www.mathcentre.ac.uk/.../5.10%20The%20addition%20formulae.mp4

There are 6 addition formula.
I'm not going to prove them.

But I am going to do is start
with one of them.

And derive a second one.

Then I'm going to take another
one is given and derive a second

one from that, and then we're
going to use those four to help

us derive the final two. So this
is the one we're going to start

with. The sign of a

plus B. This is where they get
their name from addition formula

because here we have a sub and
we're going to find it sign.

This breaks down.
Assign a Cosby

Plus calls a

sign be. Notice in terms
of remembering it, we keep the

A&B in the same order and the
signs and the cosines alternate.

So what about the
sign of a minus

B? Well, I just have
a think about this minus B.

It can be the sign
of a plus minus B.

And now. The formula that we

had here. Can be exactly the
same As for this one.

But in this one we can replace

B. By minus be
so let's do that sign.

A. Calls
of minus B.

Plus calls a.

Sign of minus

B. Well, the
sign is OK.

And the cause of minus B is
just cause big.

The cause a is OK, but the sign
of minus B is minus sign B, so

that's going to change that plus
sign into a minus sign and I can

just write sign be at the end.

So that's the second.

Of our addition

formula. If we've got these
for sign, it seemed reasonable

to expect that will have the
same things for cause for

cosine, so let's have a look
cause of A+B.

Well, this is cause a
caused B minus sign, a

sign be and that's our
starting one. So let's do

the same as we did
here cause of A-B.

Will rewrite the minus B
as a plus minus B.

And then we can replace the bees
in here in this first formula

with the minus be there so will
have cause a.

Cause of minus B.

Minus sign, a sign
of minus B.

Now the cause of minus B is
just cause be, so we have caused

a caused B.

And then we have minus sign a
Times by sign of minus B.

But the sign of minus B is a
minus sign be, so we have two

minus signs together, giving us
a plus sign plus sign a.

Sign be.

So we've now got 4 addition
formula. Let me turn over and

write those down as a group.

Sign of A+B?

Is sign a?

Call speedy close calls
a sign be.

Sign of a minus
B.

Is sign a
Cosby minus cause

a sign be?

Cause of

A+B? Is cause
a calls B?

Minus sign a

sign B. The
cause of A-B is

cause a caused B
plus sign a sign.

So there are our four.

Addition formula. Now we did
promise 6 but these are the fall

really basic ones. And really
these are the four that you've

got to learn.

The other two will presumably

be. Tangent tan of

A+B? Well, we
can derive these from the

others. Tan is sign over
cause so we have the

sign of A+B divided by
the cost of A+B.

So we can now make the
replacement for sign of A+B by

its expansion here.

And cause of A+B by
its expansion here?

Sign a caused

B. Close calls a

sign be. All
over.

Calls a calls B.

Minus. Sign a

sign be. Now this looks
very unwieldy and it would be

nice if in the same way that
this is in terms of signs and

causes and this ones in terms of
signs and causes. I could have

tan of A+B somehow in terms of
tangent and possibly cotangent,

but certainly I want to have
some tangents in there. So what

can I do? Well, look at this
term here cause a Cosby.

Supposing I divided everything
by that term cause a Cosby.

Well, that would give me one
here and I'd have this over

cause A cause be of course I'd
have sign over cause sign over

cause for each of A&B so I have
tan a tan be there.

Would I get anything nice on the
top? Well, let's write it down

in full. So we're
going to divide

everything by calls

a Cosby. And
I have to divide

everything by this because

I've got. A pair of
equal signs here a balance. So

what I do to one side I must do
the other. I must do everything

to all of the terms to preserve

the equality. Now that looks
absolutely awful. Absolutely

massive algebra. So
how can we

make it any
simpler? Well, let's

just go back
to the denominator

here. This bottom

term. Cause a over cause
a cancels down Cosby over. Cosby

cancels down. This is just one.

Sinai over 'cause I is Tanay and
sign be over. Cosby is tan be.

So this is just tan a tan B.

Have a look at this. Well,
there's a common factor on top

and bottom here of Cosby. I can
cancel that out. Leaves me with

sign a over cause a tan a.

Here the causes go out.

Sign be over. Cosby
leaves me with Tan

B so. What we end

up with. Turn of
A+B is 10A.

Plus 10B.

