Find all x values in the interval [0, 2π] that satisfies: 2 cos(x) − 1 > 0

To solve the inequality \(2 \cos(x) - 1 > 0\) for \(x\) in the interval \([0, 2\pi]\), follow these steps:

### Step 1: Solve the inequality
\[
2 \cos(x) - 1 > 0
\]
Add 1 to both sides:
\[
2 \cos(x) > 1
\]
Divide by 2:
\[
\cos(x) > \frac{1}{2}
\]

### Step 2: Determine the values of \(x\) where \(\cos(x) > \frac{1}{2}\)
The cosine function is positive in the first and fourth quadrants. Specifically, \(\cos(x) > \frac{1}{2}\) when:
\[
x \in \left( -\frac{\pi}{3}, \frac{\pi}{3} \right) \cup \left( \frac{5\pi}{3}, \frac{7\pi}{3} \right)
\]

### Step 3: Adjust the intervals to match the given interval \([0, 2\pi]\)
We need to find the \(x\) values within the interval \([0, 2\pi]\). The relevant portions of the solution are:
\[
x \in \left[ 0, \frac{\pi}{3} \right) \cup \left( \frac{5\pi}{3}, 2\pi \right]
\]

### Final Answer
The values of \(x\) that satisfy \(2 \cos(x) - 1 > 0\) in the interval \([0, 2\pi]\) are:
\[
x \in \left[ 0, \frac{\pi}{3} \right) \cup \left( \frac{5\pi}{3}, 2\pi \right]
\]

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Here are some related questions:
1. How do you solve the inequality \( \sin(x) > \frac{1}{2} \) over the interval \([0, 2\pi]\)?
2. What is the general solution to the equation \( \cos(x) = \frac{1}{2} \)?
3. How do you find the maximum and minimum values of the function \( f(x) = 2 \cos(x) - 1 \)?
4. What is the graphical interpretation of \( \cos(x) > \frac{1}{2} \)?
5. How do the solutions of \( \cos(x) = \frac{1}{2} \) change if the interval is \( [0, 4\pi] \)?

**Tip:** When solving trigonometric inequalities, it's useful to first determine the critical points where the inequality changes sign and then consider the behavior within each interval defined by these points.

Learn how to solve the trigonometric inequality 2 cos(x) - 1 > 0 for x in the interval [0, 2π]. Understand the steps involved, including adding and dividing by constants, and determining cosine ranges. Suitable for high school students studying trigonometry.

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