A triangle. Has
a unique shape.
Anyone triangle given 3 pieces
of information forms a unique
shape. Having said that, there
are two exceptions. Let's think
about a triangle.
There are six pieces
of information available. Angles
ABC. And the sides, the clay.
It'll be a little see notice
that I've labeled the side
little B that is opposite the
angle be the side that is
little. A is the one that's
opposite the angle A and the
same the Anglesey Little sees
the side. That's opposite the
Anglesey. Now. If I
take three of these six
pieces of information.
With two exceptions, I will get
a unique triangle.
Let's just get rid of the
exceptions. The first exception
is the three angles. If I have a
triangle with three given
angles. Let's say that one. Then
I can draw another triangle.
That is the same shape.
Only bigger.
So I haven't fixed on a
unique triangle.
There is another way in which
that can happen where we can get
more than one triangle. Suppose
we are given.
That side and an angle.
Here. Got an angle here? Then
the side could go on and on and
on like that.
Say I was given the length of
this side. And say I was given a
land that was that long.
While there's also another
triangle up here that has the
same. Length. There and there,
and So what I've got there are
two triangles. I've got a big
one and I've got a little one
and I've got those out of three
pieces of information. The
length of that side. The length
of this one, and that angle.
But those are the only two cases
where if I take any three of
these pieces of information, I
will get a unique triangle.
These are the two exceptions,
OK? Because that means a
triangle is fixed. Once I've got
these three pieces of
information, it means that if
you given three piece of
information, you ought to be
able to calculate the rest and
what we're going to be looking
at is formally that enable us to
do that. So we will begin with
a set of formula that are
known as the cosine formula.
First one is cause
a is equal to
B squared plus C
squared minus a squared
all over 2 BC.
So what we've got here at the
right hand side as Givens are
the three sides of the triangle.
So we use these formula.
When we've got three sides of a
triangle given to us and they
enable us to workout the angles.
I use that in the plural there,
but I've only written down one
formula. Let's write down the
others cause B is equal to C
squared plus A squared minus B
squared. All over to
see a. And notice
that I've cycled the letters
through the formula.
Calls of a has minus a squared
over 2 BC cause of B has minus B
squared over 2 CA, and so we
ought to be able to predict what
the cause of C's going to be.
That will be A squared plus B
squared minus C squared.
All over 2A B. So whilst
they look complicated.
They're very easy to remember,
so we use.
When given three sides.
To find
Angles. Now we
can actually rearrange these, so
let's take this first one. Do a
little bit of algebra.
And see how we can make use
of the result to generate some
more equations. So be squared
plus C squared minus a squared
all over 2 BC. First of all,
we can multiply up by the two
BC. So we have two BC cause
a is equal to B squared plus
C squared minus a squared.
And now can add an A squared
to each side and take this lump
away from both sides and that
will give me a squared is equal
to B squared plus C squared
minus two BC cause a.
That is a formula for getting
the side. A.
What do the other ones look
like? Well, be squared is going
to be equal to C squared plus
a squared minus, two CA Cosby,
and notice. We've cycled the
letters through the formula
again and see squared is going
to be equal to A squared plus
B squared minus two AB cause
see. So the cosine formula
actually made up of 6 Formula
One or each of the angles.
That's three altogether, and
another one for each of the
sides. And again, that's another
three. When would we use these
formula? What are we being given
on this side? Well, obviously
we're beginning two of the sides
A&B in this case and an angle.
Let's just repeat the diagram
again. Well, we've got the
angle. A angle B Anglesey, and
the labeling little a. It'll be
a little C.
And let's have a look at this
formula. Here were finding, see
when we've been given a.
When we being given B and when
we've been given the Anglesey.
So we're using these formula.
Defined
aside.
See, when we're given
two sides.
And
the
angle
between.
Or the angle included.
Between. The
two sides. So
those are the six cosine
formula. You only need to
learn two of them, one
for the angle.
One for the side and then just
cycle the letters through to
find the others.
Another formula
is the
sign.
Formula. The sign
formula looks like this a
over sign a is equal
to B over sign B
is equal to see over
sign, see. Is equal
to. Two
are. What Earth is our?
Where did that suddenly come
from? Well, that's just a
complete the formula and what
our is equal to R is the radius
of the circum circle, so the
Circum Circle is the circle that
we can draw that will go through
all the points.
Of the triangle, and that's
our so where are.
Is the
radius.
Of
the
circum Circle, and
that's the circle that goes
through all the points of the
triangle. Because we can write a
over sign a is be over sign BC
over Sciences to our if we just
leave off the two are we can
turn that upside down and write
it assign a over a equal sign B
over B. Equal sign C over C and
we can use it that way up as
well. And when do we
use this? Well, if we
just look at that bit.
