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Math Help - Trigonometry problem ; Help?

  1. #1
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    Question Trigonometry problem ; Help?

    I'm trying to find the value of this question :

    Cos β / (sinα.sinγ) + cos α / (sin β. sin γ)+ cos γ/ (sin α. Sin β)
    Provided that α, β, γ is the angle in a triangle

    Please help me to do this

    The answer in the book says that the value is 2, but how on earth are we going to find "2"

    I already tried like this
    = Cos β / (sinα.sinγ) + cos α / (sin β. sin γ)+ cos γ/ (sin α. Sin β)
    = cos β sin β + sin α cos α + sin γ cos γ / sin α sin β sin γ
    = sin 2β + sin 2α + sin 2 γ/ 2 (sin α sin β sin γ)

    Using identities sin A + sin B = 2 sin (A+B/2) cos (A-B/2)
    And 2 sin A sin B =sin (A+B) sin (A-B)
    We have

    = 2 sin β cos 0 + 2 sin α cos 0 + 2 sin γ cos 0 / (sin ( α+β ) sin ( α- β)) sin γ
    = 2 ( sin α + sin β + sin γ) / (sin γ( 2 sin β cos α))

    Then 2 sin β sin γ becomes 2 sin γ cos β

    = 2 (sin α + sin β + sin γ) / 2 cos α cos β sin γ
    = sin α + sin β + sin γ / cos α cos β sin γ
    = ( (sin α + sin β / sin γ) x 1/(cos α cos β) + 1/ cos α cos β
    = 1/cos α cos β((sin α + sin β / sin γ) +1)
    2cos A cos B = cos (A+B) + cos (A-B)
    = (2 / (cos (α + β) + cos (α- β) ) ((sin α + sin β / sin γ) +1)

    Please help me ! thanks
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  2. #2
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    Re: Trigonometry problem ; Help?

    first add all the fractions to get one : with common denominator sinasinbsinc and numerator cosbsinb+cosasina+coscsinc
    then multiply the fraction by two both numerator and denominator...to get 2sinasinbsinc as denominator and sin(2a)+sin(2b)+sin(2c) as numerator.
    then leave the denominator as it is and try to factorize the numerator using sum to product formulas .Dont forget that 2a+2b+2c=360 now...and you have to prove that
    sin2a+sin2b+sin2c =4sinasinbsinc this proves your relation.....
    so start the job right now....good luck...
    Thanks from Deci
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  3. #3
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    Re: Trigonometry problem ; Help?

    thanks MINOANMAN for your help,
    What do you mean by sum to product formula? is it




    I have tried to prove it, but still can not find. Would you please help me?
    Last edited by Deci; January 23rd 2014 at 01:39 AM.
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  4. #4
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    Re: Trigonometry problem ; Help?

    yes these are the formulas
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  5. #5
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    Re: Trigonometry problem ; Help?

    Okay, got it. Thanks for your help
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