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Math Help - three sides of a triangle

  1. #1
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    three sides of a triangle

    Let a ,  b ,  c be the three sides of a triangle , and let

    \alpha ,  \beta  ,  \gamma , be the angles opposite them. If a^2+b^2=1989  c^2,

    find

    \frac{\cot \gamma}{\cot \alpha+\cot \beta}
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  2. #2
    Super Member wingless's Avatar
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    Huh.. that was long..

    a^2 + b^2 = 1989c^2
    First thing I thought about this equation was the law of cosines. We'll use it later as expected.

    \frac{\cot \gamma}{\cot \alpha+\cot \beta} = ?
    Let's start of with reducing this.

    \frac{\cot \gamma}{\cot \alpha+\cot \beta}

    \frac{\frac{\cos \gamma}{\sin \gamma}}{\frac{\cos \alpha}{\sin \alpha}+\frac{\cos \beta}{\sin \beta}}

    \frac{\frac{\cos \gamma}{\sin \gamma}}{\frac{\cos \alpha \sin \beta + \cos \beta \sin \alpha}{\sin \alpha \sin \beta}}

    \frac{\frac{\cos \gamma}{\sin \gamma}}{\frac{\sin (\alpha + \beta)}{\sin \alpha \sin \beta}}

    As \alpha + \beta + \gamma = \pi,
    \alpha + \beta = \pi - \gamma

    \frac{\frac{\cos \gamma}{\sin \gamma}}{\frac{\sin (\pi - \gamma)}{\sin \alpha \sin \beta}}

    \frac{\frac{\cos \gamma}{\sin \gamma}}{\frac{\sin \gamma}{\sin \alpha \sin \beta}}

    \frac{\cos \gamma \sin \alpha \sin \beta}{\sin \gamma \sin \gamma}

    Law of sines tell us that \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}

    From here we get \frac{\sin \alpha}{\sin \gamma} = \frac{a}{c} and \frac{\sin \beta}{\sin \gamma} = \frac{b}{c}.

    By using them,
    \frac{\cos \gamma \not\sin \not\alpha \not\sin \not\beta}{\not\sin \not\gamma \not\sin \not\gamma}\cdot \frac{a}{c} \cdot \frac{b}{c}

    \cos \gamma \cdot \frac{a.b}{c^2}

    It's simple enough now. Time to find \cos \gamma by the law of cosines..

    a^2 + b^2 = 1989c^2

    By the law,
    c^2 = a^2 + b^2 - 2a.b \cos \gamma

    c^2 = 1989c^2 - 2a.b \cos \gamma

    1988c^2 = 2a.b\cos\gamma

    994c^2 = a.b\cos\gamma

    \cos\gamma = \frac{994c^2}{a.b}

    We got \cos\gamma now. By using it,

    \cos \gamma \cdot \frac{a.b}{c^2} = ?

    \frac{994c^2}{a.b} \cdot \frac{a.b}{c^2}

    \boxed{994}
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