Question

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

• A. $(0,\frac{\pi }{2})$
• B. $(0,\pi )$
• C. $(\frac{\pi }{4},\frac{\pi }{2})$
• D. $(0,\frac{\pi }{4})$

Answer 1

The correct option is D $(0,\frac{\pi }{4})$

As $(sin\theta ,cos\theta )$ and (3, 2) lie on the same side of x+y-1=0 , they should be of same sign.
$\therefore sin\theta +cos\theta -1>as3+2-1>0$
$\Rightarrow \sqrt{2}sin(\theta +\frac{\pi }{4})>1$
$\Rightarrow sin(\theta +\frac{\pi }{4})>\frac{1}{\sqrt{2}}\Rightarrow 0<\theta <\frac{\pi }{4}$.