www.mathcentre.ac.uk/.../5.7Cosine%20Formulae%20sine%20formulae%20area%20of%20a%20triangle%20formula.mp4

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.