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Algebraic form of any trygonometrical function

  1. Nov 16, 2009 #1
    sin x = 1/10, or any other number that can't be found in math tables - how to know what x is?
     
  2. jcsd
  3. Nov 16, 2009 #2

    Hurkyl

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    What do you mean by "know"? For a great many purposes, the equation
    sin x = 1/10​
    is sufficient to "know" x.

    If by "know" you mean a decimal approximation, well it's easy enough to get that from a calculator with the arcsine function.

    If by "know" you mean finding an algebraic number* y such that
    [tex]\sin \pi y = 1/10[/tex]​
    then I very, very strongly suspect it's impossible -- y would have to be transcendental, not algebraic.

    *: An algebraic number is any number that is the root of a polynomial with integer coefficients. Numbers that can be expressed in terms of integers, +, -, *, /, and taking of roots are kinds of algebraic numbers.
     
  4. Nov 16, 2009 #3
    sin 17 = x

    I want to know algebraic form of x

    (its 17 grades)
     
  5. Nov 16, 2009 #4

    Hurkyl

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    You almost surely don't want to know anything about x aside from the fact it is the sine of 17 gradians, and is approximately 0.26387305. What are you trying to do? :confused:
     
  6. Nov 16, 2009 #5
    sin of 3 degrees can be written with just square roots, but sine of 17 degrees, 17 not a multiple of 3, requires solving a cubic equation, in addition to several quadratic equations.

    According to Maple, sine 17 degrees is a zero of the polynomial

    [tex]
    281474976710656\,{x}^{48}-3377699720527872\,{x}^{46}+18999560927969280
    \,{x}^{44}
    -66568831992070144\,{x}^{42}+162828875980603392\,{x}^{40}-295364007592722432\,{x}^{38}
    +411985976135516160\,{x}^{36}-
    452180272956309504\,{x}^{34}+396366279591591936\,{x}^{32}
    -280058255978266624\,{x}^{30}+160303703377575936\,{x}^{28}-74448984852135936\,{x}^{26}
    +28011510450094080\,{x}^{24}-8500299631165440\,{x}^{22}+2064791072931840\,{x}^{20}
    -397107008634880
    \,{x}^{18}+59570604933120\,{x}^{16}-6832518856704\,{x}^{14}+583456329728\,{x}^{12}
    -35782471680\,{x}^{10}+1497954816\,{x}^{8}-39625728\,{x}^{6}
    +579456\,{x}^{4}-3456\,{x}^{2}+1
    [/tex]
     
  7. Nov 16, 2009 #6

    Hurkyl

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    I assume he meant 17 gradians, which is [itex]17 \pi / 200[/itex] radians or 15.3 degrees.

    I'm pretty sure this requires solving a quintic too. (Only one quintic -- sin 72 can be expressed in terms of square roots)
     
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