The set of all possible values of θ in the interval 0 , π for which the points 1 ,   2 and…

We know that if $A\left(x_1,y_1\right) \& B\left(x_2,y_2\right)$ 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 $A\left(1,2\right) \& B\left(\sin \theta ,\cos \theta \right)$ lies on the same side of the line $x+y-1=0$.

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

$\Rightarrow \sin \theta +\cos \theta >1$

$\Rightarrow \sin \theta \times \frac{1}{\sqrt{2}}+\cos \theta \times \frac{1}{\sqrt{2}}>1\times \frac{1}{\sqrt{2}}$

$\Rightarrow \sin \theta .\cos \frac{\mathrm{\pi }}{4}+\cos \theta .\sin \frac{\mathrm{\pi }}{4}>\frac{1}{\sqrt{2}}$

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

$\Rightarrow \frac{\mathrm{\pi }}{4}<\left(\theta +\frac{\pi }{4}\right)<\frac{3\mathrm{\pi }}{4}$

$\Rightarrow 0<\theta <\frac{\mathrm{\pi }}{2}$

$\Rightarrow \theta \in \left(0,\frac{\pi }{2}\right)$