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Math Help - Combination of Positive Integers

  1. #1
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    Combination of Positive Integers

    Please help. I'm taking an honors geometry class and my teacher gave us this problem as extra credit. I unfortunately am having a problem understanding exactly where to go with this one. I understand combination of positive integers but when the cos is thrown in I am confused.

    Thanks

    ---Susan

    Express cos(3m) as a combination of positive integer powers of cos(m). Use this expression to obtain a cubic polynomial p(x) with rational number coefficients such that x = cos(m) is a root. Graph the resulting polynomial for m = 20 degrees that displays all three of the roots.
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  2. #2
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    Quote Originally Posted by atvfan View Post
    Please help. I'm taking an honors geometry class and my teacher gave us this problem as extra credit. I unfortunately am having a problem understanding exactly where to go with this one. I understand combination of positive integers but when the cos is thrown in I am confused.

    Thanks

    ---Susan

    Express cos(3m) as a combination of positive integer powers of cos(m). Use this expression to obtain a cubic polynomial p(x) with rational number coefficients such that x = cos(m) is a root. Graph the resulting polynomial for m = 20 degrees that displays all three of the roots.
    identities you'll need ...

    1. sum identity for cosine

    \cos(a+b) = \cos{a}\cos{b} - \sin{a}\sin{b}

    2. double angle identities for cosine and sine

    \cos(2x) = 2\cos^2{x} - 1

    \sin(2x) = 2\sin{x}\cos{x}

    3. pythagorean identity

    \cos^2{x} + \sin^2{x} = 1


    ... start with the first identity, and the fact that \cos(3m) = \cos(2m+m)
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  3. #3
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    cos2m= 2cos^2 m -1
    sin2m=2sin(m)cos(m)
    cos (A+B)= cosAcosB-sinAsinB

    Cos(2m+m)=cos3m=(2cos^2m-1)cos(m) - sin(m)(2sin(m)cos(m))
    =2cos^3(m) - cos(m) - 2(sin^2m)cos(m)
    =2cos^3(m) - cos(m) - 2(1-cos^2(m))cos(m)
    =2cos^3(m) - cos(m) -2cos(m) +2cos^2(m)
    =4cos^3(m)-3cos(m)

    Plz thank if u like this.
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