Question

Given $\sin \theta =\frac{1}{2}$ and $\cos \varphi =\frac{1}{3}$ are acute angles.Find range of $\theta +\varphi$

• A. $(\frac{\pi }{3},\frac{4\pi }{3})$
• B. $[\frac{\pi }{3},\frac{2\pi }{3}]$
• C. $(\frac{\pi }{2},\frac{2\pi }{3})$
• D. $(\frac{2\pi }{3},\frac{4\pi }{3})$

Answer 1

Answer: (C)

$(\frac{\pi }{2},\frac{2\pi }{3})$

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

$\Rightarrow \theta =\frac{\pi }{6}Now,0<\frac{1}{3}<\frac{1}{2}\Rightarrow \cos \frac{\pi }{2}<\cos \varphi <\cos \frac{\pi }{3}i.e.,\frac{\pi }{3}<\varphi <\frac{\pi }{2}So,\frac{\pi }{3}+\frac{\pi }{6}<\varphi +\frac{\pi }{6}<\frac{\pi }{2}+\frac{\pi }{6}\Rightarrow \frac{\pi }{2}<\varphi +\theta <\frac{2\pi }{3}\therefore \varphi +\theta ϵ(\frac{\pi }{2},\frac{2\pi }{3})$