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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Trignometry
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If $\tan\alpha=\large\frac{1}{7}$ and $\sin\beta=\large\frac{1}{\sqrt {10}}$ where $0\;<\alpha,\beta\;<\;\large\frac{\pi}{2}$ then $2\beta$ is equal to:

$(a)\;\large\frac{\pi}{4}$$-\alpha\qquad(b)\;\large\frac{3\pi}{4}-$$\alpha\qquad(c)\;\large\frac{\pi}{8}$$-\alpha\qquad(d)\;\large\frac{3\pi}{8}-\frac{\alpha}{2}$

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1 Answer

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If $\tan\alpha =\large\frac{1}{7}$ and $\sin\beta=\large\frac{1}{\sqrt{10}}$
$\sin 2\beta=2\sin\beta\cos\beta$
$\qquad\;\;=2.\large\frac{1}{\sqrt{10}}.\frac{3}{\sqrt{10}}$
$\qquad\;\;=2.\large\frac{3}{10}$
$\qquad\;\;=\large\frac{3}{5}$
$\tan(\alpha+2\beta)=\large\frac{\tan\alpha+\tan 2\beta}{1-\tan\alpha\tan 2\beta}$
$\qquad\qquad\;\;\;=\large\frac{\Large\frac{1}{7}+\Large\frac{3}{4}}{1-\Large\frac{1}{7}.\frac{3}{4}}$
$\qquad\qquad\;\;\;=\large\frac{25}{25}$
$\qquad\qquad\;\;\;=1$
$2\beta=\large\frac{\pi}{4}$$-\alpha$
Hence (a) is the correct answer.
answered Oct 8, 2013 by sreemathi.v
 

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