The range of values of `theta` in the interval `(0, pi)` such that the points (3,5) and `(sin theta , cos theta)` lie on the same side of the line `x+y-1=0`, is

Options

A. `0 lt theta lt (pi)/(4)`
B. `0 lt theta lt (pi)/(2)`
C. `0 lt theta lt pi`
D. `(pi)/(4) lt theta lt (3pi)/(4)`

Explanation

To solve the problem, we need to determine the range of values of \( \theta \) in the interval \( (0, \pi) \) such that the points \( (3, 5) \) and \( (\sin \theta, \cos \theta) \) lie on the same side of the line defined by the equation \( x + y - 1 = 0 \).

### Step-by-Step Solution:

1. **Identify the line equation**: The line is given by the equation \( x + y - 1 = 0 \). We can rewrite this as \( y = -x + 1 \).

2. **Determine the position of the point (3, 5)**:
- Substitute \( x = 3 \) and \( y = 5 \) into the line equation:
\[ 3 + 5 - 1 = 7 > 0 \]
This means that the point \( (3, 5) \) lies above the line.

3. **Condition for the point \( (\sin \theta, \cos \theta) \)**:
- For the point \( (\sin \theta, \cos \theta) \) to lie on the same side of the line as \( (3, 5) \), we need:
\[ \sin \theta + \cos \theta - 1 > 0 \]
This simplifies to:
\[ \sin \theta + \cos \theta > 1 \]

4. **Manipulate the inequality**:
- To analyze \( \sin \theta + \cos \theta > 1 \), we can multiply and divide by \( \sqrt{2} \):
\[ \frac{1}{\sqrt{2}}(\sin \theta + \cos \theta) > \frac{1}{\sqrt{2}} \]
- This can be rewritten using the angle addition formula:
\[ \sin \theta + \cos \theta = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin \theta + \frac{1}{\sqrt{2}} \cos \theta \right) = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
- Thus, we have:
\[ \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) > 1 \]
- Dividing both sides by \( \sqrt{2} \):
\[ \sin\left(\theta + \frac{\pi}{4}\right) > \frac{1}{\sqrt{2}} \]

5. **Find the angles**:
- The inequality \( \sin x > \frac{1}{\sqrt{2}} \) holds for:
\[ x \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \]
- Therefore, we set:
\[ \theta + \frac{\pi}{4} \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \]
- This leads to:
\[ \frac{\pi}{4} < \theta + \frac{\pi}{4} < \frac{3\pi}{4} \]
- Subtracting \( \frac{\pi}{4} \) from all parts:
\[ 0 < \theta < \frac{\pi}{2} \]

6. **Conclusion**:
- The range of values of \( \theta \) in the interval \( (0, \pi) \) such that the points \( (3, 5) \) and \( (\sin \theta, \cos \theta) \) lie on the same side of the line \( x + y - 1 = 0 \) is:
\[ \theta \in \left( 0, \frac{\pi}{2} \right) \]