Final answer:
To find the exact value of sin (3 θ), we can use the trigonometric identity sin(3θ) = 3sin(θ) - 4sin³(θ). Given that cos(θ) = 1/3 for 0 < θ < π/2, we can plug in this value to find sin(θ) and then substitute it into the sin(3θ) equation to find the exact value. The exact value of sin(3θ) is 6√2/3 - (32√2/27).
Step-by-step explanation:
To find the exact value of sin (3 θ), we can use the trigonometric identity sin(3θ) = 3sin(θ) - 4sin³(θ). Given that cos(θ) = 1/3 for 0 < θ < π/2, we can use the Pythagorean identity: sin²(θ) = 1 - cos²(θ) to find sin(θ). Since cos(θ) = 1/3, we have sin²(θ) = 1 - (1/3)² = 8/9. Therefore, sin(θ) = √(8/9) = 2√2/3. Plugging this into the identity sin(3θ) = 3sin(θ) - 4sin³(θ), we get sin(3θ) = 3(2√2/3) - 4(2√2/3)³ = 6√2/3 - 4(8√2/3³) = 6√2/3 - 4(8√2/27) = 6√2/3 - (32√2/27). Therefore, the exact value of sin(3θ) is 6√2/3 - (32√2/27).