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

Verified step by step guidance

1. Start with the given equation: \(2 \tan x - 1 = 0\).
2. Isolate \(\tan x\) by adding 1 to both sides and then dividing by 2, giving \(\tan x = \frac{1}{2}\).
3. 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.
4. 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.
5. 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.

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values within a given interval that satisfy the equation. This often requires algebraic manipulation and understanding the periodic nature of trig functions.

Properties of the Tangent Function

The tangent function, tan(x), is periodic with period π and is undefined at odd multiples of π/2. Knowing its period and behavior helps find all solutions within the interval by adding integer multiples of π to the principal solution.

Interval Notation and Solution Restrictions

When solving trig equations over a specific interval like [0, 2π), solutions must be restricted to that range. This means identifying all valid solutions within the interval and expressing them as exact values or decimal approximations as required.