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Using trig to find solutions to a quintic

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Given that

    [itex]cos5θ=16cos^{5}θ-20cos^{3}θ+5cosθ[/itex]

    Find the 3 solutions to

    [itex]16x^{5}-20x^{3}+5x+1=0[/itex]

    2. Relevant equations

    3. The attempt at a solution

    I have let [itex]x=cosθ[/itex] and [itex]θ=cos^{-1}x[/itex]
    and I know that [itex]cos5θ=-1[/itex]
     
  2. jcsd
  3. Feb 22, 2012 #2

    tiny-tim

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    hi conorordan! :smile:
    that's right! :smile:

    so what are the 3 solutions to cos5θ = -1 ? :wink:
     
  4. Feb 22, 2012 #3
    Well there are infinite solutions to it, how am I to know which 3 will satisfy the quintic above?
     
  5. Feb 22, 2012 #4

    tiny-tim

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    agreed :smile:

    but don't lose the plot :wink:

    you're only interested in cosθ (=x), and there aren't infinitely many values of that! :smile:
     
  6. Feb 22, 2012 #5
    Are you suggesting I plot graphs of cos x and the horrific quintic above? It must be much simpler, this is only a 4 mark question!
     
  7. Feb 22, 2012 #6

    tiny-tim

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    no :biggrin:

    write on one line the infinitely many solutions for θ

    write on the next line the values of cosθ for those values of θ

    on the next line, cross out any duplicates …

    how many are left? :wink:
     
  8. Feb 22, 2012 #7
    Okay I see now, so x=cosθ for π/5, 5π/3 and π will give me my 3 solutions?
     
  9. Feb 22, 2012 #8

    tiny-tim

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    (you mean 3π/5 :wink:)

    yup! :biggrin:
     
  10. Feb 22, 2012 #9
    Indeed! Thanks for the help!
     
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