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Math Help - Help to verify & solve this determinants problem!

  1. #1
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    Help to verify & solve this determinants problem!

    Verify the system of linear equations in cosA, cosB, and cosC.

    _______ c cosB + b cosC = a
    c cosA ______ + a cosC = b
    b cosA + a cosB _______ = c

    Then use Cramer's Rule to solve for cosC, and use the result to verify the Law of Cosines, c^2 = a^2 + b^2 - 2ab cosC
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by test2k6 View Post
    Verify the system of linear equations in cosA, cosB, and cosC.

    _______ c cosB + b cosC = a
    c cosA ______ + a cosC = b
    b cosA + a cosB _______ = c

    Then use Cramer's Rule to solve for cosC, and use the result to verify the Law of Cosines, c^2 = a^2 + b^2 - 2ab cosC

    c cosB + b cosC = a
    c cosA + a cosC = b
    b cosA + a cosB = c

    write this in the form: Ax = b, where A, x, b are matrices
    so we have:

    |0......c......b| |cosA|.......|a|
    |c......0......a| |cosB|...=...|b|
    |b......a......0| |cosC|.......|c|

    Now detA = 0 + abc + abc - 0 - 0 - 0 = 2abc

    By Cramer's rule:

    cosC = det |0......c......a| divided by detA = 2abc
    ................|c......0......b|
    ................|b......a......c|

    => cosC = (0 + cb^2 + ca^2 - 0 - 0 - c^3)/2abc
    => cosC = (cb^2 + ca^2 - c^3)/2abc
    => cosC = (b^2 + a^2 - c^2)/2ab ..............factored out and canceled the c's
    => 2abcosC = a^2 + b^2 - c^2
    => c^2 = a^2 + b^2 - 2abcosC which is the cosine rule
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by test2k6 View Post
    Verify the system of linear equations in cosA, cosB, and cosC.

    c cosB + b cosC = a
    c cosA + a cosC = b
    b cosA + a cosB = c
    I must admit I'm curious as to where these equations come from, myself.

    -Dan
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