The sine and cosines obtained using the double angle formula are;
(a) sin(2·θ) = 8·√(65)/81
(b) cos(2·θ) = 49/81
The details of the above solution are as follows;
(a) sin(θ) = 4/9, 0 < θ < π/2
The double angle formula indicates that we get;
sin(2·θ) = 2·sin(θ)·cos(θ)
The Pythagorean identity indicates that we get;
cos(θ) = √(1 - (4/9)²)
√(1 - (4/9)²) = √((81 - 16)/81)
cos(θ) = (√65)/9
sin(2·θ) = 2 × 4/9 × (√65)/9
2 × 4/9 × (√65)/9 = 8·(√65)/81
sin(2·θ) = 8·(√65)/81
(b) cos(2·θ) = cos²(θ) - sin²(θ)
cos(2·θ) = ((√65)/9)² - (4/9)²
cos(2·θ) = (65/81) - (16/81)
cos(2·θ) = 49/81