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If \( sin\theta =\frac{1}{2}\), \( cos\phi =\frac{1}{3}\), then \( \theta +\phi \) belongs to, where \( 0\lt \theta , \phi \lt \frac{\pi }{2}\)( )
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

B
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Solution

Given:\(\sin\theta=\frac{1}{2},\cos\phi=\frac{1}{3}\) and \(\theta\in(0,\frac{\pi}{2})\).
\(\implies \theta=\frac{\pi}{6}\cdots(i)\)
Since,\(0\lt\frac{1}{3}\lt\frac{1}{2}\)
We have \(\frac{\pi}{3}\lt \phi\lt\frac{\pi}{2}\)
Adding \(\theta\) on each sides of the inequality, we get
\(\frac{\pi}{3}+\theta\lt \phi+\theta\lt\frac{\pi}{2}+\theta\)
From (i) \(\frac{\pi}{3}+\frac{\pi}{6}\lt \phi+\theta\lt\frac{\pi}{2}+\frac{\pi}{6}\)
\(\implies (\theta+\phi )\in (\frac{\pi}{2},\frac{2\pi}{3})\).
Hence, correct option is B
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