Question
The range of values of $$\theta \, \in \left[ {0,\,2\pi } \right]$$ for which $$\left( {1 + \cos \,\theta ,\,\sin \,\theta } \right)$$ is an interior point of the circle $${x^2} + {y^2} - 1$$ is :
A.
$$\left( {\frac{\pi }{6},\,\frac{{5\pi }}{6}} \right)$$
B.
$$\left( {\frac{{2\pi }}{3},\,\frac{{5\pi }}{3}} \right)$$
C.
$$\left( {\frac{\pi }{6},\,\frac{{7\pi }}{6}} \right)$$
D.
$$\left( {\frac{{2\pi }}{3},\,\frac{{4\pi }}{3}} \right)$$
Answer :
$$\left( {\frac{{2\pi }}{3},\,\frac{{4\pi }}{3}} \right)$$
Solution :
For point $$\left( {1 + \cos \,\theta ,\,\sin \,\theta } \right)$$ to be interior point of the circle $${x^2} + {y^2} = 1$$
$$\eqalign{
& {\left( {1 + \cos \,\theta } \right)^2} + {\sin ^2}\theta - 1 < 0 \cr
& \Rightarrow 1 + 1 + 2\cos \,\theta - 1 < 0 \cr
& \Rightarrow \cos \,\theta < - \frac{1}{2} \cr
& \Rightarrow \theta \in \left( {\frac{{2\pi }}{3},\,\frac{{4\pi }}{3}} \right) \cr} $$