Evaluate sin ( 2 cos ^-1 sqrt 2/2 )

Answer 1

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.