www.mathcentre.ac.uk/.../acosxbsinxrcosxa.mp4

In this video, we're going to be
having a look at a particular

form of trigonometric function.
So in order to get into this

form, let's just consider 3
cause X +4 sign X.

Now it's two terms.

And ideally, if we were going to
solve an equation or do

something useful with it, it
might be better if it was one

term. So the question is, how
can we bring these together?

Let's have a look at the graph
of this function.

So we'll get into the right mode
on the Calculator and graph and

will graph three cars.

X.

Plus four sign X.

And now we've graphed it.

We can see that it's looking
like a periodic function.

Looks In other words, rather
like sign or cause.

So if we look at the little
ticks on the Y axis, we can see

that it's approximately 5 high.

On the peaks and approximately 5
deep on the troughs.

So if it looks like causal
sign, let's graph.

Cause. X. So
there's cause X plowing along.

You can see they both look
periodic and what we want to do

is to change cause X into.

What we had to begin with.

So let's remember that the
height of the peaks was five and

the depth of the troughs was
five. So let's multiply cosine

by 5, so we'll graph.

5 calls X.

And what we see is, we've gotten
almost exact copy. It's just

displaced a little bit.

Things are happening a little
bit too soon.

OK, that being the case, how
much too soon it seems to come

if we look at the ticks on the X
axis and look at the distance

between the two peaks.

We seem to be about 1 unit on
the X axis early.

So let's correct that and
let's graph.

5.

Kohl's

Of X minus one.

And that's almost correct. It's
just covering over.

The graph and making it a little
bit thicker, so it seems that we

can write this as 5 cause X

minus one. Now my Calculator
was working in radians, so

this one is one Radian.

3 calls X +4 sign. X
is 5 cause X minus one.

Well, 3, four and five or a
pythagoreans triple.

3 squared +4 squared
equals 5 squared.

So that kind of size to us. If
we had any function, A cause X

Plus B cynex, could we bring
this function? These two

functions together as one
function like that, and if we

did it with the number that went
in front with the square root of

A squared plus B squared.

Cause of X minus something,
let's call it Alpha. And

then what is Alpha?

OK, all we've done is some graph
work, made some guesses what we

have to do now is work with
algebra and work with our trig

functions and our trig
identities to try and see if

this little bit here at the

bottom. Can actually be made
to work, so that's our next

step. So let's start with where
we want. We want to end

up a number are multiplying
cause of X minus Alpha.

Equals are now cause of
X minus Alpha is one

of our addition formula, so
we can expand it as

cause X cause Alpha plus

sign. X

sign Alpha That's
multiply out that bracket are

cause X cause Alpha plus
our sign X sign Alpha.

Now I want to rewrite this
so it's very clear what I

am multiplying cause X by. So
here I am multiplying cause X

by our cause Alpha and I'm
multiplying synex buy our sign

Alpha. So let's make that very
clear off 'cause Alpha times by

cause X plus our sign Alpha
times by sign.

Thanks and what we want is
that this should be a cause

X Plus B sign X. In
other words, this our cause

Alpha will be A and this
are sign Alpha will be B

so we can make that
identification are cause Alpha

is a. And B is
our sign Alpha.

Now, how can we bring these
together? Well, one way is to

notice that cost square plus
sign squared is one. So let's

Square this one, A squared
square. This one and add the two

together. A squared plus B
squared is R-squared cost

squared Alpha plus R-squared,
sine squared Alpha.

And there's a common factor of
R-squared that we can take out,

leaving us Cos squared Alpha
plus sign squared Alpha.

This is one of our basic
identities, Cos squared plus

sign squared at the same angle
is always one and so that gives

us our squared. So the thing
that we said was a possibility,

in fact is a result that if we
take A squared plus B squared,

take the square root, we do
indeed end up with are the

number that goes in front here
of the cosine.

But what about the Alpha? Have
we got enough information at

this point here to find Alpha?
Well, let's bright these two

equations a equals our cause.
Alpha and B equals R. Sign Alpha

down again, but this time let me
write them above one another.

So we write these down
again, but this time.

Going to write.

The top and the other one

directly underneath. By doing it
like that, it suggests the

possibility of dividing this
equation into this equation and

so we have B over A is equal
to R sign Alpha over our cause

Alpha. We can divide top and
bottom by R and we're just left

with sign out for over 'cause
Alpha, which we know is Tan

Alpha. And so now we
can workout what Alpha is,

because Alpha is the angle whose
tangent is B over A.

So what we've got now is
that we can write any expression

of the form a Cos X
Plus B Sign X.

