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.