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.