Solve the equation tan(x) = √3 for x in the interval [0, 2π].

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Question: Solve the equation tan(x) = √3 for x in the interval [0, 2π].

Options:

Correct Answer: π/3

Solution:

The solutions are x = π/3 and x = 4π/3 in the interval [0, 2π].

Solve the equation tan(x) = √3 for x in the interval [0, 2π].

Solve the equation tan(x) = √3 for x in the interval [0, 2π].

Correct Answer: x = π/3 and x = 4π/3

Step 1: Understand the equation tan(x) = √3. This means we need to find angles x where the tangent of x equals √3.
Step 2: Recall that the tangent function is positive in the first and third quadrants.
Step 3: Find the reference angle where tan(x) = √3. This occurs at x = π/3 (or 60 degrees).
Step 4: Since tangent is positive in the third quadrant as well, we need to find the angle in that quadrant. This angle can be found by adding π to the reference angle: π/3 + π = 4π/3.
Step 5: Now we have two angles: x = π/3 and x = 4π/3.
Step 6: Check if both angles are within the interval [0, 2π]. Both π/3 and 4π/3 are within this interval.

Trigonometric Functions – Understanding the properties and values of the tangent function, particularly its periodicity and specific angle values.
Inverse Trigonometric Functions – Using the inverse tangent function to find angles that correspond to given tangent values.
Interval Notation – Identifying and restricting solutions to a specific interval, in this case, [0, 2π].