Question

If both $\theta$ and $\varphi$ are acute angles such that $\sin \theta =\frac{1}{2}$ and $\cos \varphi =\frac{1}{3}$, then $(\theta +\varphi )$ belongs to

• A. $(\frac{\pi }{3},\frac{5\pi }{12})$
• B. $(\frac{\pi }{2},\frac{2\pi }{3})$
• C. $(\frac{2\pi }{3},\frac{5\pi }{6})$
• D. $(\frac{5\pi }{12},\frac{\pi }{2})$

Answer 1

The correct option is B $(\frac{\pi }{2},\frac{2\pi }{3})$

Given that $\sin \theta =\frac{1}{2}$ and $\cos \varphi =\frac{1}{3}$; $\theta$ and $\varphi$ are acute angles.

$\sin \theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{6}$
We know that, $0<\frac{1}{3}<\frac{1}{2}$
$\Rightarrow \cos \frac{\pi }{2}<\cos \varphi <\cos \frac{\pi }{3}$
$\Rightarrow \frac{\pi }{3}<\varphi <\frac{\pi }{2}$
$\therefore \frac{\pi }{3}+\frac{\pi }{6}<\varphi +\theta <\frac{\pi }{2}+\frac{\pi }{6}$
$\Rightarrow \frac{\pi }{2}<\theta +\varphi <\frac{2\pi }{3}$
$\Rightarrow \theta +\varphi \in (\frac{\pi }{2},\frac{2\pi }{3})$