To solve the problem, we need to determine the values of \( \theta \) such that the points \( A(5/\sqrt{2}, \sqrt{3}) \) and \( B(\cos^2 \theta, \cos \theta) \) lie on the same side of the line given by the equation \( 2x - y = 1 \).
### Step-by-Step Solution:
1. **Identify the line equation**:
The line can be rewritten in the standard form:
\[
2x - y - 1 = 0
\]
Here, \( a = 2 \), \( b = -1 \), and \( c = -1 \).
2. **Substitute point A into the line equation**:
Substitute \( A(5/\sqrt{2}, \sqrt{3}) \) into the line equation:
\[
2\left(\frac{5}{\sqrt{2}}\right) - \sqrt{3} - 1
\]
Simplifying this:
\[
= \frac{10}{\sqrt{2}} - \sqrt{3} - 1 = 5\sqrt{2} - \sqrt{3} - 1
\]
3. **Substitute point B into the line equation**:
Substitute \( B(\cos^2 \theta, \cos \theta) \) into the line equation:
\[
2(\cos^2 \theta) - \cos \theta - 1
\]
4. **Determine the condition for same side**:
For points \( A \) and \( B \) to be on the same side of the line, the product of the results from substituting both points into the line equation must be greater than zero:
\[
(5\sqrt{2} - \sqrt{3} - 1) \cdot (2\cos^2 \theta - \cos \theta - 1) > 0
\]
5. **Evaluate the sign of \( 5\sqrt{2} - \sqrt{3} - 1 \)**:
Calculate \( 5\sqrt{2} - \sqrt{3} - 1 \):
- Approximate values: \( \sqrt{2} \approx 1.414 \) and \( \sqrt{3} \approx 1.732 \)
- Thus, \( 5\sqrt{2} \approx 7.07 \)
- Therefore, \( 5\sqrt{2} - \sqrt{3} - 1 \approx 7.07 - 1.732 - 1 \approx 4.338 \) (which is positive).
6. **Set up the inequality**:
Since \( 5\sqrt{2} - \sqrt{3} - 1 > 0 \), we need:
\[
2\cos^2 \theta - \cos \theta - 1 > 0
\]
7. **Factor the quadratic**:
The quadratic can be factored:
\[
2\cos^2 \theta - \cos \theta - 1 = (2\cos \theta + 1)(\cos \theta - 1)
\]
8. **Determine the intervals**:
The product \( (2\cos \theta + 1)(\cos \theta - 1) > 0 \) gives us two conditions:
- \( 2\cos \theta + 1 > 0 \) implies \( \cos \theta > -\frac{1}{2} \)
- \( \cos \theta - 1 > 0 \) implies \( \cos \theta > 1 \) (which is not possible).
Thus, we only consider the first condition:
- \( \cos \theta > -\frac{1}{2} \)
9. **Find the angles for \( \theta \)**:
The cosine function is greater than \(-\frac{1}{2}\) in the intervals:
\[
\theta \in [\frac{5\pi}{3}, 2\pi] \cup [0, \frac{\pi}{3}]
\]
However, since we are restricted to \( [\pi, 2\pi] \), we focus on:
\[
\theta \in [\frac{5\pi}{3}, 2\pi]
\]
### Final Answer:
The values of \( \theta \) in the interval \( [\pi, 2\pi] \) such that points \( A \) and \( B \) are on the same side of the line \( 2x - y = 1 \) are:
\[
\theta \in \left[\frac{5\pi}{3}, 2\pi\right]
\]
