>> We're going to determine the exact value of sine pii over 12 using a half angle identity.
The most common two half angle identities for a sine and cosine are given here,
though some textbooks do refer to these power distinct formulas also as half angle identities.
For this example, we'll be applying this half angle identity for sine,
though in the next video, we'll find the same sine function value using this equivalent
identity given here.
Let's begin by determining the value of a as well
as whether the sine function value will be positive or negative.
It might be tempting to convert pii over 12 times 180 degrees.
Of course, we could do this.
Pii over 12 times 180 degrees divided by pii.
Notice how the pii just simplifies out.
We'll be left with 180 degrees divided by 12 equals 15 degrees.
For practice, let's go ahead and leave this in terms of radians.
So notice that a over 2 must equal pii over 12 radians so if a divided by 2 is equal
to pii divided by 12 radians to solve for a, which we need on the right side
of our identity we would multiply both sides by 2 so a is equal to 2
or 2 over 1 times pii over 12 radians.
We have a common factor of 2.
There's 1, 2 and 2, and six 2's and 12.
So a would be equal to pii over 6 radians.
So, applying the identity, we know that sine pii over 12 is equal
to the sine of pii over 6 divided by 2.
And now before we apply this identity,
let's determine whether the sine function value is going to be positive or negative.
Well, since pii over 12 radians or 15 degrees is in the first quadrant where both x
and y are positive and sine theta is equal to y over r, we know r is always positive
and now we know that y is positive and therefore the sine function value is going to be positive.
And as we know, all of the trig function values are positive in the first quadrant.
So we now will have a positive square root, and then we'll have one minus cosine a
which now we know is cosine pii over 6 and all this is divided by 2.
Notice how the denominator of 2 is underneath the square root.
And now for the next step we'll determine the value of cosine pii over 6
so we can use either a reference triangle or the unit circle and let's show both.
Pii over 6 radians is equal to 3 degrees so instead of position,
this is the initial side of the angle.
The terminal side would be approximately here
where again the reference angle is pii over 6 radians or 30 degrees.
So if we sketch a reference triangle, we need to recognize
that we have a 30, 60, 90 reference triangle.
So we can label these short leg one to hypotenuse 2
and the longer leg square root of 3.
So using a reference triangle cosine of pii over 6 radians is equal to the ratio
of the adjacent side of the hypotenuse, which would be square root of 3 divided by 2.
And of course we can check this trig function value on the unit circle.
Here's the terminal side of pii over 6 radians or 30 degrees,
and because we have the unit circle x equals cosine theta, so cosine pii over 6
or cosine 30 degrees is square root 3 divided by 2, which means we now have the square root
of 1 minus square root of 3 divided by 2 all over 2.
Now, let's simplify this.
Looking at just the numerator, we can clear the denominator of 2 by multiplying by 2,
but if we multiply the numerator of our fraction by 2,
we must also multiply the denominator by 2.
Notice how this is equivalent to multiplying by 2 over 2 underneath the square root.
So now we have the square root of--
we distribute, we have 2 minus square root of 3 divided by 2 times 2.
The 2 would simplify to 1 leaving us with just minus square root 3.
And our denominator would be 2 times 2 over 4, and now because we have a fraction
under the square root, we can write this as the square root of the numerator
over the square root of the denominator which is the square root of 4, and the square root
of 4 simplifies to 2, so this leaves us with the square root
of 2 minus square root 3 divided by 2.
Now, let's check this on the calculator.
First, convert this to a decimal and then approximate the value
of sine pii over 12 on the calculator.
Let's put the numerator in parentheses so we have open parenthesis second x squared
for square root and then 2 minus square root 3 right arrow,
right arrow close parentheses divided by 2, enter.
And now we'll compare this to sine of pii divided by 12.
Let's first make sure the calculator's in radian mode so press the mode key,
go down to the third row, highlight radian, press enter.
And second mode for quit and then we'll press sine pii divided by 12
and notice how this verifies our work is correct.
There's one more thing I do want to point out though before we go.
We did find the exact value of sine pii over 12 or sine 15 degrees using a [inaudible] as well.
Though the result did look different, it is the same exact value.
So, let's do a quick comparison.
Again, we know that pii over 12 radians is equal to 15 degrees.
It's also equal to 45 degrees minus 30 degrees.
So, if we compare the difference identity for sine given here in red, we'd have the difference
of these two products which if we simplify would the quantity square root
of 6 minus square root 2 divided by 4.
So using the difference identity for sine, this was the exact function value,
and when we use the half angle identity, this was the exact trig function value.
Okay, they look quite different.
So, if we can [inaudible] decimal, we can see they are the same.
So, depending on what method we use to find the exact value of sine 15 degrees
or sine pii divided by 12 radians, they're different but the function values are the same.
I hope you found this helpful. ------------------------------68710dff3f5f--