Question

If $\theta ,\varphi$ are acute, $\sin \theta =\frac{1}{2},\cos \varphi =\frac{1}{3},$ then $(\theta +\varphi )\in$

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

Answer 1

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

Ans. $\left(b\right)$.

$\sin \theta =\frac{1}{2}$

$\therefore \frac{\pi }{6}=\theta =\frac{\pi }{6}$ .............$\left(1\right)$

$\cos \varphi =\frac{1}{3}$

Now $\frac{1}{2}>\frac{1}{3}>0$ and $\cos \frac{\pi }{3}>\cos \varphi >\cos \frac{\pi }{2}$

$\therefore \frac{\pi }{3}<\varphi <\frac{\pi }{2}$ .......$\left(2\right)$

Adding $\left(1\right)$ or $\left(2\right)$, $\frac{\pi }{2}<\theta +\varphi <\frac{2\pi }{3}$