www.mathcentre.ac.uk/.../5.11The%20double%20Angle%20Formula.mp4

In this video, we're going to be
looking at the double angle

formula. But to start with,
we're going to start from the

addition formula. Not all of
them, just the ones that deal

with A+B. So let's just write
those down to begin with sign of

A+B, we know is sign a.

Cause B. Post
cause a sign

be. Next, we want
the cause of A+B, which

will be cause a calls

B. Minus sign, a
sign be and finally

the tan one tan
of A+B, which will

be 10 A plus

10B. Over
1 -

10 a
10B.

So those are three of
our addition formula.

And each one is to do with A+B.

So what happens if we let
a be equal to be?

In other words, instead of
having a plus B, we have a plus

a. So that would be
sign of A plus a would

be signed to A.

What does that do to
this right hand side? Well

gives us sign a cause,
A plus cause a sign

a. In other words, these two
at the same, so we can just add

them together. So sign of 2A is
2 sign a cause A and that's our

first double angle formula
double angle because it's 2A

where doubling the angle.

So what is it sign and so
on. Let's do the same with

cause. Let's put a equal to
be. So will have cars to a

is equal to what it was
cause a Cosby. It's now

going to be cause A cause a
witches caused squared A.

Sign a sign be when it's now
going to be sign a sign a

which is sine squared minus sign

squared A. And that's how

a second. Double angle formula.

Doing the same with Tan
Tan 2A is equal to.

10A plus 10 B this is now
angle a so it's 10A Plus 10A

which is 2 Tab A.

All over 1 - 10, eight and be.
But this is now a instead of B,

so it's tanae.

10 eight times by
10 A is 10 squared.

1 - 10 squared A.

And here are our three
double angle formula again

to be learned to be
recognized and to be used.

Let's just have a look at this
one cause to A.

White pick out this one. Well
this right hand side which is

the bit that interests because
it's got cost squared and sign

squared in it and there is an

identity. That's to do with cost
squared. Plus sign squared

equals 1. What that means is we
can replace the sine squared.

And get everything in terms of
Cos squared. Or we can do it the

other way round.

So I just have a
look at that one.

Cause to a
cost squared, A

minus sign squared

a butt. Cost
squared A plus sign,

squared A equals 1.

In other words, sign squared a
is 1 minus Cos squared a so

we can replace the sine squared
here in our double angle formula

by one minus Cos squared, so
will have cause to A.

Is cost squared A minus
one minus Cos squared A?

Using the brackets, notice
to show I'm taking away all

of it and now let's remove
the brackets.

Minus one. Minus
minus gives me a plus

cause squared a, so I
now have two cost squared,

A minus one, so that's
another double angle formula for

cost to a.

Now because I replaced the sine
squared here by one minus Cos

squared, I can do the same again
and replace the cost squared by

one minus sign squared and what
that will give main is cause 2A

is 1 - 2 sine squared A.

So lot of formally there.
Let's just write them all

down again. Sign
to a

IS2. Find a

Kohl's A. Calls

to a. Is
cost squared A minus sign,

squared A and we can
rewrite that as two cost

square day minus one or
as 1 - 2 sine

squared A. And then
turn to a is

equal to 2.

Tam a over
1 - 10

squared A. So there are
our double angle formula

formula to be learned
formally to be remembered

and most importantly
recognized and used when

we need them.

So let's have a look at how we
can make use of these double

angle formula. So sign of three
X. Is it possible to write

sign of 3X all in terms
of sine X?

Well. Let's try and break this

3X up. 3X is 2X Plus X,
so we can write this a sign of

2X Plus X.

OK, this means we can
use our addition formula sign

of two X cause X.

Plus cause of two
X sign X.

Now I can use my double
angle formula here sign of two

X is 2 sign X Cos
X still to be multiplied by

Cos X Plus.

Now I have a choice.

There are three double angle
formula for cause 2X, so my

choice is got to be governed by
what it is I'm trying to do and

we're trying to write sign 3X
all in terms of sine X.

