If `cos theta ge 1//2` in the first quadrant, which one of the following is correct ?

Options

A. `theta le pi/3`
B. `theta ge pi/3`
C. `theta le pi/6`
D. `theta ge pi/6`

Answer

To solve the problem, we need to analyze the condition given: \( \cos \theta \geq \frac{1}{2} \) in the first quadrant.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We know that \( \cos \theta \) represents the x-coordinate of a point on the unit circle corresponding to the angle \( \theta \). The condition \( \cos \theta \geq \frac{1}{2} \) means we are looking for angles where the cosine value is at least \( \frac{1}{2} \).

2. **Identifying the Angle**:
The angle where \( \cos \theta = \frac{1}{2} \) is \( \theta = \frac{\pi}{3} \) radians (or 60 degrees).

3. **Analyzing the First Quadrant**:
In the first quadrant, the cosine function decreases as the angle increases. Therefore, if \( \cos \theta \) is greater than or equal to \( \frac{1}{2} \), then \( \theta \) must be less than or equal to \( \frac{\pi}{3} \).

4. **Conclusion**:
Thus, the correct interpretation of the condition \( \cos \theta \geq \frac{1}{2} \) in the first quadrant is that \( \theta \) must satisfy:
\[ \theta \leq \frac{\pi}{3} \]

### Final Answer:
The correct statement is that \( \theta \) is less than or equal to \( \frac{\pi}{3} \).

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