Solve the equation 3cos^2(x) - 1 = 0.
Correct Answer: x = π/3 and 2π/3
- Step 1: Start with the equation 3cos^2(x) - 1 = 0.
- Step 2: Add 1 to both sides of the equation to get 3cos^2(x) = 1.
- Step 3: Divide both sides by 3 to isolate cos^2(x), resulting in cos^2(x) = 1/3.
- Step 4: Take the square root of both sides to find cos(x). This gives cos(x) = ±√(1/3).
- Step 5: Simplify √(1/3) to 1/√3 or √3/3.
- Step 6: Find the angles x where cos(x) = 1/√3 and cos(x) = -1/√3.
- Step 7: The angles for cos(x) = 1/√3 are x = π/6 and x = 11π/6.
- Step 8: The angles for cos(x) = -1/√3 are x = 5π/6 and x = 7π/6.
- Step 9: Combine all the solutions: x = π/6, 11π/6, 5π/6, and 7π/6.
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