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Math Help - In any given triangle prove that...

  1. #1
    Super Member bigwave's Avatar
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    In any given triangle prove that...

    In any given triangle prove that...

    tan\frac{1}{2}(B-C) = tan(45\,^{\circ}-\theta)cot\frac{1}{2}A
    where tan\theta=\frac{c}{b}

    find, \frac{1}{2}(B-C) if b=321, c=436, A=119\,^{\circ}15'


    the answer is... -5\,^{\circ}5'; (\theta = 53\,^{\circ}38')

    not sure how they got this...
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello bigwave
    Quote Originally Posted by bigwave View Post
    In any given triangle prove that...

    tan\frac{1}{2}(B-C) = tan(45\,^{\circ}-\theta)cot\frac{1}{2}A
    where tan\theta=\frac{c}{b}

    find, \frac{1}{2}(B-C) if b=321, c=436, A=119\,^{\circ}15'


    the answer is... -5\,^{\circ}5'; (\theta = 53\,^{\circ}38')

    not sure how they got this...
    A few preliminary results:
    \cot\tfrac12A=\cot\tfrac12(180^o-[B+C])=\cot(90^o-\tfrac12[B+C])=\tan\tfrac12(B+C) (1)

    \tan 45^o=1 (2)

    Using the Sine Rule: \frac{c}{b}=\frac{\sin C}{\sin B}=\frac{\sin \tfrac12C\cos \tfrac12C}{\sin \tfrac12B\cos\tfrac12 B} (3)
    Then, starting with the LHS:
    \tan(45^o-\theta)\cot\tfrac12A
    =\frac{\tan45^o-\tan\theta}{1+\tan45^o\tan\theta}\cdot\tan\tfrac12  (B+C) using (1)

    =\frac{1-\dfrac{c}{b}}{1+\dfrac{c}{b}}\cdot\tan\tfrac12(B+C  ) using (2)

    =\frac{1-\dfrac{\sin \tfrac12C\cos \tfrac12C}{\sin \tfrac12B\cos\tfrac12 B}}{1+\dfrac{\sin \tfrac12C\cos \tfrac12C}{\sin \tfrac12B\cos\tfrac12 B}}\cdot\frac{\tan\tfrac12B+\tan\tfrac12C}{1-\tan\tfrac12B\tan\tfrac12C} using (3)

    =\frac{\sin \tfrac12B\cos\tfrac12 B-\sin \tfrac12C\cos \tfrac12C}{\sin \tfrac12B\cos\tfrac12 B+\sin \tfrac12C\cos \tfrac12C}\cdot\frac{\tan\tfrac12B+\tan\tfrac12C}{  1-\tan\tfrac12B\tan\tfrac12C}

    =\frac{\tan \tfrac12B-\tan \tfrac12C}{\tan \tfrac12B+\tan \tfrac12C}\cdot\frac{\tan\tfrac12B+\tan\tfrac12C}{  1-\tan\tfrac12B\tan\tfrac12C}

    =\tan\tfrac12(B-C)
    I agree with the numerical answer given. This is a straightforward use of a calculator, using the result we've just proved. Is there a problem with this?

    Grandad
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