>> Hi, this is Julie Harland
and I'm Your Math Gal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
>> We're going derive the half
angle Trig [background
noise]' identities.
We're going
to use these double angle
identities
for the cosine of 2X.
So one of them is cosine
of 2X equals 2 cosine squared
minus 1, and the other one
we'll be using is cosine
of 2X equals 1 minus 2 sine
squared X. So let's start
with this first one,
cosine of 2X equals 2 cosine
squared X minus 1.
All right, what I'm going
to do is try to solve
for cosine squared X,
all right?
So this part right here --
so what would I do?
One thing I could do is
rewrite the equation.
So we have the 2 cosine
squared X minus 1
on the left side
and that's equal
to cosine of 2X.
So I'm just [background
writing equation] switching
the order, what's on each side
of the equal sign and now
to solve for the cosine
of squared X. I'm going
to add 1 to both sides,
so this gives me cosine
squared X equals,
1 plus the cosine of 2X
and then, we would divide both
sides by 2.
So here we go dividing by 2
and I've solved
for the cosine squared X. Now
I'm going to solve
for the cosine of X.
To do that,
I'll need to take the square
roots of both sides
and you have to remember
when you take the square roots
of both sides, you have a plus
or minus in front
of the square root.
So on the right hand side,
I'll have plus
or minus the square root
of 1 plus the cosine
of 2X all over 2.
Now this is a little
bit tricky.
You're saying how could the
cosine of X be plus or minus?
Because if I give you a value
for X, the cosine
of that value is just only one
thing -- it's either going
to be positive the square root
of something
or negative square root
of something.
And that depends
on where X is,
which quadrant it's in.
So you have
to choose the sign based
on the quadrant
that X lies in.
[ Background noise ]
Okay. Now,
notice I said I was deriving a
half-angle identity
and you don't see any
half-angle identity
but in a way it's there.
Notice here I have the cosine
of 2X and over here
on the right I have just the
cosine of X. So this X is half
of this 2X.
To see it as a half-angled
identity which is how most
people like to see it,
we do a little substitution.
We're going to let --
let's call this 2X,
instead of calling it 2X,
let's call it U, okay?
So instead of 2X, we're going
to write U. Then what would
X be?
If I wanted
to divide both sides by 2 --
I'm sorry --
yes, divide both sides by 2,
then X would be U over 2.
So I am going to replace the X
over here with U over 2,
which now looks
like a half-angle,
and I'm going
to replace this 2X
with a U. Okay?
So that's the replacement.
Where I see a 2X,
I'm going to write a U.
Where I see an X,
I'm going to write a U over 2.
And so this formula now
becomes the cosine of X
which we're going to write
as U over 2
and you could think of that
as a half-angle is equal
to plus or minus square root
of 1 plus,
the cosine of 2X all over 2,
and that is a half-angle
identity for cosine.
So what it's saying --
oops I forgot
to replace the 2X
with U. I said that I did it
but I didn't actually
write it.
Here we go, cosine of U,
there we go.
So what this is saying
if I wanted to find the cosine
of half of an angle,
I can figure it
out if I know the cosine
of the figure angle.
So for instance,
if I know the cosine
of 30 degrees,
then I could use this formula
to figure out the cosine of 30
over 2 or 15 degrees, right?
And that's the idea.
Okay, now let's say I was
finding the cosine
of 15 degrees.
I know that 15 degrees is
in Quadrant 1,
I would use the
positive value.
If I was figuring
out the cosine
of let's say 112.5 degrees,
that is an angle
in Quadrant 2,
so I would use the negative
value because the cosine is
negative in Quadrant 2.
We'll do some problems;
actually work
through some problems
in another video.
In this video I'm just going
to derive the formulas for --
the half-angle identities.
So this is the formula
for cosine of U over 2.
Now you could go back and call
that X over 2 or Y over 2,
just whatever angle you want
over 2.
