Transforming a polynomial into a trigonometric format tia sal2

Discussion in 'Maple' started by sal2, Apr 28, 2008.

  1. sal2

    sal2 Guest

    Transforming a polynomial into a trigonometric format tia sal2

    Greetings All

    I'm using maple 11 and I have a polynomial and would like to convert it
    into
    a Trigonometric format. Is this possible?

    Example:
    I have a polynomial
    0.00154991- 4.01371 x + 1.81197 x^2 + 8.00183 x^3 - 9.3462 x^4

    How can I transform this into a trigonometric format
    Example:
    0.00596679 Cos[6.98132 x] + 0.00358397 Cos[7.21403 x] +
    2.25013 Sin[0.232711 x] - 4.51511 Sin[0.465421 x]

    Note: these aren't correct answers I just wanted to include and example

    tia sal2
     
    sal2, Apr 28, 2008
    #1
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  2. sal2

    Axel Vogt Guest

    This way it makes no sense: your later expression can
    not be a polynom, since it does not go to +- infinity
    for large values.

    However you can express it in terms of certain orthogonal
    polynomials, for example Chebychev polynomials written as
    cos(n*arccos(x)):

    0.00154991- 4.01371 *x + 1.81197 *x^2 + 8.00183 *x^3 - 9.3462 *x^4;
    OrthogonalSeries:-ChangeBasis(%,ChebyshevT(n,x));
    OrthogonalSeries:-ConvertToSum(%);
    convert(%,cos);
    eval(%);

    -2.597290090+1.987662500*x-3.767115000*cos(2*arccos(x))+
    2.000457500*cos(3*arccos(x))-1.168275000*cos(4*arccos(x))

    The command 'expand' should give you back your polynom.

    Note that for 1 < abs(x) there is some complex stuff done
    in the background, which makes the difference to your above
    suggestion ( arccos(x) is not longer real, which gives some
    chances for cos ):

    x = 1/2+1/2*cos(2*arccos(x)), now take x = 10, then
    arccos(x) = ln(10+3*11^(1/2))*I ( ~ 2.993222846*I) and
    cos(of that) = ... = 10 ( ~ 9.999999999 numerical).
     
    Axel Vogt, Apr 28, 2008
    #2
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