Question: Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
Options:
π/3
2π/3
4π/3
5π/3
Correct Answer: π/3
Solution:
The solutions are x = π/3 and x = 4π/3.
Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
Practice Questions
Q1
Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
π/3
2π/3
4π/3
5π/3
Questions & Step-by-Step Solutions
Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
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: Identify the angles where tan(x) = √3. These angles are x = π/3 and x = 4π/3.
Step 4: Identify the angles where tan(x) = -√3. These angles are x = 2π/3 and x = 5π/3.
Step 5: List all the solutions: x = π/3, x = 4π/3, x = 2π/3, and x = 5π/3.
Step 6: Since we only need the solutions in the interval [0, 2π], we keep all four solutions.
Trigonometric Equations – The question tests the ability to solve equations involving trigonometric functions, specifically the tangent function.
Periodic Nature of Trigonometric Functions – Understanding the periodicity of the tangent function is crucial for finding all solutions within a specified interval.
Quadratic Form in Trigonometry – Recognizing that tan^2(x) can be treated as a quadratic equation helps in solving for x.
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