A line makes an angle `theta` both with x-axis and y-axis. A possible range of `theta` is

Options

A. `[0, pi/4]`
B. `[0,pi/2]`
C. `[pi/4,pi/2]`
D. `[pi/6,pi/3]`

Explanation

To solve the problem of finding the possible range of the angle \( \theta \) that a line makes with both the x-axis and y-axis, we can follow these steps:

### Step 1: Understanding Direction Ratios
The direction ratios of a line making angles \( \theta \) with the axes can be expressed using the cosine of these angles. If a line makes an angle \( \theta \) with the x-axis and \( \phi \) with the y-axis, then the direction ratios can be represented as:
- \( \cos \theta \) with respect to the x-axis
- \( \cos \phi \) with respect to the y-axis
- \( \cos \lambda \) with respect to the z-axis

### Step 2: Using the Cosine Rule
From the properties of direction cosines, we know that:
\[ \cos^2 \theta + \cos^2 \phi + \cos^2 \lambda = 1 \]
This is the fundamental equation governing the relationship between the angles a line makes with the coordinate axes.

### Step 3: Setting Up the Equation
Since the line makes the same angle \( \theta \) with both the x-axis and y-axis, we can set \( \phi = \theta \). Thus, we rewrite the equation as:
\[ \cos^2 \theta + \cos^2 \theta + \cos^2 \lambda = 1 \]
This simplifies to:
\[ 2\cos^2 \theta + \cos^2 \lambda = 1 \]

### Step 4: Expressing \( \cos^2 \lambda \)
From the equation above, we can express \( \cos^2 \lambda \) as:
\[ \cos^2 \lambda = 1 - 2\cos^2 \theta \]

### Step 5: Finding the Range for \( \cos^2 \lambda \)
Since \( \cos^2 \lambda \) must be non-negative (as it represents a squared value), we have:
\[ 1 - 2\cos^2 \theta \geq 0 \]
This leads to:
\[ 2\cos^2 \theta \leq 1 \]
or
\[ \cos^2 \theta \leq \frac{1}{2} \]

### Step 6: Solving for \( \theta \)
Taking the square root gives us:
\[ |\cos \theta| \leq \frac{1}{\sqrt{2}} \]
This implies:
\[ \cos \theta \leq \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos \theta \geq -\frac{1}{\sqrt{2}} \]

### Step 7: Finding the Angles
The angles corresponding to \( \cos \theta = \frac{1}{\sqrt{2}} \) are:
\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{3\pi}{4} \]
Thus, \( \theta \) can take values in the ranges:
\[ \frac{\pi}{4} \leq \theta \leq \frac{3\pi}{4} \]

### Conclusion
The possible range of \( \theta \) is:
\[ \theta \in \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \]