Question

If $\theta$ and $\varphi$ are acute angles such that $\sin \theta =\frac{1}{2}$ and $\cos \varphi =\frac{1}{3}$, then $\theta +\varphi$ lies in

• A. $[\pi /3,\pi /2]$
• B. $[\pi /2,2\pi /3]$
• C. $[2\pi /3,5\pi /3]$
• D. $[5\pi /6,\pi ]$

Answer 1

Answer: (B)

$[\pi /2,2\pi /3]$

$\sin \theta =1/2\Rightarrow \theta =\pi /6$
and $\cos \varphi =1/3\Rightarrow \pi /3<\varphi <\pi /2$
So that $\frac{\pi }{2}<(\theta +\varphi )<\frac{2\pi }{3}$.