Question

Given both $\theta and \phi are acute angles and sin \theta=\frac{1}{2}, cos \phi=\frac{1}{3},$ then the value of $\theta+\phi $ belongs to

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

Answer 1

Answer: (B)

Since, $sin \theta=\frac{1}{2} and cos \phi =\frac{1}{3}$
$\Rightarrow \theta=\frac{\pi}{6} and 0 < \Bigg(cos \phi =\frac{1}{3}\Bigg) < \frac{1}{2} \Bigg[as 0 < \frac{1}{3} < \frac{1}{2}\Bigg]$
$\Rightarrow \theta=\frac{\pi}{6} and cos^{-1} (0) > \phi > cos^{-1}\Bigg(\frac{1}{2}\Bigg)$
$\Bigg[\text{the sign changed as cos x is decreasing between}\Big(0, \frac{\pi}{2}\Big)\Bigg]$
$\Rightarrow \pi =\frac{\pi}{6} and \frac{\pi}{3} < \phi < \frac{\pi}{3}$
$\Rightarrow \frac{\pi}{2} < \theta + \phi < \frac{2\pi}{3}$
$\therefore \theta \in \Bigg(\frac{\pi}{2},\frac{2\pi}{3}\Bigg)$