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Cos(x) = x

  1. Apr 1, 2005 #1
    The solution to this equation is approximately 0.739085.

    Does anyone know how to express the solution exactly
    in terms of contants like pi, e, phi, etc?

    (phi = golden ratio = 1/2 + sqrt(5)/2)
     
  2. jcsd
  3. Apr 1, 2005 #2
    In all likelyhood it's impossible to do so in a simple way.
     
  4. Apr 1, 2005 #3

    dextercioby

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    Sure it is.If "x" is a solution to the equation,then be can expressed as

    [tex] x=\frac{x}{\pi e\varphi} \pi e\varphi [/tex]

    Daniel.
     
  5. Apr 1, 2005 #4
    You're right, of course. I took some license in my interpretation of his question. I'll be more precise:

    It's very likely impossible to express the solution in terms of a finite number of products, extractions of roots, additions, exponentiations, and divisions of elements of the set [tex]\{e, \pi, \phi\} \cup \mathbb{Z}[/tex]

    ~
     
    Last edited: Apr 1, 2005
  6. Apr 1, 2005 #5

    dextercioby

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    Let's tell Antiphon that not all transcendental numbers can be written using only [itex] e[/itex] and [itex] \pi [/itex] and the set of algebraic numbers...

    Daniel.
     
  7. Apr 4, 2005 #6
    I suspected this, but I asked the question assuming it was possible.

    So then you think it's impossible or you're not sure in this case?

    Perhaps then I should assign it a greek letter!
     
  8. Apr 4, 2005 #7

    dextercioby

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    1.It is impossible.

    2.You should.

    Daniel.
     
  9. Jan 3, 2007 #8
    The solution of the equation cos (x) = x can be given as applying the cosine function infinite nubmer of times to a starting point ..

    x = cos cos cos ...... cos (a)

    In other words , the solution can be expressed as :

    [tex]x = \lim _ { n \to \infty } \cos ^ { \circ n } ( a ) [/tex]


    That came from the Contraction Mapping Theorem .
     
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