Question

Find the range of $\theta$ in the interval $(0,\pi )$ such that the point $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y–1=0.$

Answer 1

$3+5-1=7>0$

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

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

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

$\Rightarrow 0<\theta <\frac{2\pi }{4}$

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