As a function which is our
cause X minus Alpha.

And we know that all will be the
square root of A squared plus B

squared, and the Alpha will be
the angle whose tangent is B

over a. And that is a
very very useful tool to have

at ones disposal. It reduces two
trig functions to one trick

function makes it so much easier
to deal with.

So let's have a look at how we
can use this to solve equations

and also how we can use it to
determine Maxima and minima of

functions. So will begin with an

equation. Route 2.

Cause X. Plus
sign X equals

1. Now if we
compare this with the standard

form we've been working with, A
cause X Plus B Sign X and

a is equal to two and B
is equal to 1. So we can

calculate are. All is the square
root of A squared plus B

squared. So that's the square
root of 2 all squared plus one

squared. 2 all squared is 2.

1 squared is just one, so this
is Route 3.

So what we've got is
now our cause of X

minus Alpha, so we've got
Route 3 cause of X

minus Alpha equals 1.

What we have to do is we've got
to workout what Alpha is so.

We know that Tan Alpha
is equal to B over

A. And so that's
one over Route 2.

And so we need to know what does
that tell us about Alpha?

Well, one of the things we
haven't specified is what's the

range of values of X and what
units are we working in. So

let's say we're going to be
working in radians, and that we

want X to be between plus and

minus pie. So if X is in radians
then Alpha's got to be in

radians. I will now bringing the
Calculator in order to work that

out. So I want the angle

whose Tangent. Is.

1. Divided by.

Square root of.

2.

And that angle is nought.
.615 lots of other figures,

but will stop there for
the moment radiance.

So now I know what our is and I
know what Alpha is.

I'm not going to write down
North Point 615 for a little

while. I'm going to keep it as

Alpha. But the equation that
we're now trying to solve is

Route 3. Cause of X
minus Alpha is equal to 1

and we know that we want
X to be between pie and.

Minus pie. Let's have a look
at What is this cosine want to

do that? We're going to divide
both sides by Route 3.

And so we have cause of
X minus Alpha is one over

Route 3. So what is X
minus Alpha?

Let's just draw a little graph.

To help us see what we're

looking at. Between
plus and minus pie.

The cosine graph.

Looks like that.

There's plus pie.

As minus pie, this is minus π by
two, and this is π by 2.

So we're looking for two
answers across here.

Come down. To there.

So let's have a look for one of
them. An hour Calculator will

tell us this one here.

So the angle that
I want is the

angle whose tangent is

one. Divided by the square
root of 3.

And that is.

Nought

.95 5.

Now if that's not .955
there, remember our

Calculator will give us this
one. Then the other one by

symmetry is minus N .955.

Well, now we've got X minus
Alpha. What we need is X, so

this will be X equals nought .9.

55 plus
Alpha or

minus nought.
.955 plus

Alpha. And if
you remember Alpha, we

calculated as nought. .615. So
we can write the value

of Alpha rent.

And work out these two.

Values of X, so
we add these two

together. We're going to
get 1.570.

And if we take these two.

We're going to get minus
nought .340 so if we

work to two decimal places,

that's 1.57. And minus
nought .3 four radians.

So let's take another example.

It's time you have cause
X Minus Route 3.

Sign X is equal to 2.

And we'll take X to be in
degrees and between North.

And 360.

OK. Well, slight
difference here. We've got a

minus sign and not a plus sign,
so let's see what this means. A

must be equal to 1 because we've
got one cause X&B must be equal

to minus three.

And this is going to make a
difference to what happens.

Let's leave it on one side for
the moment and go ahead with our

calculation. Are will be the
square root of A squared plus B

squared. Which will be the
square root of 1 squared plus

minus 3 squared.

1 squared is one.

Minus Route 3 squared is 3, so
it's a square root of 1 + 3,

which is 4 and so are is 2.

So what we've got now is
2 cause X minus Alpha equals

2 and we can divide
throughout by these two to

give us cause of X minus
Alpha equals 1.

This should be relatively easy
to solve provided we can workout

what Alpha is. So what is
Alpha now? Remember we calculate

Alpha's tan Alpha is B over
a, which in this case is

minus Route 3 over 1, which
is just minus Route 3.

This is a problem to us.

It's a problem because
previously all the answers have

been positive and we have had
positive A and positive be and

we've gone straight for the

first quadrant. Let's just
remind ourselves what we mean by

Quadrant. I'll write down again.

10A. Which
is calculated as B over a.

Let's just remind ourselves what
these are that are sign Alpha

and our cause Alpha.

And in this question then minus
Route 3 over 1.