All over 1
- 10 a

10B.

Now we can do the same
again for tan.

All A-B and we've got two ways
of approaching it, first of all.

We could work through this again
X set with tan of A-B is the

sign of a minus B over the cause

of A-B. Or we could work through

it again. By making the same
replacement as we did before,

and rewriting this as the time
of A plus minus B.

Let's be assured that what it is
going to give us, which ever way

we do it.

Is going
to be

that. And So what we've got now
our our six.

Addition formula.

Sign of A+B sign of A-B cause of
A+B cause of A-B. These are the

ones you must know and learn
from those four you can derive

these. It's a help if you know
them. If you can learn them. The

important thing is to be able to
recognize them when you see them

and recognize when you need to

use them. Let's have a look at
three fairly typical examples of

the use of these. In many ways,
it's practiced that we're

getting at recognizing these
particular formula, 'cause we

may need them more often when
we're doing other, more

complicated manipulations. So
first of all, let's have a look

at this particular problem.

If we know that sign of

A. Is 3/5.

And that the cause.

Of B is 5

thirteenths. Then
What's the sign

of A+B?

What's the cause

of A-B?
OK.

We've got sign a but we don't
know anything about cause a

seemingly. We've got Cosby.

And we really don't know
anything about sign be.

We don't know much about A and
be really because a could be

either an obtuse angle or an
acute angle. So we really need a

little bit more information. So
let's say that A&B are acute.

In alerts that both less than 90

degrees. Well, if the boat less
than 90 degrees, one of the

things we can do is represent

the angle. And the sign with a
right angle triangle. So I just

have a look at that.

Right angle triangle.

Right angle there the angle a
sign a is 3/5 opposite over

hypotenuse, so that's three. The
side opposite the angle A and

that's five the hypotenuse which
is always opposite the right

angle, always the longest side.

Pythagoras tells us that this
other side has to be 4.

And so now that we know the
adjacent side, we know

everything about angle a. We
know it sign, we know it's

cosine and if we want we can
find its tangent.

Let's do the same.

The angle be.

Here is the angle be in its
right angle triangle and were

told that Cosby is 5 over 13.

Cosine is adjacent over
hypotenuse. This is the side

that's adjacent, so there's
five. This is the hypotenuse,

the longest side, the side
opposite the right angle, so

that's 30. So using Pythagoras
theorem, this side is 12.

Now notice I said using
Pythagoras Theorem, but it was

as though I knew these three
455-1213. I do know them and

you've got to get to know them
as well. Pythagorean triples,

simple triples of integers,

whole numbers. That a Bay,
Pythagoras's theorem if you have

them at your fingertips, can we
call them easy? Easily? You find

this so much better in working
in trigonometry 'cause these

numbers are used an awful lot.
OK, then, let's have a look at

what we can do. Sign of A+B? The
addition formula tells us they

sign a. Cause B.

Close calls a sign

be. Equals.
Sign a we know is 3/5.

Times, Cosby, and we know
that one. It's five thirteenths

Plus cause a we can
read cause a off here

it's adjacent over hypotenuse so
it's 4 over 5.

Times by sign B and we
can read, sign be off this

triangle it's opposite over
hypotenuse 12 over 13.

Do the arithmetic 3 fives
are 15 and five 1365,

so that's 15 over 65
+ 4 twelve 48.

And 5:13's again are 65 and
so that gives me a fraction

63 over 65.

We had cause of

A-B. So let's approach that in
exactly the same way.

Will quote our
expansion. Our

addition formula cause
of a Cosby.

Plus sign of a sign

of B. Substitute our
values cause a That's four

over 5 adjacent over hypotenuse.

Times, Cosby. That's five
over 13, adjacent over

hypotenuse.

Plus sign a That's opposite
over hypotenuse, three over 5.

Times by sign B.
That's 12 over 13,

opposite over hypotenuse.

Now here we've got some
fractions and it might be

tempting to cancel the fives
here. Five goals, five goes, but

five times by 13 is the 65 and
five times by 13 is the 65 to

have the same denominator.

So I'm not going to cancel.
4 fives are 20 over five

1365. Plus

Three twelves
are 36 over that

65 again giving
me 56 over 65.

So that's one way in which these
can be used.

