>> We're going to determine the exact value of sine, 112.5 degrees using a half angle identity.
The most common half angle identities for a sine and cosine are given here so in this example,
we'll use this half angle identity for sine.
However, some textbooks do use these two identities
for half angle identities also often referred to as power reducing formulas.
So the next video will show how to do the same problem using this equivalent identity
given here.
Let's begin by determining the value of A, as well as the sine of our sine function value.
Notice for this example, a divided by 2 must equal 112.5 degrees so if a divided
by 2 must equal 112.5 degrees, if we multiply both sides of the equation by 2,
we would have a equals 2 times 112.5 degrees or a is equal to 225 degrees so we've discovered
that sine 112.5 degrees is equal to the sine of 225 degrees divided by 2.
And now it's in the correct form to apply this given half angle identity.
But again, before we apply this, let's determine whether it's going
to be a positive or a negative value.
Remember sine theta is equal to y divided by r and 112.5 degrees
in center position would have initial side here
and the terminal side approximately here in the second quadrant.
And because y coordinates are positive in the second quadrant,
sine function values are positive in the second quadrant and therefore, we know we're going
to have a positive square root when applying this identity.
So the exact sine function value is equal to the positive square root of 1 minus cosine a,
which in our case is 1 minus cosine 225 degrees divided by 2.
Notice how the denominator of 2 is also under the square root and now we want
to find the value of cosine 225 degrees.
So we can either use reference triangles or we can use a unit circle
because we already have the coordinate plane here, let's go ahead and sketch 225 degrees
in center position and use a reference triangle.
So the initial side would be along the positive x axis
and then we would rotate half a ration would be 180 degrees.
We have to rotate 45 degrees more to here and notice how the reference angle is 45 degrees
so if we sketch a reference triangle, we'll have a 45, 45, 90 reference triangle.
So we should recognize that a 45, 45, 90 reference triangle,
the two legs would have a length of 1 and the hypotenuse would have a length of square root 2.
But because we're in the third quadrant where both the x and y coordinates are negative,
this would be negative 1 and so would this.
Now using a reference triangle, cosine 225 degrees would be equal to the ratio
of the adjacent side of the hypotenuse or negative 1 divided by square root 2.
And of course to rationalize this, we multiply by the square root of 2 over the square root
of 2, which would give us negative square root 2 divided by 2.
And of course we can also find this cosine function value on the unit circle
where here is the terminal side of 225 degrees and the x coordinate is equal
to the cosine function value because we're on the unit circle,
verifying we have the correct cosine function value of negative 2 divided by square root 2,
which means the exact sine function value would be equal to the square root of 1 minus --
and then we have negative square root 2 divided by 2 all over 2.
And now let's go ahead and simplify this.
If wanted to clear the denominator of 2 from the numerator, we have to multiply through by 2,
which we can do as long as we also multiply the denominator by 2.
So we got to multiply this fraction underneath the square root by 2 over 2 to simplify.
So notice when we do this, we would have the square root of --
in the numerator when we distribute, we would have 2.
Notice here we have minus a negative, which becomes plus, and then square root 2
over 2 times 2, the 2s just simplify to 1, leaving us with plus square root 2.
And notice now the denominator would 4.
Notice here we have the denominator of 4 and this 4 is under the square root
and so the square root of 4 is equal to 2, we can simplify again.
This would be equal to the square root of 2 plus square root 2, all divided by 2.
So this would be the exact value of sine, 112.5 degrees.
Let's go ahead and verify this by converting this exact value to a decimal approximation
and then also find the sine function value on the calculator.
Let's first convert this exact value to a decimal approximation so we would have
in the numerator, the square root of 2 plus the square root of 2, right arrow,
right arrow, close parentheses divided by 2.
Now let's use our calculator to find the decimal approximation for sine 112.5 degrees.
We first want to verify we are in degree mode so we'll press the mode key.
Notice how we are in degree mode.
Let's go back to the home screen and we'll enter sine 112.5 degrees, closed parentheses and enter
and notice how this does verify our exact value is correct.
I hope you found this helpful. ------------------------------fdca48d3e07f--