Revise what we know about
quadrants. This is the first

quadrant. The second quadrant.

The third quadrant, the 4th
quadrant going round

anticlockwise. Now in the first
quadrant, all of our trig

ratios, a for all that all

positive. In the second
quadrant, only the sine ratio is

positive. In the third
quadrant, only the tangent

ratio is positive, and in the
fourth quadrant only the

cosine ratio is positive.

Now we have 10 Alpha is
minus Route 3 over 1.

So it's negative, which means
that we must either be in the

second quadrant or the 4th
quadrant, which is where tangent

is negative. So we have a choice
to make our angle Alpha is in

here or it's in here.

Let's have a look what we can
see from here are sign Alpha is

minus Route 3 are cause, Alpha

is one. That means sign Alpha
must be negative and cause Alpha

must be positive.

And it's in this 4th quadrant
where cause Alpha is positive

and sign Alpha is negative.

So our tan Alpha is minus Route
3 is in here somewhere.

So let's draw a picture, this
time looking the same, perhaps,

but where's the angle? Well,
we've got to have minus Route

3 and one.

This is our angle here.

Alpha

So that what we can see is that
if tan of Alpha is equal to

minus Route 3, then Alpha must
be equal to. Now the angle whose

tangent is plus Route 3 is 60
degrees, so this one must be

minus 60 degrees.

OK, we've got Alpha sorted out.
Let's just remember what our

equation was. Our equation. We
had got down to cause of X

minus Alpha was equal to 1.

So now we need to put these two
pieces of information together

to help us solve the full
equation. Help us find the value

of X so it's turnover.

And write these down. Alpha we
know is minus 60 degrees and

we're looking to solve the
equation cause of X minus Alpha

is equal to 1.

And we know that X is to
be between North and 360

degrees, so let's sketch our
cosine curve between North and

360 degrees. We know that it
looks like that there's 360.

There's 180, and here we have
90, and here 270 to squeezing

that in there. We know that this
goes down as far as minus one.

And these maximum points are up
at one. So what we can see

there is that the angle whose
cosine is one is either zero

degrees or 360 degrees, so X
minus Alpha is equal to 0

degrees or 360 degrees.

What is Alpha? Well, it's minus
60, so X minus minus 60

is equal to 0.

All 360 degrees.

That's X plus 60 is equal
to 0 or 360 degrees and

now we can take 60 away
from both of these. So X

is minus 60 degrees or 300

degrees. Remember the original
range of values of X was between

Norton 360, so this is not an
answer minus 60 degrees. This

one is the only answer within
our range of values of XX is

equal to 300 degrees.

Let's turn over.

Now let's have a look at what
might be termed a calculus type

example. In other words, a
question that's going to involve

us with Maxima and minima.

So what about this function
F of X is 4?

Cause X +3.

Sign X. Minus 3.

Water, it's turning points. What
are its maximum and minimum

values? Well. Knowing what
we do know from the previous

work that we've done, we know
that we can express this bit for

cause X plus three sign X as
5 cause of X minus Alpha.

And then the minus three where.

Tan Alpha is 3
over 4B over A.

Now this function is now very
easy to deal with because we

know all about a cosine

function. The maximum
value of cause.

Is

warm. Never
gets any bigger than one, and so

the maximum value.

All F of X.

Is. Well, it must
be 5 times by one.

Minus three, which is just two.

And when does that occur? Well,
let's think the maximum value of

cosine is one when.

The angle. Equals

0. Well, in this case
the angle will equal 0.

When X equals Alpha, so very
quickly we found out exactly

where this maximum value is
exactly at what point?

This function F of X has its

maximum value. What about the

minimum value? The least

value. Well, let's think about
Co sign the least value of

cosine is minus one never goes
any lower than minus one. When

does that happen? That happens
when the angle is.

Pie. Or 180.

So what we know there is that
the minimum value.

Of F of X most occur when
this is at its minimum. In other

words, minus one, so it be 5
times by minus 1 - 3 which

is just minus 8.

And when will that happen?
Will happen when the angle

X minus Alpha equals π.
In other words, when X

equals π plus Alpha.

So very quickly and with a
minimum amount of work, we've

established a maximum value
under minimum value and based

upon this information we could
go ahead and rapidly sketch the

curve. So again, we see that
this form our cause X minus

Alpha is a very powerful form
for us to know how to use and

for us to be able to formulate
from expressions such As for

cause X plus three sign X.

It's a form that you really
do need to work with, and a

form that you really do need
to practice. Initially. It's

not so easy to get your mind
around, but it is possible

to do so and it will pay
great dividends if you can.