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Thread: If b\cos{\beta} = c\cos{\gamma} prove that

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    Super Member bigwave's Avatar
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    Cool If b\cos{\beta} = c\cos{\gamma} prove that

    If $\displaystyle b\cos{\beta} = c\cos{\gamma}$

    prove that either $\displaystyle b=c$ or $\displaystyle A=90^o$

    tried for an hour trying to prove this with the law of cosines but that didn't seem to go anywhere

    a,b,c are the sides of any triangle
    $\displaystyle
    A,\beta,\gamma$ are the corresponding angles
    Last edited by bigwave; Dec 17th 2009 at 03:32 PM. Reason: more info
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  2. #2
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    Quote Originally Posted by bigwave View Post
    If $\displaystyle b\cos{\beta} = c\cos{\gamma}$

    prove that either $\displaystyle b=c$ or $\displaystyle A=90^o$

    tried for an hour trying to prove this with the law of cosines but that didn't seem to go anywhere

    a,b,c are the sides of any triangle
    $\displaystyle
    A,\beta,\gamma$ are the corresponding angles
    Sine rule: $\displaystyle \frac b{\sin \beta} = \frac c{\sin\gamma}$. Deduce that $\displaystyle \sin\beta\cos\beta = \sin\gamma\cos\gamma$, hence $\displaystyle \sin2\beta = \sin2\gamma$. If two angles have the same sine then they are either equal or supplementary.
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