In this video, we will use a half-angle identity
to determine the exact value of cosine
pi divided by 12 radians,
which is equal to cosine 15 degrees.
Looking at the notes below,
cosine of A divided by two
equals plus or minus the square root
of the quantity one plus cosine A divided by two.
We have a plus or minus sign here because
we need to determine the sign of the cosine function value
based upon the given value.
Also, notice how this too is under the square root.
The first step is to find the value of A
where A divided by two equals the given angle.
So if we double this angle and divide it by two,
we will still have 15 degrees,
and therefore we begin by writing cosine 15 degrees
as cosine of 30 degrees
divided by two.
Notice 30 degrees divided by two is still 15 degrees.
In this form, we can see
that A is 30 degrees.
And again, we need angle A
for the right side of the identity.
Applying the formula,
because 15 degrees in quadrant one,
and all the trig function values
are positive in quadrant one,
we will have a positive square root.
So we have the square root.
And then under the square root we have the quantity
one plus cosine A divided by two,
where A is 30 degrees,
which gives us one plus cosine 30 degrees
and all this is divided by two
under the square root.
And now we find the exact value of cosine 30 degrees.
In standard position, the terminal side
intersects the unit circle at this point here,
where on the unit circle, x equals cosine theta
and y equals sine theta,
which indicates that cosine 30 degrees
is equal to square root three divided by two,
which gives us under the square root,
the quantity one plus square root three divided by two
all over two.
So this is the exact value of cosine 15 degrees,
but let's simplify this complex fraction
under the square root by clearing this denominator of two.
To do this, we multiply the numerator and denominator
of the complex fraction by two,
which is equivalent to multiplying by two over two,
which is equal to one.
Distributing the two here,
we have two times one which is two,
and then we have two times square root three divided by two
or two over one times square root three over two,
which notice how we can simplify before multiplying.
There's one two and two here and here,
and therefore the product is square root three.
So we have two plus the square root of three
all over two times two which is four.
And now, let's take the square root of the numerator
and denominator.
This is equal to the square root
of two plus square root three
divided by the square root of four,
and the square root of four is equal to two.
This simplifies to the square root
of two plus square root three divided by two,
which, if we want, we can think of having a one here,
and write this as one half times the square root
of the quantity two plus square root three.
And this is the form requested in the homework.
So this is the exact value of cosine 15 degrees
or cosine pi divided by 12 radians.
Before we go though,
let's verify this on a calculator.
Let's get a decimal approximation for cosine 15 degrees
and then a decimal approximation for this exact value.
The calculator is already in degree mode,
so I will enter cosine
15 close parenthesis enter.
And I'll compare this value to this exact value,
which is one half or one divided by two
and then we have times the square root
which is second x squared.
Under the square root we have two plus square root three,
so two plus second x squared for the square root,
three and then enter.
Notice how we do get the same decimal approximations,
which verifies our work is correct.
I hope you found this helpful.