The set of all possible values of

Question:

The set of all possible values of $\theta$ in the interval $(0, \pi)$ for which the points $(1,2)$ and $(\sin \theta, \cos \theta)$ lie on the same side of the line $x+y=1$ is :

  1. (1) $\left(0, \frac{\pi}{2}\right)$

  2. (2) $\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$

  3. (3) $\left(0, \frac{3 \pi}{4}\right)$

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


Correct Option: 1

Solution:

Let $f(x, y)=x+y-1$

Given $(1,2)$ and $(\sin \theta, \cos \theta)$ are lies on same side.

$\therefore f(1,2) \cdot f(\sin \theta, \cos \theta)>0$

$\Rightarrow 2[\sin \theta+\cos \theta-1]>0$

$\Rightarrow \sin \theta+\cos \theta>1 \Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}$

$\Rightarrow \theta+\frac{\pi}{4} \in\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right) \Rightarrow \theta \in\left(0, \frac{\pi}{2}\right)$

 

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