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

sin(3π7) \sin \left ( \frac{3\pi}{7} \right )

Unparseable latex formula:

\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 :
64s6112s4+56s27=0 \displaystyle 64s^6 -112s^4+56s^2-7=0 .
Isn't that great??
(edited 8 years ago)

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Original post by Ano123
sin(3π7) \sin \left ( \frac{3\pi}{7} \right )

=73456+73843i3+714473456+73843i3+712 \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
@Zacken



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



get ur chops around that, big lad


lol i'll summon @Student403 and @13 1 20 8 42
TSR Ouija board

XD
Original post by thefatone
lol i'll summon @Student403 and @13 1 20 8 42


Erm what do you want me to do here?
Reply 6
This is the kind of crap that made me take physics instead of further maths
Reply 7
Original post by Student403
Erm what do you want me to do here?

Praise the maths God.
Original post by notnek
Praise the maths God.


That's not how you spell DFranklin
Original post by Student403
Erm what do you want me to do here?


maybe solve it? i was just interested in the latex xD
WtF?!
Reply 11
Only a real mathematician (or future mathmetician) can appreciate finding exact values like this - regardless of how ugly they may look.
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.

:biggrin:
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.
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.

:biggrin:


ezpz
Reply 15
I don't know if there is a quick way of obtaining these answers for all of the sin(kπ7) \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.
Reply 16
Original post by C-rated
This is the kind of crap that made me take physics instead of further maths


How dare you :biggrin:
Original post by Student403
Why don't you and I'll check your answer


my calculator doesn't give me an exact answer
Reply 18
Original post by Ano123
How dare you :biggrin:


If it helps I got an E in physics and had to drop it at A2 :burnout:
Reply 19
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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