Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
2 tan x -1 = 0

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Start with the given equation: \(2 \tan x - 1 = 0\).

Isolate \(\tan x\) by adding 1 to both sides and then dividing by 2, giving \(\tan x = \frac{1}{2}\).

Recall that \(\tan x = \frac{\sin x}{\cos x}\) and that the tangent function has a period of \(\pi\), so solutions repeat every \(\pi\) radians.

Find the principal solution \(x_1\) by taking the arctangent: \(x_1 = \arctan\left(\frac{1}{2}\right)\), which will be in the first quadrant since \(\frac{1}{2}\) is positive.

Find the second solution \(x_2\) in the interval \([0, 2\pi)\) by adding \(\pi\) to the principal solution: \(x_2 = x_1 + \pi\). These two values are the solutions to the equation in the given interval.

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