If we need to find one of these
four things, the side a, the
angle a, the side B, or the
angle be. Just need to find one
of those four things we've got
to know the three others, so we
have to know two angles.
And the side, and it is the non
included side. If we look at
this one. And we want to find
one of these four things. Say we
want to find the angle a. We
have to know the two sides and
the angle be and it will be the
non included angle. So we can
use this either to find aside
given two angles and aside
notices were given two angles,
we actually know all the angles
because the angles of a triangle
add up to 180 or we can use it
given. Two sides and an angle to
find a second angle. Now let's
have a look at some examples.
So we'll take.
An example where we begin
with a is 5.
The is 7.
And see is 10, so we are given
all three sides of the triangle
and having been given all three
sides of the triangle, what
we've got to do to solve the
triangle is find the three
angles. So that's going to be
our cosine formula, so we'll
start with cause a, which is
going to be B squared plus C
squared minus a squared all
over. 2 BC.
So we can put our
numbers into their B
squared. That's 7
squared plus 10
squared minus 5
squared all over 2 *
7 * 10.
Some settings are
49. 10 squared is
105 squared is 25.
All over 140 two sons
of 14 and times by
10. Arithmetic we've 100
add on 49 takeaway
25. That's 124 over
140. And what we need
to do is to work this out and
find out what the angle is.
The angle a is the angle
whose cosine is 124 over 140,
and for that we need to use a
Calculator.
So let's set a power
Calculator.
Turn it on.
Choose the right mode.
I mean radians. Normally in
doing these calculations we
would want to have our
Calculator in degrees, so will
just switch that into degrees.
Now we can work this out.
We want the angle
whose cosine is.
124 Divided
by 140. And
that is 27.7 degrees
working to one decimal
place. Well, that's one angle of
the triangle. I can go ahead and
use the formula again and find
the second angle of the
triangle, and then I can use
that information to find the
third one by adding the two that
I know together and taking them
away from 180 degrees.
So now let's take another
example where we need to use a
different set or formally will
take B equals 10.
C equals 5 and the angle
a equals 120 degrees. Let's
sketch this first of all so
we can see exactly what
information we've got. So here's
the angle a.
120 degrees.
B and C2 angles that
we don't know.
And here this is the side
littleby which is equal to 10.
The side little see which is
equal to five. This is the side
that we want to be able to find
little A. Well, we've got
two sides. And the angle between
them. So that suggests to us
that we want to use a squared
equals B squared plus C squared
minus two BC cause a because
this is what we're given. We
know be we know CB&C and we know
A and this will help us to find
a. So let's put the numbers
in. B squared B is 10, so
that's 10 squared.
C squared CS 5 so
that's 5 squared minus two
times B which is 10.
Time see which is 5.
Times the cosine of
120 degrees. So
this is 100.
Plus 25 five squared.
Minus 2 * 10 is 20 *
5 is 100.
Times and the cosine
of 120 is minus
nought .5. So this
is 100 + 25, that's
125. 100 times by minus
1/2 is minus 50, but we've got
this minus sign here, so that's
minus minus 50 is plus 50
altogether gives us 175. It's a
that we're after, so we need to
take the square root of 175 and
the square root of that is 13.2
three and will give the answer.
The two places of decimals.
So now let's take
a third example.
And in this case
will have seen this
8. B is
12. And the
angle C is 30 degrees.
OK, first we need a sketch.
What information have we been
given? So label are triangle
ABC. Label the sides with clay.
It'll be little C and put the
information on so see is 30.
Be. 12
And the side little C is 8.
So we've got two sides
and an angle.
We don't have the angle included
between the two sides, so this
is the sine formula.
So remember that.
A over sign a is B over.
Sign B is C over sign. See
or we can use it the other
way up. Sign a over A is
signed B over B is sign C.
Oversee now which bits do we
want? Well, we've got bees 12.
So we've got that one.
We've got little C is 8, that's
that one, and we've got the
Anglesey is 30. That's that one.
So it looks as though it's this
box that we're going to be using
and the angle we're going to be
finding is B, so let's work with
these because the sign B is on
the top. So let's write that
down separately. Sign B over B
is equal to sign C over C,
and let's put some numbers in.
This is signs B over 12 is
equal to sign of 30 degrees over
8. And so sign
B is 12 times
sign 30 degrees over
8. That's fairly complicated and
I could use a Calculator
straight away, but one of the
things that I do recognize here
is that sign 30 is 1/2.
