Question

The range of values of $\theta$ in the interval $(0,\pi )$ such that the points $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y-1=0$, is

• A. $(0,\frac{\pi }{2})$
• B. $(0,\frac{\pi }{4})$
• C. $(\frac{\pi }{4},\frac{\pi }{2})$
• D. $(\frac{\pi }{4},\frac{3\pi }{4})$

Answer 1

The correct option is A $(0,\frac{\pi }{2})$

Since,
$3+5-1>0\therefore \sin \theta +\cos \theta -1>0$
$\Rightarrow \sin \theta +\cos \theta >1\Rightarrow \sin (\frac{\pi }{4}+\theta )>\frac{1}{\sqrt{2}}$
$\Rightarrow \frac{3\pi }{4}>\theta +\frac{\pi }{4}>\frac{\pi }{4}\Rightarrow \frac{\pi }{2}>\theta >0$
Hence, $\theta \in (0,\frac{\pi }{2})$