Find the values of x that satisfy 3cos^2(x) - 1 = 0.

Find the values of x that satisfy 3cos^2(x) - 1 = 0.

Correct Answer: x = π/3 + 2kπ, 2π/3 + 2kπ (k ∈ Z)

Step 1: Start with the equation 3cos^2(x) - 1 = 0.
Step 2: Add 1 to both sides of the equation to isolate the term with cos^2(x). This gives you 3cos^2(x) = 1.
Step 3: Divide both sides by 3 to solve for cos^2(x). This results in cos^2(x) = 1/3.
Step 4: Take the square root of both sides to find cos(x). Remember to consider both the positive and negative roots. This gives you cos(x) = sqrt(1/3) and cos(x) = -sqrt(1/3).
Step 5: Find the angles x that correspond to cos(x) = sqrt(1/3) and cos(x) = -sqrt(1/3).
Step 6: The angles for cos(x) = sqrt(1/3) are x = π/3 and x = 5π/3 (since cos is positive in the first and fourth quadrants).
Step 7: The angles for cos(x) = -sqrt(1/3) are x = 2π/3 and x = 4π/3 (since cos is negative in the second and third quadrants).
Step 8: Combine all the solutions: x = π/3, 2π/3, 4π/3, and 5π/3.

Trigonometric Equations – The question tests the ability to solve equations involving trigonometric functions, specifically cosine.
Quadratic Form in Trigonometry – The equation is transformed into a quadratic form, requiring knowledge of how to manipulate and solve such equations.
Periodic Solutions – Understanding that trigonometric functions are periodic and finding all equivalent angles that satisfy the equation.