Evaluate sin ( 2 cos ^-1 sqrt 2/2 )
Let's start by using the double angle identity for sine:
sin(2θ) = 2sinθcosθ
Here, θ = cos^-1(sqrt(2)/2)
Now, let's find sin(θ) and cos(θ):
sin(θ) = sin(cos^-1(sqrt(2)/2)) = sqrt(1 - (sqrt(2)/2)^2) = sqrt(1 - 1/2) = sqrt(1/2) = sqrt(2)/2
cos(θ) = cos(cos^-1(sqrt(2)/2)) = sqrt(2)/2
Now, we can substitute these values into the double angle identity:
sin(2θ) = 2(sin(θ)cos(θ)) = 2 * ((sqrt(2)/2) * (sqrt(2)/2)) = 2 * (2/4) = 1/2
Therefore, sin(2 cos^-1(sqrt(2)/2)) = 1/2.