Let's have a look at another

way. Supposing we were asked to
find what's sign 75, but no

don't reach for your Calculator,
you've got to workout sign 75 as

an expression, not as a set of
decimals, but as an expression.

What we've got to think about is
how can we make 75 from angles

who sign and cosine's? We know

very well. Well, 45
+ 30 gives us

75. I'm 45 and 30
are two of the angles that

we know sines and cosines for
exact sines and cosines.

So this would be sign
of 45 cause of 30.

Close calls of
45. Sign of

30. Sign 45
that's one over Route

2. Times by cost 30
will cost 30 is Route 3

over 2.

Plus cause of 45 is one
over Route 2.

Times by sign of 30, which
is just a half.

So we have Route 3. That's one
times by Route 3 over 2 Route 2.

Plus one over and again two
times route 2 is 2 Route 2.

We've got the same denominator,
so 2 Route 2.

To Route 2 and on the
top Route 3 + 1.

Now leave not won't like that we
might try and get rid of this

Route 2 in the denominator, but
just for the moment, let's leave

it like that. There's nothing to
be gained by doing it at this

stage, and let's take another

example. Let's have a look.

An 50 What we
need to be able to do is to find

15 in terms of angles that we
know lots of things about.

And of course.

One of the ways of doing this,
but only one is to do 60 - 45.

So that means that we need the
addition formula that's to do

with tangent and the one that's
to do with the tan of A-B.

So this is 1060 - 1045.

Over 1
+ 10

sixty 10:45.
So now let's put the numerical

values in. Now the tangent of 60
is Route 3 and the tangent of

45 is one.

Over. One plus
Route 3.

Times by one.

So we'll have root 3 - 1
over Route 3 times by one is

just Route 3.

Plus one. Now that's
all right, and it's correct.

Doesn't look very nice and we
tend to have a tradition of not

leaving things like root 3 plus
one in the denominator.

So one of the ways of tidying

that up. Is to multiply top
and bottom of this fraction by

the same thing?

Multiplying top and bottom by
the same thing keeps it the same

value, but what should we
multiply it by? Well, I'm going

to choose to multiply by Route 3

- 1. Not be cause that's what's
in the numerator there, but

because this is the difference
of two squares A+B, A-B and

A+B times by A-B, let's just
write that down up here.

Is A squared minus B

squared? So that means that
on the bottom I have Route 3

times by Route 3A squared.

Which is just three minus B
squared. B was just one, so

that's just minus one and 3 - 1
is 2 and integer not absurd. Not

one of these things with a
square root sign attached to it.

What have we got on top? We two
brackets that we're multiplying

together route 3 by Route 3 is
3, route 3 by minus one is minus

Route 3. 1 by minus one
by Route 3 is minus Route 3

and minus one with minus one is

plus one. 3 plus One is 4 -
2 RT. Threes were taking away

all over 2 and now there's a
common factor of 2 on the top

and on the bottom 2 into 4 goes
to an two into minus 2. Route 3

goes minus Route 3.

And two into two goes wall, so
we needn't worry about that one.

So there we've got a second
example to add to the first one

that we did. Let's take one more

type. Of example, where we can
make you solve the addition

formula and this is where we get
to simplify an expression. So

supposing we've got the sign of
90 plus A.

Could we have this simpler?
Could we just have, say, sign a

or minus sign a or cause a or
something like that? Does it

have to be 90 plus a? Well,
let's have a look. It's sign of

90. Calls of

a. Plus the
cause of 90

sign of A.

Sign 90 is one.

Cause I just cause I so first of
all we have one times cause a

which is just cause a.

But cause of 90.

Is 0. Times by sign a
well, anything times by zero is

0 so we just left with cause a.

So an expression such a sign
of 90 plus a reducers tool

cause a what about something
like the cosine of 180 minus

a? Well, this
is cause 180

cause a.
Plus sign

180 sign

a. The cosine of
180. That's minus one, so minus

one times by cause a is minus
cause a sign of 180 zero and

sign is just sign a but zero
times by anything is 0.

So we're just left with minus
calls a so we can see that these

addition formula help us to very
quickly simplify and get in

terms of this angle a.

Possibly quite complicated

expressions. But the basic thing
is that these 6 addition

formula. Again, I stress it. You
have to learn you have to know.

But most importantly, you've got
to recognize them and recognize

their use when you see them and
when you see the situation.