever saw you explain things clearly for some stupid reason I took an online math class and you're basically a teacher thanks so much

Nio SmissonPost author>Working counter-clockwise on a triangle to make it SSA

The high school teacher FEARS the ASS triangle.

Noa NirPost authori wonder if mckenzie is doing good

Ariona MaurerPost authorFucking love you.

miranda bPost authorthank you

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B3ATSPost authorThanks

Salem ZenatiPost authorThank you so much for making this video. It helped me so much! I am doing so bad at trig, you are the only thing keeping me up. Thanks again!

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wuzjrPost authorWhy are there girls in a math class?

Daniel LeePost authorThanks for the video. It really helped me review for the SAT. Also a quick shortcut to see if there is an ambiguous case is to look at the angle value of the first angle you solved for. If it is greater than the given angle then there is an ambiguous case.

Drishti ModiPost authorWonderful video

Abdullah HussainPost authorJust a short question, where are you teaching? Is it in a secondary school? Or a college?or a university?

Salman TahirPost authorWhat if we are given only one side and one angle ?

Reyana 285Post authorDUH HELLO 😂 who else saw it ?

Karl KennethPost authorGreat teaching!

Riley CavanaughPost authorBut what does it look like when there is only one triangle? How do you decide/prove that.

Valerie JoycePost authorThank you so much for this! It’ll be easy to remember this, cause of how it’s spelled backwards. Lol, but honestly that’s how I’ll remember! Thank you so much for your videos though! I watched your videos last semester, and this semester as well. Whenever I search help for math on YouTube, I click on yours over all the others…every time! Wish you were my teacher.

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HARPO !Post authorThis is way simpler than the shitty way my teacher tried

gangstalishisPost authoryou are a life and money saver, thank you so much. you are the only person know who can explain something so convoluted into something simple

Yordanos HalekaPost authorHow did u get the 72 I can't seems to get it

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Ethan RowePost authorThis man saves lives

william thibeauPost authorLove your videos. Hate the way you hold the marker on the low part of the board.

BellaPost authorMaking test corrections today after school because i didn’t know how to do SSA or the ambiguous case. Thank you for saving my grade.

Lucas ArgandoñaPost authorI have a final in 30 minutes. This is hard

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Marcus ZhangPost authori thought the law of sines was the opposite of what you have. is it correct if i flip what you have on the board or does it have to be what you wrote on the board?

Elite Legendary GamerPost authorHe is the only person with this content and he is really useful!

Shaik PashaPost authorwhere did you get 107.79 degrees from?

Samiya MohamedPost authorI have an math exam tomorrow!! thank you so much

NaveNozidPost authorBrian McLogan goes Sicko Mode

ssj dannyPost authori WISH you were my teacher. My teacher just says, "this is the law of sines, sometimes there's two triangles, GO."

G I O V A N N IPost authorThank you so much! You really are going to save me from my upcoming test!

Tarah Beatrice PlacidesPost authorthank you so much. you really help me understand these trig lessons that my teacher doesn’t teach well.

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Gilberto MessiasPost authorHelpful😩

Madison MelwaniPost authorIs there a specific reason why he used angle B for his case study?? If he used angle A would it work also?

Brändøn P.Post authorThanks this is going to help for the test im going to have in 8 hours 🤣🤘

John BrownPost authoryou're a lifesaver

Nikhil MathurPost authorThanks man

Chris LinPost authorThank you!

geleta dedefiPost authorI was totally lost confused when reading the book but after watching this video it is all clear now. Thanks for your help!

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Mottled BubblePost authormckenzie = all of us

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Lethal BaconPost authorthanks for making these vids they are extremely helpful

Deepanshu RanaPost authorRatio of the areas of both the acute and obtuse triangles should be equal to the ratio of their bases

I.e. 10.26:3.29

After 15:09

Michael SaltonPost authoryour way better than my math teacher

Chloe GoudaPost authorthat case 2 triangle look like booty

ZamorePost authorDuh Hello!

Leo VolontPost authorJeez, THIS GUY IS RETARDED! He is SO confused. He can't draw anything approaching scale. In what Country is this guy teaching Math? Also, if he would Teach By Lesson Plan, he wouldn't have to be poking his Calculator during Class. I bet the guy is American. they say that America is behind all of the Developed Countries of the World. Well, here is some News. America is behind most every Country that is even keeping records. Honestly, America is dropping back at such an obvious rate that the United Nations is thinking of a New Category: Receding Nations. You know, America WAS Developed, but it's just not THERE anymore. Like this guy McLogan. Not exactly America's Highest Point, is he?

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Trevor BodnarPost authorThis video helped allot. I was studying for my final and my textbook is terrible at explaining SSA problems

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Rohan PatelPost authorDuh hello

Oscar EscobarPost authorTHUMBS UP!!

Shadi RezaeiPost authorso every time that we have SSA when find the second angle we are going ti use sin to find the other angle?

Yeyint MoePost authorThank you, sir. It was a great explanation; clear and understandable. Please do more videos.

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redPost authorTHANK YOU😭

Nonster87Post authorGood video, helped a lot, thanks.

kboisPost authorthank you SO MUCH

Yogi BearPost authorYou look like Randy Orton, especially in this particular video