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.