Final answer:
To find the value of sin(θ/2) given sin(θ) = 2sqrt2/5 and π/2 < θ < π, we used the half-angle formula and determined sin(θ/2) to be sqrt((5 + sqrt(17))/10).
Step-by-step explanation:
We are given that sin(θ) = 2sqrt2/5, with θ restricted to the interval (π/2, π). To find sin(θ/2), we can use the half-angle formula:
sin(θ/2) = ±√((1 - cos(θ))/2)
To determine the sign of sin(θ/2), we need to consider the given interval. Since π/2 < θ < π, we know θ/2 will be in the second quadrant, where sine is positive. Next, we need to find cos(θ). Using the Pythagorean identity:
cos^2(θ) = 1 - sin^2(θ)
cos^2(θ) = 1 - (2sqrt2/5)^2
cos(θ) = -√(1 - 8/25)
cos(θ) = -√(17/25) = -√17/5
Since we are in the second quadrant, cos(θ) is negative. Finally, we apply the positive sign to the half-angle formula as the angle θ/2 falls in the first quadrant:
sin(θ/2) = √((1 - (-√17/5))/2)
sin(θ/2) = √((1 + √17/5)/2)
sin(θ/2) = √(5 + √17)/10