In this video,
we will use a half-angle identity
to determine the exact value of sine pi divided
by eight radians, which is equal
to sine 22.5 degrees.
Looking at the half-angle identities below,
sine A divided by two is equal to plus
or minus the square root of the quantity,
one minus cosine A divided by two.
We have a plus or minus sign here
because we need to determine the sine
of the trig function value based
upon the given angle.
Also notice how this two is under the square root.
The first step is to find the value
of A where A divided by two is equal
to the given angle, which in our case is 22.5 degrees.
Well, if we double this angle then divide
by two, we will still have 22.5 degrees.
22.5 times two is 45 and therefore,
we write the sine of 22.5 degrees
as the sine of 45 degrees divided
by two.
Notice in this formula, you can easily see
the angle A is 45 degrees.
This is the angle we use on the right side
of the identity.
So applying the formula, we have the sine
of 22.5 degrees where the sine of 45 degrees divided
by two is equal to, because 22.5 degrees is
in the first quadrant where all the trig function values
are positive, we are going to have a positive square root.
Under this square root, we have one minus cosine A,
which is one minus cosine 45 degrees.
And all this is divided by two.
Where, again, the two is under the square root.
The next step is to find the value
of cosine 45 degrees.
In center position,
the terminal side intersects the inner circle
at this point here where both the x
and y coordinates are the square root
of two divided by two.
And we know on the unit circle,
cosine theta equals x and sine theta equals y
The cosine of 45 degrees is square root
of two divided by two.
So now we have the square root
of one minus square root two divided
by two all over two.
So this is the exact trig function value
of sine 22.5 degrees.
Let's go ahead and simplify this
by clearing this denominator of two
from the numerator.
To do this, we multiply the numerator
and denominator of this complex fraction
by two, which is equivalent to multiplying
by one.
This is equal to the square root
of distributing the two here, we have two times one,
which is two.
Now we have minus two times square root
of two divided by two.
Well, notice how the two is to simplify
out, giving us just square root
of two in there for this product simplify.
So two minus square root two.
In the denominator, we still have two times two,
which is four.
So the next step, let's take the square root
of the numerator and the denominator.
This is equal to the square root
of two minus square root two divided
by the square root of four.
And the square root of four simplifies
to two, which gives us the square root
of two minus square root two all
over two.
While if we wanted to, we could think
of having a one here and write this as one half times
the square root of two minus square root two,
which in our homework, is the form that is requested.
Before we go, though, let's check this
on the calculator.
Let's compare the decimal approximation
for the sine of 22.5 degrees and the decimal approximation
for this exact value.
The calculator's already in the green mode,
so I will enter sine 22.5, closed parentheses,
enter.
And now we'll approximate this exact value,
which is one half or one divided
by two times the square root.
Under the square root, we have two minus square root
of two.
So two minus second x squared, brings up the square root.
Two and then enter.
And notice how we do get the same decimal approximation,
verifying our work is correct.
I hope you found this helpful.