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

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

Step 1: We know that sin x = 1/2. This means we need to find the value of sin(2x).
Step 2: We will use the double angle formula for sine, which is sin(2x) = 2 * sin x * cos x.
Step 3: Since we have sin x = 1/2, we can substitute this value into the formula: sin(2x) = 2 * (1/2) * cos x.
Step 4: Now, we need to find cos x. We can use the Pythagorean identity: cos^2 x + sin^2 x = 1.
Step 5: Substitute sin x into the identity: cos^2 x + (1/2)^2 = 1.
Step 6: This simplifies to cos^2 x + 1/4 = 1.
Step 7: Now, subtract 1/4 from both sides: cos^2 x = 1 - 1/4 = 3/4.
Step 8: Take the square root of both sides to find cos x: cos x = √(3/4) = √3/2.
Step 9: Now we can go back to our formula for sin(2x): sin(2x) = 2 * (1/2) * (√3/2).
Step 10: Simplify this: sin(2x) = 1 * (√3/2) = √3/2.

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