Solve the equation tan^2(x) = 3.

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Question: Solve the equation tan^2(x) = 3.

Options:

Correct Answer: x = π/3

Solution:

The solutions are x = π/3 and x = 4π/3.

Solve the equation tan^2(x) = 3.

Solve the equation tan^2(x) = 3.

Step 1: Start with the equation tan^2(x) = 3.
Step 2: Take the square root of both sides to get tan(x) = ±√3.
Step 3: Find the angles where tan(x) = √3. This occurs at x = π/3 + kπ, where k is any integer.
Step 4: Find the angles where tan(x) = -√3. This occurs at x = 2π/3 + kπ, where k is any integer.
Step 5: Combine the solutions from Steps 3 and 4. For k = 0, the solutions are x = π/3 and x = 2π/3.
Step 6: For k = 1, the solutions are x = π/3 + π = 4π/3 and x = 2π/3 + π = 5π/3.
Step 7: List the solutions: x = π/3, 4π/3, 2π/3, and 5π/3.

Trigonometric Functions – Understanding the properties and values of the tangent function, particularly its squares and periodicity.
Inverse Trigonometric Functions – Using the inverse tangent function to find angles that satisfy the equation.
Periodic Solutions – Recognizing that trigonometric equations can have multiple solutions due to their periodic nature.