Question

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

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

Answer 1

The correct option is D

$\left(0,\frac{\pi }{4}\right)$

Explanation for the correct answer:

Finding the range of the given expression:

Let $L$ be the line given by $x+y–1=0$

$(3,2)$ lies in line $L$

$L_1\ge 3+2–1=4>0$

$(\sin \theta ,\cos \theta )$lies in line $L$

$L_2\ge \sin \theta +\cos \theta –1>0$

Since points lies on same side, $L_1{L}_2>0$

$$\begin{gathered} \Rightarrow 4(\sin \theta +\cos \theta –1)>0 \\ \Rightarrow \sin \theta +\cos \theta >1 \\ \Rightarrow \sqrt{2(\sin \theta +45°)}>1 \\ \Rightarrow \sin \theta +\frac{\pi }{4}>\frac{1}{\sqrt{2}} \\ \Rightarrow \frac{\pi }{4}<\theta +\frac{\pi }{4}<\frac{\pi }{2} \\ \therefore 0<\theta <\frac{\pi }{4} \end{gathered}$$

Hence, the correct answer is option (D).