Solve the equation 3cos^2(x) - 1 = 0.

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Question: Solve the equation 3cos^2(x) - 1 = 0.

Options:

Correct Answer: x = π/3, 2π/3

Solution:

Rearranging gives cos^2(x) = 1/3, so x = π/3 and 2π/3.

Solve the equation 3cos^2(x) - 1 = 0.

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.

Trigonometric Identities – Understanding and applying the identity cos^2(x) + sin^2(x) = 1 to solve for angles.
Quadratic Equations – Recognizing and solving quadratic forms in trigonometric equations.
Periodic Functions – Considering the periodic nature of the cosine function when finding all solutions.