Given sin(theta)=2/3 and 90<theta<180, determine the exact values of sin(2theta), tan(2theta), and sin(theta/2)

Answer 1

sin(2θ) = 2sinθcosθ; cosθ = - (1 - sin²θ)^1/2 = - (1 - 4/9)^1/2 = - √5/3;

sin(2θ) = 2·2/3(- √5/3) = - 4√5/9

cos(2θ) = 1 - 2sin²θ = 1 - 8/9 = 1/9;

tan(2θ) = sin(2θ)/cos(2θ) = - (4√5/9)/(1/9) = - 4√5;

sin(θ/2) = √(1 - cosθ)/2 = √(1 + √5/3)/2 = √(3 + √5)/6