If x = cos^(-1)(1/2), what is the value of sin(x)?

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Q: If x = cos^(-1)(1/2), what is the value of sin(x)?

Solution: Using the identity sin(x) = sqrt(1 - cos^2(x)), we have sin(x) = sqrt(1 - (1/2)^2) = √3/2.

Steps: 7

Step 1: Understand that x = cos^(-1)(1/2) means we are looking for an angle x whose cosine value is 1/2.
Step 2: Recall that cos(x) = 1/2 corresponds to the angle x = 60 degrees or x = π/3 radians.
Step 3: Use the identity sin(x) = sqrt(1 - cos^2(x)) to find sin(x).
Step 4: Calculate cos^2(x): Since cos(x) = 1/2, then cos^2(x) = (1/2)^2 = 1/4.
Step 5: Substitute cos^2(x) into the identity: sin(x) = sqrt(1 - 1/4).
Step 6: Simplify the expression: 1 - 1/4 = 3/4, so sin(x) = sqrt(3/4).
Step 7: Further simplify: sin(x) = sqrt(3)/sqrt(4) = sqrt(3)/2.