Find the exact value using half angle identity sin(22.5 degrees)

Answer 1

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.