Question
Given both $$\theta $$ and $$\phi $$ are acute angles and $$\sin \theta = \frac{1}{2},\,\cos \phi = \frac{1}{3},$$ then the value of $$\theta + \phi $$ belongs to
A.
$$\left( {\frac{\pi }{3},\frac{\pi }{2}} \right]$$
B.
$${\left( {\frac{\pi }{2},\frac{{2\pi }}{3}} \right)}$$
C.
$$\left( {\frac{{2\pi }}{3},\frac{{5\pi }}{6}} \right]$$
D.
$$\left( {\frac{{5\pi }}{6},\pi } \right]$$
Answer :
$${\left( {\frac{\pi }{2},\frac{{2\pi }}{3}} \right)}$$
Solution :
Given that $$\sin \theta = \frac{1}{2}\,{\text{and}}\,\cos \phi = \frac{1}{3}\,{\text{and }}\theta \,{\text{and }}\phi $$ both are acute angles
$$\eqalign{
& \therefore \,\,\theta = \frac{\pi }{6}\,{\text{and 0 < }}\frac{1}{3} < \frac{1}{2} \cr
& {\text{or }}\cos \frac{\pi }{2} < \cos \phi < \cos \frac{\pi }{3}\,{\text{or }}\frac{\pi }{3} < \phi < \frac{\pi }{2} \cr
& \therefore \,\,\frac{\pi }{3} + \frac{\pi }{6} < \theta + \phi < \frac{\pi }{2} + \frac{\pi }{6}{\text{ or }}\frac{\pi }{2} < \theta + \phi < \frac{{2\pi }}{3} \cr
& \Rightarrow \,\,\theta + \phi \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} \right) \cr} $$