The set of all possible values of $\theta$ in the interval $(0, \pi)$ for which the points $(1,2)$ and…
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 .
$\left(0, \frac{\pi}{2}\right)$
$\left(0, \frac{\pi}{4}\right)$
$\left(0, \frac{3 \pi}{4}\right)$
$\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$
Solution
for lying on the same side $(\sin \theta+\cos \theta-1)(1+2-1)>0$
$\begin{aligned}
& \Rightarrow \sin \theta+\cos \theta>1 \\
& \Rightarrow \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta>\frac{1}{\sqrt{2}}
\end{aligned}$
$\begin{aligned} & \Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}} \\ & \Rightarrow \frac{\pi}{4} < \theta+\frac{\pi}{4} < \frac{3 \pi}{4} \\ & \Rightarrow 0 < \theta < \frac{\pi}{2}\end{aligned}$
$\begin{aligned} & \Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}} \\ & \Rightarrow \frac{\pi}{4} < \theta+\frac{\pi}{4} < \frac{3 \pi}{4} \\ & \Rightarrow 0 < \theta < \frac{\pi}{2}\end{aligned}$