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

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

Correct Answer: 1

Step 1: Identify the given value. We have sin(θ) = 1/√2.
Step 2: Recall the double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ).
Step 3: We already know sin(θ) = 1/√2. Now we need to find cos(θ).
Step 4: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1.
Step 5: Substitute sin(θ) into the identity: (1/√2)² + cos²(θ) = 1.
Step 6: Calculate (1/√2)², which is 1/2. So, 1/2 + cos²(θ) = 1.
Step 7: Solve for cos²(θ): cos²(θ) = 1 - 1/2 = 1/2.
Step 8: Take the square root to find cos(θ): cos(θ) = 1/√2 (we take the positive root).
Step 9: Now substitute sin(θ) and cos(θ) back into the double angle formula: sin(2θ) = 2(1/√2)(1/√2).
Step 10: Calculate the product: 2(1/√2)(1/√2) = 2 * (1/2) = 1.

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