Now we're going to start
with the other formula
for the cosine of 2X
which is the cosine
of 2X equals 1 minus 2 sine
squared of X,
to derive the formula
for the sine of a half-angle.
And we're going
to do a similar problem
but for fun,
let's do our substitution
with U first this time.
It really doesn't matter
when you do it,
but remember what we said?
We said that 2X was U
and then we said that means
that X is U over 2.
So you could either do your
replacements right now
or you could do it at the end
of the problem.
To show you that it's okay
to do it at the beginning,
I'm going to work this
out by replacing those
values now.
So instead of 2X,
I'm going to write U
and instead of X,
I am going to write U over 2.
And then my goal here is to --
and that's not a four
that is a U in there --
my goal is to solve
for the sine of U over 2.
So it might be easier
to just bring that over
to the left side
of the equation
by adding 2 sine squared U
over 2 to both sides.
So I get 2 sine squared U
over 2 and at the same time
I'm going to subtract cosine
of U. So this becomes 1 minus
cosine of U. All right,
now I'm trying to solve
for the sine squared first,
so I need to divide both sides
by 2, all right?
It should look similar
to that last problem I did.
So we'll be able to cancel
out those 2s
and then I've got the sine
squared of U over 2,
so now we get
to take the square roots
of both sides,
sine of U over 2
and that means we're going
to take the square root
of the right hand side
and remember you need
to put plus or minus
in front of it.
And again, depending on what U
over 2 is, what quadrant U
over 2 is,
that will help you decide
whether you're going
to use the positive
or the negative value.
And when we actually work
problems, we'll see how we
do that.
So now we've got the formula
for the sine of U over 2.
Okay, so you've done the
cosine of U over 2
and the sine of U over 2.
As a reminder,
this was the cosine
of U over 2.
We had plus [background
writing equation]
or minus 1 plus the cosine
of U, over 2.
Okay, now remember the whole
fraction is underneath the
square root, okay?
So I didn't rationalize the
denominator here.
This just usually how
it's written.
So we've got the sine,
we've got the cosine and now
for the tangent,
we know the tangent is the --
of X is sine of X over cosine
of X. So if I wanted
to get the tangent,
I just need
to put this fraction
over this other fraction.
So the tangent of U
over 2 is plus
or minus the square root of --
well actually, let's write
down all the steps here.
It's the sine of U over 2,
divided by the cosine
of U over 2.
So this will be --
now, again,
you're not sure whether it's
going to be plus or minus
but I'm going to write those
in and depending on where U
over 2 is,
again this is how you're going
to deal with it
until where the function is
positive --
whether to use the plus
or minus sign.
So we've got --
let's see, for the sine,
that's 1 minus cosine of U all
over 2, and the cosine is plus
or minus 1 plus, cosine of U
over 2, and you're going
to have to do a little bit
of Algebra to simplify this.
So if we have this square root
divided by this square root,
it's the numerator divided
by the reciprocal
of the denominator
which is actually --
and I'm going to just put plus
or minus in front
of the whole thing.
I'm going to have the square
root of 1 minus cosine U
over 2 times 2,
over 1 plus cosine U.
And when you put these two
together, this is the square
root of the whole thing
multiplied together,
these two will cancel out.
So our final answer here will
be just 1 minus cosine U,
over 1 plus cosine U. That's
all under the square root sign
and then we've got the plus
or minus sign in front of it.
So our three formulas for now,
here they are, the sine of U
over 2, the cosine of U
over 2, and the tangent
of U over 2.
Now there're actually two
other formulas for the tangent
of U over 2 that you can use
so I'm going
to derive those two.
But -- so these are formulas
we have so far.
There's just two other ways
of writing the tangent
of U over 2.
All right,
to get the other formulas,
we start off with this.
The tangent of U
over 2 could be defined
as the sine of U over 2 over,
the cosine over U over 2,
and instead
of writing those formulas,
we could do a little
trick here.
We can multiply the numerator
and denominator by 2 cosine
of U over 2.
So watch what happens
when I do that.
