Ambiguous case law of sines two triangles SSA

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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