If points `A(5/sqrt2,sqrt3)` and `(cos^2theta, costheta)` are the same side the line 2x - y = 1, then find the values of `theta` in `[pi,2pi]`.

Answer

To solve the problem, we need to determine the values of \( \theta \) in the interval \([ \pi, 2\pi ]\) such that the points \( A( \frac{5}{\sqrt{2}}, \sqrt{3} ) \) and \( ( \cos^2 \theta, \cos \theta ) \) are on the same side of the line defined by the equation \( 2x - y = 1 \).

### Step-by-Step Solution:

1. **Identify the line equation**: The line can be rewritten in the standard form:
\[ 2x - y - 1 = 0 \]

2. **Determine the position of point A**: Substitute the coordinates of point \( A \) into the line equation:
\[ 2\left(\frac{5}{\sqrt{2}}\right) - \sqrt{3} - 1 \]
Calculate:
\[ = \frac{10}{\sqrt{2}} - \sqrt{3} - 1 = 5\sqrt{2} - \sqrt{3} - 1 \]
Since \( 5\sqrt{2} \approx 7.07 \) and \( \sqrt{3} \approx 1.73 \), we have:
\[ 5\sqrt{2} - \sqrt{3} - 1 \approx 7.07 - 1.73 - 1 = 4.34 > 0 \]
Thus, point \( A \) is above the line.

3. **Determine the position of point \( ( \cos^2 \theta, \cos \theta ) \)**: Substitute the coordinates into the line equation:
\[ 2(\cos^2 \theta) - \cos \theta - 1 \]
We want this expression to be greater than 0 for the point to be on the same side as point \( A \):
\[ 2\cos^2 \theta - \cos \theta - 1 > 0 \]

4. **Solve the quadratic inequality**: Factor the quadratic:
\[ 2\cos^2 \theta - \cos \theta - 1 = 0 \]
Using the quadratic formula:
\[ \cos \theta = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
\[ = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm 3}{4} \]
This gives us:
\[ \cos \theta = 1 \quad \text{or} \quad \cos \theta = -\frac{1}{2} \]

5. **Determine intervals**: The roots divide the number line into intervals. We check the sign of \( 2\cos^2 \theta - \cos \theta - 1 \) in the intervals:
- For \( \cos \theta < -\frac{1}{2} \) (e.g., \( \cos \theta = -1 \)): The expression is positive.
- For \( -\frac{1}{2} < \cos \theta < 1 \): The expression is negative.
- For \( \cos \theta > 1 \): The expression is positive (not possible since \( \cos \theta \) ranges from -1 to 1).

Therefore, the valid interval is:
\[ \cos \theta < -\frac{1}{2} \]

6. **Find \( \theta \) values**: The values of \( \theta \) corresponding to \( \cos \theta = -\frac{1}{2} \) are:
\[ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} \]
Since we need \( \theta \) in the interval \( [\pi, 2\pi] \), we take:
\[ \theta \in \left[ \frac{4\pi}{3}, 2\pi \right] \]

### Final Answer:
The values of \( \theta \) in the interval \([ \pi, 2\pi ]\) such that the points are on the same side of the line are:
\[ \theta \in \left[ \frac{4\pi}{3}, 2\pi \right] \]