So I'm going
to multiply both the numerator
and denominator by 2 cosine
of U over 2...
[ Background noise ]
>> ...and so what I really
have in the numerator now is 2
sine U over 2,
times the cosine of U over 2,
and in the denominator,
I have 2 cosine squared,
[background writing equation]
U over 2.
Now we don't want
to go canceling out these
because we'd be back
to square one.
The idea is we actually have
two identities right here.
You might not recognize it
immediately
but the numerator --
see how you see,
like 2 times the sine
of something times the cosine
of the same thing?
So what would that be?
Well, that would be 2 times
the sine of twice that angle.
So the numerator is actually 2
times -- I'm sorry,
it's the sine
of 2 times, U over 2.
Okay? Now the denominator,
you might not recognize
as readily.
So I'm going
to alter it a little bit.
We know that 2 cosine squared
X minus 1 is an identity,
but I don't have that minus 1.
But how about if I just add 1
and subtract 1,
that should be okay.
So I start out with plus 1,
I'm going to write plus,
2 cosine squared U over 2
which is really there,
and then I'm going
to go minus 1.
So basically that's still just
2 cosine squared U over 2,
right, because I have a 1
and a minus 1
and hopefully you recognize
that this is an identity
right here.
So that will give me --
what's happening
in the numerator?
Well the 2s cancel
and I have just the sine of U,
and in the denominator I have
this 1 plus --
now what's this identity?
Recognize that one?
That's the cosine of 2 times
that angle.
So I have 2 --
the cosine of 2 times U
over 2, what will that be?
U. So this is an alternate
identity for the tangent of U
over 2 and notice there's no
square roots
and there's no plus
or minus signs.
So if you know both the sine
of U and you know the cosine
of U, that might be an easier
identity to figure
out what the tangent of U
over 2 is.
There's one more formula
where we start off
with the tangent of U over 2
and I already have the steps
written out because this
video's getting so long.
So I can write that as sine
of U over 2 over cosine of U
over 2, and the trick I'm
going to employ here is
multiplying the numerator
and denominator by negative 2,
sine of U over 2,
which will give me the
negative 2 sine squared U
over 2, all
over negative 2 sine U
over 2 cosine U over 2
and in the numerator,
the trick I'm going
to do here is I'm going
to add 1 and subtract 1, okay?
So in the numerator,
if I put these together,
you could think of that
as 1 minus sine squared U
over 2 which is a formula
for the cosine of 2 times
that angle, or cosine U
which you'll see down here.
And I solve
that minus 1 hanging
out at the end.
So the numerator,
hopefully you'll see ends
up being cosine U minus 1.
What about the denominator?
Now the minus sign in front
of 2 sine of something times
the cosine of the same thing,
so that's just the sine
of 2 times that angle
and of course, the angle's U
over 2, 2 times
that angle is going to be U.
So the denominator is minus
sine U, so I have cosine U
minus 1, over negative sine U,
and we're just going
to simplify this a little bit
by factoring out --
let's see this minus sign
of the denominator,
I'm going to put that up
as negative 1
in the numerator,
and then I'm going to multiply
that through the negative 1
cosine U minus 1 ends
up being a negative cosine
plus 1 or 1minus cosine U
over sine U. Well
that is another formula
for the half-angle,
cosine U over 2.
Again, there's no square
roots, there's no plus
or minus signs.
So now we have three formulas
for the tangent
of a half-angle
where if you know the cosine
and sine of the actual angle,
then you can figure
out the tangent of half
of that angle.
So we'll do some problems
using these formulas
on another video but for now,
let's just show all the
formulas derived
on this video.
So here are the five
half-angle formulas
and remember you could use
any letter.
I used U; you could have
that as sine of X over 2,
sine of theta over 2,
et cetera.
The idea is
if you know the sine and
or cosine of an angle,
you can figure out the sine,
cosine, or tangent of half
that angle using
these formulas.
[ Silence ]
>> Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.