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.