# The set of all possible values of θ in the interval

Question:

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:

1. $\left(0, \frac{\pi}{4}\right)$

2. $\left(0, \frac{3 \pi}{4}\right)$

3. $\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$

4. $\left(0, \frac{\pi}{2}\right)$

Correct Option: , 4

Solution:

Given that both points $(1,2) \&(\sin \theta, \cos \theta)$ li on same side of the line $x+y-1=0$

So, $\left(\begin{array}{l}\text { Put }(1,2) \text { in } \\ \text { given line }\end{array}\right)\left(\begin{array}{l}\text { Put }(\sin \theta, \cos \theta \text { in } \\ \text { given line }\end{array}\right)>0$

$\Rightarrow(1+2-1)(\sin \theta+\cos \theta-1)>0$

$\Rightarrow \sin \theta+\cos \theta>1\{\div$ by $\sqrt{2}\}$

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

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

$\Rightarrow \frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{3 \pi}{4}$

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