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JEE Main & Advanced Mathematics Trigonometric Equations Question Bank Relation between sides and angles, Solutions of triangles

  • question_answer
    The perimeter of\[\Delta ABC\]is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is   [IIT Screening 1992; DCE 1999]

    A) \[\frac{\pi }{6}\]

    B) \[\frac{\pi }{3}\]

    C) \[\frac{\pi }{2}\]

    D) \[\pi \]

    Correct Answer: A

    Solution :

    We have \[a+b+c=\frac{6(\sin A+\sin B+\sin C)}{3}\] Þ \[k(\sin A+\sin B+\sin C)=2(\sin A+\sin B+\sin C)\], where \[k=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\] Þ \[k=2\], \[(\because \sin A+\sin B+\sin C\ne 0)\] \[\therefore \] \[\frac{a}{\sin A}=2\] Þ \[\sin A=\frac{1}{2}\], (\[\because \] \[a=1\]) Þ \[A=\frac{\pi }{6}\].


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