Key Concepts and Formulas

• Position of a Point with Respect to a Line: A point $(x_1, y_1)$ lies on the same side of the line $Ax + By + C = 0$ as another point if $Ax_1 + By_1 + C$ has the same sign as the expression evaluated at the other point.
• Trigonometric Identity: $a \sin x + b \cos x = R \sin(x + \alpha)$, where $R = \sqrt{a^2 + b^2}$ and $\alpha = \arctan(b/a)$.
• Solving Trigonometric Inequalities: Using the unit circle or the graph of trigonometric functions to determine the intervals where the inequality holds.

Step-by-Step Solution

1. Define the Line Function The given line is $x + y = 1$. We rewrite it as $x + y - 1 = 0$ and define $f(x, y) = x + y - 1$. Explanation: This step puts the line equation in the standard form $Ax + By + C = 0$, making it easier to evaluate.

2. Apply the Same Side Condition The points $(1, 2)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $x + y - 1 = 0$. Therefore, $f(1, 2) \cdot f(\sin \theta, \cos \theta) > 0$ Explanation: This applies the condition for points lying on the same side of a line.

3. Evaluate $f(1, 2)$ Substitute the coordinates of $(1, 2)$ into $f(x, y)$: $f(1, 2) = 1 + 2 - 1 = 2$ Since $f(1, 2) = 2 > 0$, the point $(1, 2)$ lies on the positive side of the line. Explanation: This evaluates the function at the first point to determine which side of the line it lies on.

4. Evaluate $f(\sin \theta, \cos \theta)$ Substitute the coordinates of $(\sin \theta, \cos \theta)$ into $f(x, y)$: $f(\sin \theta, \cos \theta) = \sin \theta + \cos \theta - 1$ Explanation: This evaluates the function at the second point, expressed in terms of $\theta$.

5. Formulate and Simplify the Inequality Substitute the results from steps 3 and 4 into the condition from step 2: $2 \cdot (\sin \theta + \cos \theta - 1) > 0$ Divide both sides by 2: $\sin \theta + \cos \theta - 1 > 0$ $\sin \theta + \cos \theta > 1$ Explanation: This simplifies the inequality.

6. Transform the Trigonometric Expression Using the identity $a \sin x + b \cos x = R \sin(x + \alpha)$, where $R = \sqrt{a^2 + b^2}$ and $\alpha = \arctan(b/a)$. Here, $a = 1$ and $b = 1$. So, $R = \sqrt{1^2 + 1^2} = \sqrt{2}$. And $\alpha = \arctan(1/1) = \pi/4$. Thus, $\sin \theta + \cos \theta = \sqrt{2} \sin(\theta + \pi/4)$. The inequality becomes: $\sqrt{2} \sin\left(\theta + {\pi \over 4}\right) > 1$ Explanation: This step transforms the sum of sine and cosine into a single sine function.

7. Isolate the Trigonometric Function Divide both sides of the inequality by $\sqrt{2}$: $\sin\left(\theta + {\pi \over 4}\right) > {1 \over {\sqrt 2}}$ Explanation: Isolating the sine function.

8. Determine the Domain for the Argument Since $0 < \theta < \pi$, we have: $0 + {\pi \over 4} < \theta + {\pi \over 4} < \pi + {\pi \over 4}$ ${\pi \over 4} < \theta + {\pi \over 4} < {{5\pi} \over 4}$ Let $X = \theta + \pi/4$. So, we need to solve $\sin X > 1/\sqrt{2}$ for $X \in (\pi/4, 5\pi/4)$. Explanation: This step finds the correct range for the transformed variable $X$.

9. Solve the Trigonometric Inequality for $X$ We need to find values of $X$ in $(\pi/4, 5\pi/4)$ such that $\sin X > 1/\sqrt{2}$. We know that $\sin X = 1/\sqrt{2}$ when $X = \pi/4$ or $X = 3\pi/4$. So, the solution for $X$ is $(\pi/4, 3\pi/4)$. Explanation: Identifies where the sine function exceeds the given value.

10. Solve for $\theta$ Substitute back $X = \theta + \pi/4$ into the solution for $X$: ${\pi \over 4} < \theta + {\pi \over 4} < {{3\pi} \over 4}$ Subtract $\pi/4$ from all parts of the inequality: ${\pi \over 4} - {\pi \over 4} < \theta < {{3\pi} \over 4} - {\pi \over 4}$ $0 < \theta < {{2\pi} \over 4}$ $0 < \theta < {\pi \over 2}$ So, the set of possible values for $\theta$ is $\left(0, {\pi \over 2}\right)$. Explanation: Converts back to the original variable.

Common Mistakes & Tips

• Incorrect Domain: Forgetting to adjust the domain for the transformed variable $X = \theta + \pi/4$ can lead to incorrect solutions.
• Trigonometric Errors: Mistakes in applying the trigonometric identity or solving the inequality are common. Double-check the values of $R$ and $\alpha$.
• Sign Errors: Ensure the inequality sign is correctly maintained throughout the solution.

Summary

This problem involves applying the concept of points lying on the same side of a line, transforming a trigonometric expression, and solving a trigonometric inequality within a given domain. The final solution is $\theta \in (0, \pi/2)$.

The final answer is $\boxed{\left(0, {\pi \over 2}\right)}$, which corresponds to option (D).