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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Trignometry
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Given both $\theta$ and $\phi$ are acute angles and $\sin\theta=\large\frac{1}{2},$$\cos\phi=\large\frac{1}{3}$ then the value of $\theta+\phi$ belongs to

$\begin{array}{1 1}(a)\;\big[\large\frac{\pi}{3},\frac{\pi}{2}\big]&(b)\;\big[\large\frac{\pi}{2},\frac{2\pi}{3}\big]\\(c)\;\big[\large\frac{2\pi}{3},\frac{5\pi}{6}\big]&(d)\;\big[\large\frac{5\pi}{6},\normalsize \pi\big]\end{array}$

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Given :
$\theta$ and $\phi$ both are acute angles.
0 < $\large\frac{1}{3}$ < $\large\frac{1}{2}$
$\cos\large\frac{\pi}{2}$ < $\cos \phi$ < $\cos\large\frac{\pi}{3}$
Or $\large\frac{\pi}{3}$ < $\phi$ < $\large\frac{\pi}{2}$
$\therefore\large\frac{\pi}{3}+\frac{\pi}{6}$$ < \theta+\phi < \large\frac{\pi}{2}+\frac{\pi}{6}$
Or $\large\frac{\pi}{2} $$< \theta+\phi < \large\frac{2\pi}{3}$
$\Rightarrow \theta+\phi\in \big[\large\frac{\pi}{2},\frac{2\pi}{3}\big]$
Hence (b) is the correct answer.
answered Oct 17, 2013 by sreemathi.v

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