That means the choice I have to
make here is the one that's got

signs in it, not cosines, but
the one that's got signs and

only signs, and the one that has
that is 1 - 2 sine squared X

still to be times by sign X.

So this front term is
going to be 2 sign

X cause squared X.

One times by Cynex is
plus sign X minus and

two sine squared X times.
Biosynex is sine cubed X.

Well, we're getting there. We've
got sign here sign here. Sign

cubed here. Cost squared here.

But Cost Square can be rewritten
using one of the fundamental

identity's cost square plus sign
squared is one so cost square

can be replaced by Wang.

Minus sign
squared.

And so we can see here.
Everything is now in terms of

sine X and all we need to do is

tidied up. So we multiply out
this bracket 2 sign X for

the first term, 2 sign X
times by one. Then we have

two sign X times Y minus
sign squared minus two sine

cubed X plus sign X minus
two sine cubed X.

2 sign X plus sign
X that's three sign X.

Minus two sine cubed minus two
sine cubed is minus 4 sign

cubed X and that everything is
in terms of sine X.

You can do the same with cause
as well cause 3X can be turned

into an expression that's
entirely in terms of cause X.

That's an example of using our
double angle formula in order to

reduce if we like to use that
expression and multiple angle

sign 3X is a multiple angle down
to a single angle in terms of

the sign of that angle. Let's
have a look now at solving an

equation. Let's take cause
2X is equal to

sign X and let's
take a range of

values for X.

Which puts X between plus
and minus pie.

Again, I've deliberately chosen

caused 2X. Be cause we have
a choice, we have three

possibilities. Which one do we

choose? Well, if I want to solve
an equation like this, I really

need it all in terms of one trig

function. Not two, but one.

And here I've got sine X.
Therefore makes sense here.

To replace this by
1 - 2 sine

squared, X equals sign

X. Now we have a
quadratic equation where the

variable is sign X.

Let's rearrange that so that it

equals 0. Add the two sine
squared to each side.

Plus the sign

X. And take one away
from each side.

This is now a quadratic
equation. Can I factorize it?

Let's have a look.

Two brackets, 2 sign X and sign
X when multiplied together,

these two will give me the two
sine squared I need minus one,

so let's pop a one into each
bracket, and one of them's got

to be plus and one minus.

I need plus sign X in the middle
going to make this one plus one,

so I get +2 sign X, make that
one minus so I get minus sign X

and when I combine those two
terms plus sign X there.

This says. A bracket,
a lump of algebra times by

another bracket. Another lump of
algebra is equal to 0.

And so one.

Or both of these
brackets must be equal

to 0. And so we've
reduced this fairly complicated

looking equation down to two
simple ones, and this one tells

us here. Sign X is equal
to add 1 to each side

and divide by two.

Sign X is 1/2 or this one
here tells us that sign X

is equal to minus one.

We've now got to extract the
values of X from this

information and those values of
X must be between plus and minus

pie. So let's sketch
the graph of cynex between

plus and minus pie.

There's the graph, there's pie.

Pie by 2 - π by two and
minus pie and it goes between

one and minus one. So let's take
this one. First sign X is minus

one. Well that goes across there
and down to their, so X is minus

π by two is one answer that we

get there. Sign X is 1/2, half
goes across there and we should

recognize that this is one of
those nice numbers. Sign X is

1/2 for which we've got an exact
answer, and So what we do know

is that the sign of 30 degrees
is 1/2, but where working in

radians. So in fact 30 degrees
is the same angle as pie by 6.

And this is symmetrical.
Remember the curve for sign is

symmetrical, so if that's pie by
6 in there, that's got to be pie

by 6 in there. So this, In other
words will be 5 pie by 6, and so

we have our two answers for this
one pie by 6 and five pie by 6.

So that we see that we've been
able to solve our equation using

our double angle formula and by
making the right choice,

particularly here when with cost
2X, we know that we have three

possibilities. So. That
is, have a look at another

equation again using our double
angle formula. Sign 2 X equals

sign X. And again, let's take
our value of X to lie between

plus and minus pie.

