What is the value of sin(2θ) if sin θ = 1/3?

What is the value of sin(2θ) if sin θ = 1/3?

Step 1: Start with the given value of sin θ, which is 1/3.
Step 2: Use the double angle formula for sine: sin(2θ) = 2 * sin θ * cos θ.
Step 3: To find cos θ, use the Pythagorean identity: sin^2 θ + cos^2 θ = 1.
Step 4: Calculate sin^2 θ: (1/3)^2 = 1/9.
Step 5: Substitute sin^2 θ into the identity: 1/9 + cos^2 θ = 1.
Step 6: Rearrange to find cos^2 θ: cos^2 θ = 1 - 1/9 = 8/9.
Step 7: Take the square root to find cos θ: cos θ = sqrt(8/9) = 2sqrt(2)/3.
Step 8: Now substitute sin θ and cos θ back into the double angle formula: sin(2θ) = 2 * (1/3) * (2sqrt(2)/3).
Step 9: Simplify the expression: sin(2θ) = 4sqrt(2)/9.

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