Find exact value of sin2 theta, Cos2 theta, sin (theta/2), cos (theta/2)

Question

Find exact value of sin2 theta, Cos2 theta, sin (theta/2), cos (theta/2).

When,

sin theta = 3/5 and pi/2 < theta < pi

Answer 1

This puts the angles into the second quadrant.

sin(θ) = 3/5 indicates a 3-4-5 triangle in the second quadrant, so

cos(θ) = -4/5

sin²(θ) = 9/25

cos²(θ) = 16/25

Using half angle formulas:

sin(θ/2) = ±√((1-cos(θ))/2) = ±√((1-(-4/5))/2) = ±√(9/10)

cos(θ/2) = ±√((1+cos(θ))/2) = ±√((1+(-4/5))/2) = ±√(1/10)