So I've got 12 times by
1/2. And divided by 812
times by 1/2 is 6
still to be divided by
8, which gives me 3/4
or nought .75. So my
angle that I want be
is the angle who sign
is North .75.
So let's bring up the
Calculator again.
We want the angle who sign is.
Nought .75
And we see that the angle
that we get is 48.6 degrees
working till 1 decimal place.
Now.
There is a potential
complication here.
Let's go ahead and just have a
look at the possibilities in
this particular question.
And it's to do with
these angles because B needn't
just be 48.6 degrees.
Remember that C is
30 degrees, and that's
fixed. B is 48.6
degrees or 180 -
48.6 degrees could be
either. Both have a
sign. Of North
.75 so B. Might
be that, or taking
this away from 180
one 131.4 degrees.
Now the question is, what's the
other angle? Is it possible to
have an angle a with these sets
of figures? Well, in the first
case we can have C is 30
degrees, B is 48.6 degrees, and
the angle a will be 180 minus
the sum of these two. In other
words, minus 78.6.
And so that will be
101.4 degrees. So yes, we
can have that particular
combination. What about the
other combination? See is 30
degrees be this time would
be 131.4 degrees.
And so a would
be equal to 180
minus the sum of
these 261.4. So this
gives us an angle
of 18.6 degrees. It's
still possible. And this is the
case that we came across before,
where we've got one side.
Where we've got an angle?
And where it's possible for the
other side to meet twice, once
there. And once there and still.
Produce. A triangle that works.
What this means is that we know
this side. We know that one and
that one 'cause they're the
same, so we would have two sides
to find, one for the smaller
angle A and one for the larger
angle a. A difficult one, but
we do have two distinct
triangles from the same set of
information. OK, we've dealt
with the sign formula. We've
dealt with the cosine formula.
What we want to have a look at
now is just the set of formula
which will give us the area of
a triangle. Let's just draw a
triangle.
Most people are happy with the
idea that the area of a triangle
is 1/2 times by the base times
by the height.
What does that mean in this
triangle? Well, it's this way
up, so to speak. This is the
bottom of the triangle, so this
is, let's say the base. What's
the height? The height is the
distance of the highest point
from the base, and in this case
we mean the perpendicular
distance so that that line meets
the base at right angles, and
then this is the height.
What if we are given?
Information about this triangle.
So let's label it in the same
way as we did before.
Now this means the base in my
picture is the side little A.
That would be the side
littleby and that will
be the side it will see.
Let's look at this right
angle triangle.
Here the hypotenuse is, see.
The thing that I've labeled the
height. Is the side that is
opposite to the angle be?
Let's assume that I know the
angle B and I know the side.
Little C. Then in this right
angle triangle. The
height. Divided by the
hypotenuse, C is equal to or.
Remember, height is the opposite
side and so that is going to
be sine be. So what I have
there is that the height of this
triangle is C sign be.
The base is little a, so
I have the area is 1/2.
AC sign be.
At reasonable to ask, since this
formula involves. 3 pieces of
information. Two sides in the
angle. Can I cycle through
again? Can I cycle these letters
through? Well, let's have a look
over at this side.
And again, we see that the
height is the opposite.
To Angle C&B forms the
hypotenuse, and so I can
have the area is 1/2
times the base A.
Times B sign. See because that's
what the height is in this right
angle triangle. It's not too
difficult to see that the
remaining one is going to be 1/2
BC sign A and so we have 3
formula that give us the area of
a triangle. Again we need only
learn one of them.
Because we get the other
simply by cycling through
the various letters.
So the area of that
particular triangle.
ABC
Angles
AB&C.
Area formula that we had
were a half a B
sign, see. And a
half BC sign a.
And a half see a sign
B. Let's check what information
we've got here.
AB sign CAB the angle
see so again it's two
sides and the angle between
two sides.
And the
included.
Angle.
We don't always get that sort of
information, though. One of the
things that we do know is we can
be given all three sides of a
triangle. What then? Well, an
ancient Greek, wouldn't, you
know, by the name of hero?
According to some texts or her
and according to others?
How to formula for calculating
the area of a triangle when you
know all three sides and his
formula goes like this. The area
is the square root of. You can
tell it's going to be a big
expression 'cause I put a big
bar on that square root sign S
times S minus a Times S Minus B
Times X minus C.
What's SAB&C are the lengths of
the sides, but what's S?
Where? S is
equal to a plus B
Plus C all over 2.
The semi perimeter.
Semi perimeter because apples
people see is the perimeter.
It's the distance all the way
around. We divide it by two.
