Question

If $\theta$ & $\varphi$ are acute angles & $\sin \theta =\frac{1}{2},\cos \varphi =\frac{1}{3}$, then the value of $\theta +\varphi$ belongs to the interval

• 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 1

The correct option is B $\left(\frac{\pi }{2},\frac{2\pi }{3}\right]$

$\sin \theta =\frac{1}{2}\Rightarrow \cos \theta =\frac{\sqrt{3}}{2}$
$\cos \phi =\frac{1}{3}\Rightarrow \sin \phi =\frac{2\sqrt{2}}{3}$
Now
$\sin (\theta +\phi )=\sin \theta \cos \phi +\cos \theta \sin \phi$
$\Rightarrow \sin (\theta +\phi )=\frac{1}{2}.\frac{1}{3}+\frac{\sqrt{3}}{2}.\frac{2\sqrt{2}}{3}$
$\Rightarrow \sin (\theta +\phi )=\frac{1+2\sqrt{6}}{6}\Rightarrow (\theta +\phi )ϵ(\frac{\pi }{2},\frac{2\pi }{3}]$