If cosx = `(1)/(2)`and `0 lt x lt 2pi` then solutions are

Options

A. `x=(pi)/(3),(4pi)/(3)`
B. `x=(2pi)/(3),(4pi)/(3)`
C. `x=(2pi)/(3),(7pi)/(6)`
D. `x=(2pi)/(3),(pi)/(3)`

Answer

To solve the problem where \( \cos x = -\frac{1}{2} \) and \( 0 < x < 2\pi \), we can follow these steps:

### Step 1: Identify the angles where cosine is negative
The cosine function is negative in the second and third quadrants. We need to find the reference angle where \( \cos \) is \( \frac{1}{2} \).

### Step 2: Find the reference angle
The reference angle for \( \cos x = \frac{1}{2} \) is:
\[ x = \frac{\pi}{3} \]

### Step 3: Determine the angles in the second and third quadrants
1. In the second quadrant, the angle can be found using:
\[ x = \pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

2. In the third quadrant, the angle can be found using:
\[ x = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3} \]

### Step 4: List the solutions
The solutions for \( \cos x = -\frac{1}{2} \) in the interval \( 0 < x < 2\pi \) are:
\[ x = \frac{2\pi}{3}, \quad x = \frac{4\pi}{3} \]

### Conclusion
Thus, the solutions are \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \).

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