It's the semiperimeter. So this
is heroes or herons formula for
finding the area of a triangle.
So let's have a look at an
example of each.
So in the first case.
Will take a is 5B is 7 and C
is 10 and we're trying to find
the area of a triangle and what
we've been given is the lengths
of the three sides little a
little bit and little see. So
that means we're going to have
to use herons formula. The area
is the square root of S Times S
minus a Times S Minus B times S.
Minus see
where. S equals A plus
B Plus C all over 2, and
that's got to be our first
calculation. So we 5 +
7 + 10 all over
2, five and Seven is
12 and 10 is 22
/ 2 gives us 11.
So now the area is
equal to the square root
of 11 * 11 -
5. Times 11 - 7
* 11 - 10.
Which is the square root
of 11 * 6 *
4 * 1?
Square root of 6 times by 4
times by one is 24 and what
we need is 24 times by 11.
11 four 44211 is 22 and the four
gives us 26 so the area is going
to be the square root of 264 and
again we just need the
Calculator to be able to work
that out. Turn it on.
Get into the right mode and we
want the square root.
Of. 264
And that is 16.223
significant figures, so the
area is 16.2 square
units. I didn't say what
the units were here for
the lengths of the sides,
so these are just units.
Square units for the area.
So now let's have a look at
an example using another set of
data. So in this case will take
the 10. And see to
be 5. And the angle A to
be 120 degrees and we want the
area of the triangle. So a quick
sketch. Let's just make sure we
know what we've got a BC this is
120 B we know to be 10 and
Little C we know to be 5. So
we've been given two sides and
the angle between the included
angle and so straight away we
know where. All right, to use
the area is 1/2 BC Sign
A. Put the numbers
into the Formula 1/2 *
10 * 5 times sign
of 120 degrees.
So 10 times by 5
is 50 and a half
is 25 times by the
sign of 120 degrees.
And so we need the Calculator
again to help us work this out.
So bring the Calculator up and
turn it on. And get into the
right mode and now we need 25.
Times the sign
of 120 degrees.
And that gives us an area of
21.7 working to three
significant figures. And again
this will be 21.7 square units.
I didn't specify what units
these were in. Had they been
centimeters? Then this would be
21.7 square centimeters.
So now. Let's just sum up what
we've got. We've
got our set
of cosine.
Formally.
One representative is cause a is
B squared plus C squared minus
a squared all over 2 BC,
and there are another two like
that. For the angle B and
for the angle, see and we
know how to generate them by
cycling the lettuce through.
We also know that A squared
is equal to B squared plus
C squared minus two BC cause
a. And we know that there are
another two like this.
We know how to get them. We
simply cycle the letters through
the formula. This one
we find angles.
Using three
sides. This
one we find aside.
Using two
sides. And
the included.
Angle. We
next half hour sign
formula.
A over sign a is equal to B
over sign B is equal to see over
sign C is equal to two R and
remember that are was the radius
of the circum circle.
The circle that went through all
three points of our triangle and
we can turn this the other way
up and we can say sign a over
a sign B over B is sign, see
oversee. And we use this
when were given.
Two sides plus an angle, but it
must be the non included angle
or when we're given two angles.
And aside.
So those are our cosine.
Formally, those are our sign
formula. The area formerly we've
just had. Remember herons
formula? And the other formula
for the area half a B sign. See,
that's just one representative
and the others. We cycle the let
us through. And finally, let's
see if we can connect two of
these sets of formally, that
we've just had.
Just draw a
triangle. So that we can
recall the notation. The capital
letters for the angles, little
letters on the sides opposite to
the angles for the lengths of
the sides now.
Area formally told us that
the area was equal to
1/2. AB
sign C.
1/2
BC. Find
a under half see
a sign be.
Well, let's take two of these
and actually put them equal to
each other. After all, they're
both expressions for the area of
this triangle, and so they are
in fact equal, so it write them
down a half a bee.
Sign of say is equal
to 1/2 BC sign of
A. Will immediately we see,
we've got a common factor on
each side of 1/2.
And a common factor on each side
of B so we can cancel those out
on each side. That will leave us
on this side here with a sign C.
And on this side
it leaves us with
C Sign A.
Now if I divide both
sides by sign A and
divide both sides by sign,
see, then I have a
over sign a.
Is equal to see over sign, see.
So working with the formula for
the area. We have derived a
part of the sine formula.
If I take another two of these,
say these two together, I'll get
another bit of the sign formula.
So we see that these two are
related. The area formula and
the sign formula, and we can
derive the sign formally from
the area formula very simply.