The set of all possible values of `theta` in the interval `(0,pi)` for which the points (1,2) and `( sin theta, cos theta)` lie on the same side of the line `x+y=1` is:

Options

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

Answer

To solve the problem, we need to determine the set of all possible values of \( \theta \) in the interval \( (0, \pi) \) such that the points \( (1, 2) \) and \( (\sin \theta, \cos \theta) \) lie on the same side of the line \( x + y = 1 \).

### Step-by-step Solution:

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

2. **Determine the position of the points relative to the line**: For a point \( (x_1, y_1) \) to be on the same side of the line as another point \( (x_2, y_2) \), the following condition must hold:
\[ (x_1 + y_1 - 1)(x_2 + y_2 - 1) > 0 \]
Here, \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (\sin \theta, \cos \theta) \).

3. **Substituting the points into the condition**: We substitute the points into the inequality:
\[ (1 + 2 - 1)(\sin \theta + \cos \theta - 1) > 0 \]
Simplifying this gives:
\[ 2(\sin \theta + \cos \theta - 1) > 0 \]
This simplifies to:
\[ \sin \theta + \cos \theta - 1 > 0 \]
or
\[ \sin \theta + \cos \theta > 1 \]

4. **Using the identity for sine and cosine**: We can use the identity \( \sin \theta + \cos \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \) to rewrite the inequality:
\[ \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) > 1 \]
Dividing both sides by \( \sqrt{2} \):
\[ \sin\left(\theta + \frac{\pi}{4}\right) > \frac{1}{\sqrt{2}} \]

5. **Finding the angles**: The sine function is greater than \( \frac{1}{\sqrt{2}} \) in the intervals:
\[ \theta + \frac{\pi}{4} \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \]
This translates to:
\[ \frac{\pi}{4} < \theta + \frac{\pi}{4} < \frac{3\pi}{4} \]
Subtracting \( \frac{\pi}{4} \) from all parts gives:
\[ 0 < \theta < \frac{\pi}{2} \]

6. **Conclusion**: The set of all possible values of \( \theta \) in the interval \( (0, \pi) \) for which the points \( (1, 2) \) and \( (\sin \theta, \cos \theta) \) lie on the same side of the line \( x + y = 1 \) is:
\[ \theta \in \left(0, \frac{\pi}{2}\right) \]