Find the exact value using half angle identity sin(22.5 degrees)
We can use the half angle identity for sine to find the exact value of sin(22.5 degrees).
The half angle identity for sine is given by:
sin(theta/2) = sqrt((1 - cos(theta)) / 2)
For the given angle of 22.5 degrees, we can write it as theta = 45 degrees / 2.
Using this in the half angle identity equation, we have:
sin(22.5 degrees) = sqrt((1 - cos(45 degrees)) / 2)
Now we need to find the value of cos(45 degrees).
Since cos(theta) = sin(90 degrees - theta), we have:
cos(45 degrees) = sin(90 degrees - 45 degrees) = sin(45 degrees) = 1/sqrt(2)
Substituting this value back into the original equation, we get:
sin(22.5 degrees) = sqrt((1 - 1/sqrt(2)) / 2)
To simplify this expression, we rationalize the denominator:
sin(22.5 degrees) = sqrt((2 - sqrt(2)) / (2 * sqrt(2)))
Multiplying the numerator and denominator by sqrt(2), we have:
sin(22.5 degrees) = sqrt((2 - sqrt(2)) * sqrt(2)) / (2 * sqrt(2) * sqrt(2))
= sqrt(2 - sqrt(2)) / 2
So, the exact value of sin(22.5 degrees) is sqrt(2 - sqrt(2)) / 2.