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  4. If ( sin θ, cos θ) and (3,2) lies on the same side of the line x+y=1, then θ lies between

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Q. If $(\sin \theta, \cos \theta)$ and $(3,2)$ lies on the same side of the line $x+y=1$, then $\theta$ lies between

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Solution:

Given points are $(\sin \theta, \cos \theta)$ and $(3,2)$ and a line
$x+y-1=0 .....$(i)
Since, $(3,2)$ lies on Eq. (i)
$\therefore 3+2-1>0$
and $(\sin \theta, \cos \theta)$ lies in Eq. (i)
$\therefore \sin \theta+\cos \theta-1>0$
$\Rightarrow \sin \theta+\cos \theta>1 $
$ \Rightarrow \sqrt{2}\left[\sin \left(\theta+\frac{\pi}{4}\right)\right]>1 $
$\Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}=\sin \left(\frac{\pi}{4}\right)$
image
$\Rightarrow \frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{3 \pi}{4} \Rightarrow 0 < \theta<\frac{\pi}{2}$
Hence, $ \theta \in\left(0, \frac{\pi}{2}\right)$