Question

If $\theta$ and $\phi$ are acute angles, $\sin \, \theta = \frac{1}{2} , \cos \, \phi = \frac{1}{3},$ then the value of $\theta + \phi$ lies in

• 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

Answer: (B)

$\left( \frac{\pi}{2}, \frac{2\pi}{3} \right]$

Answer 2

Now $\sin\theta = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{6}$ and $\cos \phi = \frac{1}{3} \Rightarrow \frac{\pi}{3} < \phi < \frac{\pi}{2}$ $\left(\because \frac{1}{2} > \frac{1}{3} > 0 , i.e., \cos \frac{\pi}{3} > \cos \phi > \cos \frac{\pi}{2}\right)$ Hence $\frac{\pi}{6} + \frac{\pi}{3} < \theta + \phi< \frac{\pi}{6} + \frac{\pi}{2}$