The range of values of θ in the interval (0,π) such that the points (3,2) and (cosθ,sinθ) lie on the same sides of the line x+y−1=0, is
Explanation
L : x+y−1=0
As (3,2) & (cosθ,sinθ) lies on same side of line
L:(3,2)3+2-1=4>0
So, L:(\cos\theta,\sin\theta)≡\sin\theta+\cos\theta-1>0
\Rightarrow \sqrt{2}\left[\frac{1}{\sqrt{2}}\cos\theta+\frac{1}{\sqrt{2}}\sin\theta\right]>1
\sin\left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}
\Rightarrow \frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{3\pi}{4}
\Rightarrow 0<\theta<\frac{\pi}{2}
Explanation
L : x+y−1=0
As (3,2) & (cosθ,sinθ) lies on same side of line
L:(3,2)3+2-1=4>0
So, L:(\cos\theta,\sin\theta)≡\sin\theta+\cos\theta-1>0
\Rightarrow \sqrt{2}\left[\frac{1}{\sqrt{2}}\cos\theta+\frac{1}{\sqrt{2}}\sin\theta\right]>1
\sin\left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}
\Rightarrow \frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{3\pi}{4}
\Rightarrow 0<\theta<\frac{\pi}{2}
