The range of values of $\theta$ in the interval $(0, \mathrm{\pi })$ such that the points $( 3, 2)$ and $(\cos \theta , \sin \theta )$ lie on the same side of the line $x+y-1=0$, is 

If the points $(3,2)$and $(\cos \theta , \sin \theta )$lie on the same side of $x+y-1=0,$then 

$3+2-1$ and $\cos \theta +\sin \theta -1$  must be of the same sign 

$\therefore \cos \theta +\sin \theta -1>0$

$\begin{array}{l}\Rightarrow \frac{1}{\sqrt{2}}\cos \theta +\frac{1}{\sqrt{2}}\sin \theta >\frac{1}{\sqrt{2}}\\ \Rightarrow \cos \theta \sin \frac{\pi }{4}+\sin \theta \cos \frac{\pi }{4}>\frac{1}{\sqrt{2}}\\ \Rightarrow \sin \left(\theta +\frac{\pi }{4}\right)>\frac{1}{\sqrt{2}}\\ \Rightarrow \frac{\pi }{4}<\theta +\frac{\pi }{4}<\frac{3\pi }{4}\Rightarrow 0<\theta <\frac{\pi }{2}\Rightarrow \theta \in \left(0,\frac{\pi }{2}\right)\end{array}$