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Trigonometric functions - exact values ??!!??£@&&@?! Watch

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     \sin \left ( \frac{3\pi}{7} \right )

     \displaystyle = \sqrt{ \sqrt[3]{ \frac{7}{3456} \left (-1+3\sqrt{3} i \right ) } +\frac{7}{144} \sqrt[3]{ -\frac{864}{49} \left (1+3\sqrt{3} i} \bigr )+\frac{7}{12}}

    This is a solution to the disguised cubic equation :
     \displaystyle 64s^6 -112s^4+56s^2-7=0 .
    Isn't that great??
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    (Original post by Ano123)
     \sin \left ( \frac{3\pi}{7} \right )

     \displaystyle = \sqrt{ \sqrt[3]{ \frac{-7}{3456}+\frac{7}{384\sqrt 3}i } +\frac{7}{144} \sqrt[3]{\frac{-7}{3456}+\frac{7}{384\sqrt 3}i } +\frac{7}{12}}
    this looks like one of those monstrosities i was practising my latex on
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    Zacken



    get ur chops around that, big lad
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    (Original post by Supersaps)
    Zacken



    get ur chops around that, big lad
    lol i'll summon Student403 and 13 1 20 8 42
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    TSR Ouija board

    XD
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    (Original post by thefatone)
    lol i'll summon Student403 and 13 1 20 8 42
    Erm what do you want me to do here?
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    This is the kind of crap that made me take physics instead of further maths
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    (Original post by Student403)
    Erm what do you want me to do here?
    Praise the maths God.
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    (Original post by notnek)
    Praise the maths God.
    That's not how you spell DFranklin
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    (Original post by Student403)
    Erm what do you want me to do here?
    maybe solve it? i was just interested in the latex xD
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    WtF?!
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    Only a real mathematician (or future mathmetician) can appreciate finding exact values like this - regardless of how ugly they may look.
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    (Original post by Student403)
    Erm what do you want me to do here?
    Digest the contents of this wikipedia page and then extend it to include the complex radical mentioned above.

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    (Original post by Ano123)
    Only a real mathematician (or future mathmetician) can appreciate finding exact values like this - regardless of how ugly they may look.
    True enough; but what we then want to do is to generalize in order to understand the deeper patterns. So from Wikipedia:

    "According to Baker's theorem, if the value of a sine, a cosine or a tangent is algebraic, then either the angle is rational number of degrees, or the angle is a transcendental number of degrees. That is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values."

    V. nice.
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    (Original post by thefatone)
    maybe solve it? i was just interested in the latex xD
    Why don't you and I'll check your answer
    (Original post by Kvothe the arcane)
    That's not how you spell DFranklin
    PRSOM :rofl:

    (Original post by Gregorius)
    Digest the contents of this wikipedia page and then extend it to include the complex radical mentioned above.

    ezpz
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    I don't know if there is a quick way of obtaining these answers for all of the  \displaystyle \sin \left ( \frac{k\pi}{7} \right ) values but I just formed the equation above, and solved it using the cubic formula.
    If there is a quicker way of doing these (maybe a general formula or something similar) please let me know.
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    (Original post by C-rated)
    This is the kind of crap that made me take physics instead of further maths
    How dare you
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    (Original post by Student403)
    Why don't you and I'll check your answer
    my calculator doesn't give me an exact answer
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    (Original post by Ano123)
    How dare you
    If it helps I got an E in physics and had to drop it at A2 :burnout:
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    I find it weird that you can't get rid of the imaginary terms in this exact answer, but the answer itself is purely real.
 
 
 
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