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.