Question

The range of values of $\theta ,\theta ϵ[0,2\pi ]$ for which $(\cos \theta ,\sin \theta )$ lies inside the triangle formed by $x+y-2=0,6x+2y-\sqrt{10}=0$ and $x-y-1=0$.

• A. $0<\theta <\frac{3\pi }{2}$
• B. $\theta <\frac{5\pi }{6}-{\tan }^{-1}3$
• C. $0<\theta <\frac{5\pi }{6}-{\tan }^{-1}3$
• D. None of these

Answer 1

Answer: (A)

$0<\theta <\frac{3\pi }{2}$

$S_1<0,S_2>0\&S_3>0$
$S_1<0\Rightarrow \cos \theta +\sin \theta -2<0$
$\frac{1}{\sqrt{2}}\cos \theta +\sin \theta \frac{1}{\sqrt{2}}<\sqrt{2}$
$\cos (\theta -\frac{\pi }{4})<\sqrt{2}$ True for all value of $\theta$
$S_2>0\Rightarrow 6\cos \theta +2\sin \theta >\sqrt{10}$
$3\cos \theta +\sin \theta >1.58$
Min. value of $3\cos \theta +\sin \theta$ is$-\sqrt{10}$
It is also true for all values of ${}^{\prime }{\theta }^{\prime }$
$S_3>0\Rightarrow \cos \theta -\sin \theta >1$
$\Rightarrow \cos (\theta +\frac{\pi }{4})>\cos \frac{\pi }{4}$
Hence, $0<\theta <\frac{3\pi }{2}$