We've only one choice for sign
2X, that's two.

Sign X Cos X
equals sign X.

Now. It's very, very tempting to
say our common factor on each

side. Cancel it out.

And then we've lost it.

And because we lose it, we might
lose solutions to the equation.

So what's better than canceling
out is to get everything to

one side. By taking cynex
away from each side.

And then. Take out a
common factor, and here there's

a common factor of sign X.

Which will leave us
2 cause AX minus

one. Two expressions multiplied
together give us 0.

So either or both of these
expressions is equal to 0, so

either sign X equals 0 or
two cause X.

Minus one equals 0.

Now we've managed to reduce this
equation to two smaller, simpler

equations once sign and the
other ones for cause.

But each is going to give us a
value of X. Let's take this one

first, sign of X is 0.

And let's draw a sketch up
here of sign of X. There

it is and it's zero here,
here and here on the X

axis minus Π Zero and Pi
straightaway. We've got X equals

minus π and 0, not pie,
because pie is excluded from the

range that. We've got.

But notice sign X equals 0 gave
us 2 answers for X if we have

cancelled sign X out up here and
just got rid of it, we would not

have got those two answers.

Let's go to this equation
now. 2 cause X minus one is

0, so that tells us that
cause X is equal to 1/2.

And what we need to do is sketch
the graph of cosine.

And the graph of cosine.

Looks. Like that between
pie and minus pie?

And here is the half.

And cause X equals 1/2. This is
another one of these nice

relationships and we know that
this one is 60 degrees or

because we're working in radians
Pi by three and because of

symmetry this one here has got
to be minus π by three, so

X is minus π by 3 or
π by 3.

And so we've got our four
solutions for this equation.

In doing this work with double
angles, in effect, we've been

looking at what are called
multiple angles, and here I've

been drawing sketches of a
single angle. If you like, just

sign X Cos X. So the question
is, what does the graph of sine

2X look like?

What does the graph of cause 3X
look like? Or for that matter,

what about sign of 1/2 X?

Well, let's just explore that
sign X. Let's just have a look

at its graph.

Between North And

2π So
that's sine

X. What about
sign 2X?

Over the same range.

I just think about it. What are
we doing when we're multiplying

by two? Well, we're doubling
yes, but what that means is

everything that happens on this

graph. Happened twice
as fast on this graph.

So in the time it takes this to
go from North to 2π, it's done

all of that in this space.

So that bit of graph appears
in this space so that we

have up down there. Then of
course it's periodic, so it

does it again.

So sign of 2 X.

Gets through everything twice as
quickly as sign X and sign of

three. X will get through it 3
times as quickly, so I'll have 3

copies of this graph in the same
space, not to pie.

And the same with cause 2X and
cause 3X and cost 4X.

What if I take sign of
1/2 of X?

Do that. Sign of
1/2 of X.

I'll do sign of X first just
to give us the picture again.

That's our graph between North

and 2π. So what about?

Sign. Of half of X
on the same scale.

Well, things are happening half

as quickly. 1/2 of two
pies just pie, so we will only

have got through that bit of the
curve by the time we've got to

2π, so that in effect the graph
of sign a half pie looks like

that and continues on.

In double the space.

So graphs of multiple angles
look a little bit different to

the ordinary angle.

But the thing that we have to
remember is that if we have a

multiple here, that's greater

than one. Then it's going to get
through it much quicker and

we're going to see the graph

repeated. If we've got a
multiple here, that's less than

one, it's going to take longer
to get through.

The graph and we're going to see
the graph extended and drawn

out. Let's have
a look at the

graph of cause
X and cause 2X.

Again between North.

And. 2π

So there's our graph.

What about? Mark the points in
the same positions. This is

going to go through twice as
quickly, so we're going to see

that shape repeated in this
space here, so we're going to

see that come down and go up,
and then we're going to see it

repeated again. I thought so.
Sketching these graphs of

multiple angles is quite easy.
All you need to do is look at

the original graph and judge
the number of times that it's

going to be repeated over the
given range.