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JEE Main & Advanced Mathematics Trigonometric Identities Question Bank Trigonometrical ratios of multiple and sub multiple angles

  • question_answer
    If \[\tan \alpha =\frac{1}{7}\]and \[\sin \beta =\frac{1}{\sqrt{10}}\left( 0<\alpha ,\,\beta <\frac{\pi }{2} \right)\], then \[2\beta \]is equal to

    A) \[\frac{\pi }{4}-\alpha \]

    B) \[\frac{3\pi }{4}-\alpha \]

    C) \[\frac{\pi }{8}-\frac{\alpha }{2}\]

    D) \[\frac{3\pi }{8}-\frac{\alpha }{2}\]

    Correct Answer: A

    Solution :

    Since \[\sin \beta =\frac{1}{\sqrt{10}}\Rightarrow \tan \beta =\frac{1}{3}\] Þ \[\tan 2\beta =\frac{2\tan \beta }{1-{{\tan }^{2}}\beta }=\frac{3}{4}\] \[\therefore \tan (\alpha +2\beta )=\frac{\frac{1}{7}+\frac{3}{4}}{1-\frac{1}{7}.\frac{3}{4}}=\frac{25}{25}=1\] Now,\[0<\beta <\frac{\pi }{2}\]and\[\tan 2\beta =\frac{3}{4}>0\]bothÞ \[0<2\beta <\frac{\pi }{2}\]. Again,\[0<\alpha <\frac{\pi }{2}\]and\[0<2\beta <\frac{\pi }{2}\]bothÞ \[0<\alpha +2\beta <\pi \] Thus, \[0<\alpha +2\beta <\pi \]and \[\tan (\alpha +2\beta )=1\]both Þ \[\alpha +2\beta =\frac{\pi }{4}\Rightarrow 2\beta =\frac{\pi }{4}-\alpha \].


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