If `theta` is in first quadrant and `costhetage(1)/(2)` then which one of the following is true ?

Options

A. `thetale(pi)/(3)`
B. `thetage(pi)/(3)`
C. `thetale(pi)/(6)`
D. `thetage(pi)/(6)`

Answer

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

### Step 1: Understand the cosine values in the first quadrant
In the first quadrant, the cosine function decreases as the angle increases. The values of cosine for specific angles are:
- \( \cos 0^\circ = 1 \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \approx 0.707 \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 90^\circ = 0 \)

### Step 2: Determine the range of \( \theta \)
Since \( \cos \theta \geq \frac{1}{2} \), we need to find the angles where this condition holds true. From the cosine values listed:
- \( \cos 60^\circ = \frac{1}{2} \)
- For angles less than \( 60^\circ \), the cosine value will be greater than \( \frac{1}{2} \).

Thus, the range of \( \theta \) is:
\[ 0^\circ \leq \theta \leq 60^\circ \]

### Step 3: Conclusion about the angle \( \theta \)
Since \( \theta \) is in the first quadrant and \( \cos \theta \geq \frac{1}{2} \), we conclude that:
- \( \theta \) must be less than or equal to \( 60^\circ \).

### Final Answer
The true statement is that \( \theta \) is in the range:
\[ 0^\circ \leq \theta \leq 60^\circ \]

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