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Math Help - trigonometry

  1. #1
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    trigonometry

    If A+B+C=0 then prove that sin 2A + sin 2B + sin 2C = -4 sinA sinB sinC.
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  2. #2
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    Re: trigonometry

    sin 2A + sin 2B + sin 2C = 2 sin ( 2A + 2B )/2 cos ( 2A-2B)/2 + 2 sin C cos C
    = 2 sin ( A + B ) cos ( A - B ) + 2 sin C cos C
    = 2 sin ( - C ) cos ( A - B ) + 2 sin C cos C [ A+B+C= 0 ]
    = - 2 sin C cos ( A - B ) + 2 sin C cos C
    = - 2 sin C [ cos ( A - B ) - Cos C ]
    = - 2 sin C [ - 2 sin ( A - B +C )/2 sin ( A - B - C )/2]
    = - 4 sin C [ - sin ( - 2B )/2 sin ( 2 A ) / 2 ]
    = - 4 sin C [ sin B sin A ]
    = - 4 sin A sin B sin C = RHS
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  3. #3
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    Re: trigonometry

    A+B+C  = 0 \implies 2C = -2A -2B

    \sin 2A + \sin 2B + \sin 2C

    =\sin 2A + \sin 2B - \sin (2A + 2B)

    =\sin 2A + \sin 2B - \sin 2A \cos 2B - \sin 2B \cos 2A

    =\sin 2A (1 - \cos 2B) + \sin 2B (1 - \cos 2A)

    =(2\sin A \cos A)(2 \sin^2 B) + (2\sin B \cos B)(2 \sin^2 A)

    =4\sin A \sin B(\cos A\sin B + \cos B \sin A)

    =4\sin A \sin B\sin (A + B)

    =4\sin A \sin B\sin (-C)

    =-4\sin A \sin B\sin C
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