What is the solution set of the trigonometric equation sin(2x) = cos(x) on the interval [0, 2*pi)?

Answer and explanation

Correct answer: B. pi/6, pi/2, 5pi/6, 3pi/2

  1. Step 1: Apply the double-angle identity: sin(2x) = 2sin(x)cos(x).
  2. Step 2: Substitute into the equation: 2sin(x)cos(x) - cos(x) = 0.
  3. Step 3: Factor out cos(x): cos(x) (2sin(x) - 1) = 0.
  4. Step 4: Case 1: cos(x) = 0 => x = pi/2, 3*pi/2.
  5. Step 5: Case 2: 2sin(x) - 1 = 0 => sin(x) = 1/2 => x = pi/6, 5pi/6.
  6. Step 6: Combine all solutions: x = pi/6, pi/2, 5pi/6, 3pi/2.
Answer
pi/6, pi/2, 5pi/6, 3pi/2.
Why not the others: 'pi/6, 5pi/6' omits the solutions where cos(x) = 0; 'pi/2, 3pi/2' omits the solutions where sin(x) = 1/2; 'pi/3, ...' solves for cos(x) = 1/2.

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