Question

Given both $\theta and\varphi areacuteanglesandsin\theta =\frac{1}{2},cos\varphi =\frac{1}{3},$ then the value of $\theta +\varphi$ belongs to

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

Answer 1

Answer: (B)

Since, $sin\theta =\frac{1}{2}andcos\varphi =\frac{1}{3}$
$\Rightarrow \theta =\frac{\pi }{6}and0<(cos\varphi =\frac{1}{3})<\frac{1}{2}[as0<\frac{1}{3}<\frac{1}{2}]$
$\Rightarrow \theta =\frac{\pi }{6}andco{s}^{-1}(0)>\varphi >co{s}^{-1}\left(\frac{1}{2}\right)$
$\left[\text{the sign changed as cos x is decreasing between}\right(0,\frac{\pi }{2}\left)\right]$
$\Rightarrow \pi =\frac{\pi }{6}and\frac{\pi }{3}<\varphi <\frac{\pi }{3}$
$\Rightarrow \frac{\pi }{2}<\theta +\varphi <\frac{2\pi }{3}$
$\therefore \theta \in (\frac{\pi }{2},\frac{2\pi }{3})$