The set of all possible values of θ in the interval (0, π) for...

We know that if $A\left({x}_{1}, {y}_{1}\right) \text{ and } B\left({x}_{2}, {y}_{2}\right)$ lies on the same side to line $L = 0$, then ${L}_{A\left({x}_{1}, {y}_{1}\right)}{L}_{B\left({x}_{2}, {y}_{2}\right)} > 0$.

Given, the points $A\left(1, 2\right) \text{ and } B\left(sin\theta, cos\theta\right)$ lies on the same side of the line $x + y - 1 = 0$.

$∴ \left(1 + 2 - 1\right)\left(sin\theta + cos\theta - 1\right) > 0$

$⇒ sin\theta + cos\theta > 1$

$⇒ sin\theta \times \dfrac{1}{\sqrt{2}} + cos\theta \times \dfrac{1}{\sqrt{2}} > 1 \times \dfrac{1}{\sqrt{2}}$

$⇒ sin\theta . cos\dfrac{\pi}{4} + cos\theta . sin\dfrac{\pi}{4} > \dfrac{1}{\sqrt{2}}$

$⇒ \sin \left(\theta + \dfrac{\pi}{4}\right) > \dfrac{1}{\sqrt{2}}$

$⇒ \dfrac{\pi}{4} < \left(\theta + \dfrac{\pi}{4}\right) < \dfrac{3\pi}{4}$

$⇒ 0 < \theta < \dfrac{\pi}{2}$

$⇒ \theta \in \left(0, \dfrac{\pi}{2}\right)$