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Q.
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 :
Given that both points $(1,2)\&(\sin \theta, \cos \theta)$ lie
on same side of the line $x+y-1=0$
$So\begin{pmatrix}put \left(1,2\right)in\\ given line\end{pmatrix}\begin{pmatrix}put sin\theta cos \theta in\\ given line\end{pmatrix}>0$
$\Rightarrow (1+2-1)(\sin \theta+\cos \theta-1)>0$
$\Rightarrow \sin \theta+\cos \theta>1\{\div by \sqrt{2}\}$
$\Rightarrow \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta>\frac{1}{\sqrt{2}}$
$\Rightarrow sin\left(\theta + \frac{\pi}{4}\right) > \frac{1}{\sqrt{2}}$
$\Rightarrow \frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{3 \pi}{4}$
$\Rightarrow 0<\theta<\frac{\pi}{2}$
